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World Scientific Lecture Notes in Physics V o l . 6 5

Fluctuations

UNIVERSAL FLUCTUATIONS The Phenomenology of

Hadronic Matter

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World Scientific Lecture Notes in Physics Vol. 65

UNIVERSAL FLUCTUATIONS

The Phenomenology of

Hadronic Matter

Robert Botet CNRS/Universite Paris-Sud, France

Marek Pioszajczak Grand AccSlerateur National d'lons Lourds, France

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UNIVERSAL FLUCTUATIONS The Phenomenology of Hadronic Matter

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A

CATHERINE & MARTINE

Preface

"Los pasos que da un hombre, desde el dia de su nacimiento hasta el de su muerte, dibujan en el tiempo una inconcebible figura. La Inteligencia Divina intuye esa figura inmediata-mente, como la de los hombres un tridngulo. Esa figura (acaso) tiene su determinada funcion en la economia del Universo" * [J. L. Borges (1952)].

The objective of this book is to outline a similar quest for scientific comprehension. Searching for figures just a bit more complicated than triangles. This is the quest for structures, hidden into random signals.

Intention is not new - whole modern theory of Probabilities comes from this remarkable hunt -, but consequences appear slowly.

At its flush of youth, some three centuries ago, the usual purpose of the probability calculations was gambling. Then little by little, maturity came, and we are now trying to comprehend Nature, even if gambling survives through econophysics. Understandable and reasonable evolution. For once in the history of Sciences, it is worth noting here that the way used by Probabilities to introduce themselves in Physics can be resumed in a few words : as long as classical Physics was rational and deterministic, great effort for mathematical analysis was needed but certainly not for the probabilities. And consequently, the classical iV-body physics was soon achieved for the small values of N, even if full solutions could be, or were expected

""The steps taken by a man, from his birth to his death, draw in the time an inconceivable figure. Divine knowledge understands this figure immediately, as we see a triangle. This figure (maybe) has its definite function in arrangement of the Universe"

vii

Vlll Preface

to be, mathematically tedious. Then next step in complexity consisted in playing with larger values of N, as it should for macroscopic objects built with microscopic entities. At this stage, randomness appears spontaneously, as Nature wriggles off the dull solutions of the linear equations, as interactions between individuals open new horizons of freedom, as duality engenders hesitation, as bifurcations lead to labyrinth. To write short, borders of classical physics are full of chaos. And since everything which surrounds us is built with gigantic number of quantic particles, we know why chaos should be present everywhere.

Amazingly, when looking around us, we are well forced to accept that Nature is only gently random, and chaoticity is (fortunately) not the leading behaviour. McKeen discussing this paradox, wrote with his punchy and concise style [K. McKeen (1981)] : "In a world as crazy as this one, it ought to be easy to find something that happens solely by chance. It isn't". Without going too far into anthropic principles, one can assert that if chaos is the rule for individuals, some sort of regularization hold advisedly in the collective systems.

In Probability theory, the hidden structures to achieve such regularization are the limit distribution laws. This was recognized clearly by Gauss, that fluctuations of almost any physical quantity around its mean value, were distributed according to the normal law. Beyond randomness was hidden a magic formula : exp(—z2). With his sense of the beautiful turn of phrase, he named this general phenomenon : the 'law of error', and was the first one to have intuition of the deep and practical importance of this discovery.

Such conceptual jump in the knowledge of universality of random processes did not occur for a long time. The 'law of error' was renamed as : the 'Central Limit Theorem', and its conditions were so weakened that scientists were then convinced to see the Gaussian function everywhere. At the beginning of the XX century Poincare wrote with some humour : "Tout le monde croit (la loi des erreurs) parce que les mathematiciens s 'imaginent que c'est unfait d'observation, et les observateurs que c'est un theoreme de mathematiques." * [H. Poincare (1892)]. But, by the years 30s, extension of the Central Limit Theorem was performed when its conditions are going to fail. This achievement resulted from the collective effort of the prolific

•"Everybody believe (the law of error) because mathematicians imagine that this is experimental fact, and experimentalists that this is a mathematical theorem."

Preface IX

Russian school of Probability as well as from the deep works of Levy justifying and making rigorous the powerful method of moments introduced by Markov. This spectacular progress laid to the complete and explicit characterization of the stable limit distributions, giving reliable access to a precise image of the random process. Comprehension through simple visual representation. But above all, a new complementary and more subtle problem arise spontaneously : what is the correct normalization of a random variable in order to get the limit law ?

These thoughts laid to the fundamental concept of scaling. It is difficult to trace the very beginning of this notion, since this idea of scaling seems to have emerged independently in various scientific areas. Anyway, all these results gave profusion of new concepts in the modern characterization of random processes, and made complicated things easier. The present book is about scaling of the fluctuations.

' Imagination ', in its first sense, is to replace abstract entity by an image, expected to be more directly understandable to our human intelligence. Modern Probabilities supply us exactly such imagination, to transform unreasonable tossing of a coin into a universal curve, and unconceivable Brownian motion into Hydrodynamics. "Apprivoiser I'absurdite du monde" * said the singer [M. Farmer (1999)].

Robert Botet & Marek Ploszajczak Orsay, Caen, October 2001

*" Taming absurdity of the world."

Contents

Preface vii

Chapter 1 Introduction 1

Chapter 2 Central Limit Theorem and Stable Laws 7 2.1 Central limit theorem for broad distributions 8

2.1.1 Central limit theorem for the sum of uncorrelated variables 8

2.2 Stable laws for sum of uncorrelated variables 13 2.2.1 The stability problem 13 2.2.2 Complete solution of the stability problem for

uncorrelated variables 17 2.2.2.1 The ensemble of one-dimensional stable

distributions 17 2.2.2.2 Alternative formulas for the stable

distributions 17 2.2.2.3 Range of values for p, 18 2.2.2.4 Range of values for /? 19 2.2.2.5 Gaussian distribution as a stable law 20 2.2.2.6 Moments of the stable distributions 20

2.2.3 Explicit examples of stable distributions 22 2.2.3.1 Symmetric stable distributions (/3 = 0) . . . . 22 2.2.3.2 Asymmetric stable distributions [fi = 1) . . . 24

2.2.4 The reciprocity relation for stable distributions 25 2.2.5 The tail of stable distributions 26

xi

xii Contents

2.2.6 Moments of stable distributions 26 2.2.7 Asymptotically stable laws - domains of attraction . . . 27 2.2.8 The concept of the A-scaling 29

2.3 Limit theorems for more complicated combinations of uncorrelated variables 30 2.3.1 Product of uncorrelated variables 30 2.3.2 The Kesten variable 34 2.3.3 The Gumbel distribution 35 2.3.4 The arc-sine law 38

2.4 Two examples of physical applications 39 2.4.1 The Holtsmark problem 39 2.4.2 The stretched-exponential relaxation 41

Chapter 3 Stable Laws for Correlated Variables 45 3.1 Weakly and strongly correlated random variables 46

3.1.1 Correlated random Gaussian processes 47 3.1.2 Taqqu's reduction theorem 49 3.1.3 Rosenblatt's model 50

3.2 Dyson's hierarchical model 51 3.3 The renormalization group 54

3.3.1 The renormalization group and the stability problem . . 55 3.3.2 Scaling features 56 3.3.3 e-expansion 57 3.3.4 Multiplicative structure of the renormalization group . . 59

3.4 Self-similar probability distributions 61 3.4.1 Self-similar processes 61 3.4.2 Euler theorem 62 3.4.3 Self-similarity of fractals in the renormalization

group approach 63 3.4.4 The power spectral density function 64 3.4.5 A-scaling framework 65

3.5 Critical systems 66 3.5.1 Anomalous dimension 67 3.5.2 First scaling 68 3.5.3 Second scaling 70 3.5.4 A-scaling 71

3.5.5 Studies of criticality in finite systems 73

Contents xiii

Chapter 4 Diffusion Problems 75 4.1 Brownian motion 75

4.1.1 Fick's representation 75 4.1.2 Ornstein-Uhlenbeck representation 77 4.1.3 Fokker-Planck representation 79

4.2 Random walks 81 4.2.1 Gaussian random walks and Gaussian Levy flights . . . 81 4.2.2 St. Petersburg paradox 84 4.2.3 Non-Gaussian Levy flights 86

4.2.3.1 Anomalous diffusion 86 4.2.3.2 Continuous Levy flights 89 4.2.3.3 Return to the origin of the random walk . . . 90

4.2.4 Random walk in a random environment 92 4.2.5 Sinai billiard 96

4.3 Random walks with memory 97 4.3.1 Random walks with Gaussian memory 97 4.3.2 Fractional Brownian motion 99 4.3.3 Flory's approach for linear polymers 102

4.4 Random walk as a critical phenomenon 106 4.4.1 Criticality of the Brownian motion 106 4.4.2 Criticality of the Levy flight 107 4.4.3 Criticality of the self-avoiding walk 108

4.5 Random walk as a self-similar process 109 4.5.1 Self-similarity of the Brownian motion 109 4.5.2 Anomalous diffusion in the fractal space 110

Chapter 5 Poisson-Transform Distributions 113 5.1 The class of poisson transforms 114

5.1.1 General functional relations for the Poisson transforms 116

5.1.2 Examples of Poisson transforms 117 5.1.3 Generating function for the Poisson transforms 119

5.2 Pascal distribution 120 5.2.1 Definition and moments of the Pascal distribution . . . 121 5.2.2 Recurrence relations for the Pascal distribution 121 5.2.3 Limit cases of the Poisson distribution 123 5.2.4 Stability of the Pascal distribution 123 5.2.5 Origins of the Pascal distribution 125

xiv Contents

5.2.6 Stochastic differential equation leading to the Pascal distribution 126

5.3 Stacy distribution 127 5.3.1 The generalized Gamma distribution and its

moments 128 5.3.2 Langevin and Fokker-Planck equations leading to the

generalized Gamma function 129 5.3.2.1 One-dimensional Langevin equation with

the multiplicative noise 130 5.3.2.2 Explicit physical processes leading to the

one-dimensional Langevin equation with the multiplicative noise 131

5.3.2.3 Solution of the one-dimensional Langevin equation with the multiplicative noise 132

5.3.2.4 The limit case with vanishing random force . 133 5.4 Other examples of integral transforms 135 5.5 KNO scaling limit 135

5.5.1 Extension of the KNO scaling rule 136

Chapter 6 Featuring the Correlations 139 6.1 Moments and their generating function 139

6.1.1 Moments 140 6.1.2 Cumulant moments 140 6.1.3 Factorial moments 141 6.1.4 Cumulant factorial moments 142 6.1.5 Normalized moments 142 6.1.6 Bunching parameters 144 6.1.7 Combinants 144 6.1.8 Existence of the generating functions 145

6.2 Some tools specific to the moment generating functions . . . . 146 6.2.1 Singularities of the moment generating function 146 6.2.2 The Stieltjes series 147

6.3 One example: the poisson distribution 148 6.4 Infinitely divisible distribution functions 151

6.4.1 Truncating the multiplicity distribution 153 6.5 Composite distributions 153

6.5.1 Conditional and joint probabilities 154 6.5.2 Clan structures 155

Contents xv

6.6 More about the pascal distribution 158 6.6.1 The limiting forms 159 6.6.2 High-energy phenomenology 161

Chapter 7 Exclusive and Inclusive Densities 163 7.1 Generalities and variables 163 7.2 Cumulant correlation functions 166 7.3 Scaled factorial moments 168

7.3.1 Intermittency with the scaled factorial moments . . . . 169 7.3.2 Correcting for the shape of the one-particle

distribution and the lack of the translational invariance 171

7.3.3 Unphysical correlations due to the mixing of events of different multiplicities 172

7.3.4 Dimensional projection 173 7.4 Scaled factorial correlators and bin-split moments 176 7.5 Scaled factorial cumulants 178

7.5.1 Correlation integral 180 7.6 Linked structure of the correlations 183

7.6.1 Linked pair approximation 183 7.6.2 Linked approximation in the conformal theory 184 7.6.3 Linked approximation for the A-scaling 186 7.6.4 Counts and their fluctuations 187

7.7 Erraticity concept 189 7.7.1 Wavelet representation 192

7.7.1.1 Simple examples of wavelets 195

Chapter 8 Bose-Einstein Correlations in Nuclear and Particle Physics 199

8.1 Basic features of bose-einstein quantum statistical correlations 200

8.2 Parametrization of the HBT data 202 8.2.1 The space-time structure of the multiparticle system . . 204 8.2.2 HBT measurements in condensed matter and atomic

physics 206 8.3 Bose-Einstein interference in models 208 8.4 Idealized picture of independent particle production 209

8.4.1 Monte-Carlo simulations 212

xvi Contents

8.5 Bose-Einstein correlations in high-energy collisions 214 8.5.1 Higher order cumulants in pp collisions 214 8.5.2 Small-scale Bose-Einstein correlations 217 8.5.3 Density dependence of the correlations 219

Chapter 9 Random Multiplicative Cascades 221 9.1 Multiplicative cascade models 222

9.1.1 Weak intermittency regime 223 9.1.2 Strong intermittency regime 225 9.1.3 Regularization of the scaled factorial moments in

the strong intermittency limit 226 9.2 Multifractals and intermittency 227 9.3 Correlations in random cascading 229

9.3.1 Some examples of the branching generating functions . 235 9.3.2 Link to the multifractal formalism 236 9.3.3 Relation between branching generating function and

multifractal mass exponents 238 9.4 Non-ideal random cascades: the cut-off effect 239

9.4.1 Multiscaling dependence on the cut-off parameters . . . 240 9.4.2 a-model with the cut-off at small scales 243

9.5 QCD cascade 245

Chapter 10 Random Cascades with Short-Scale Dissipation 251

10.1 Basic features of the fragmentation-inactivation binary model 254 10.1.1 Shattering transition 255 10.1.2 Scale-independent dissipation effects: the phase

diagramme 256 10.2 Various approaches to the fragmentation-inactivation binary

model 257 10.2.1 Fragmentation-inactivation binary model as a random

multiplicative cascade 257

10.2.2 Fragmentation-inactivation binary model as a mean-field branching process 258

10.2.3 Cascade equation for the multiplicity evolution 259 10.2.4 Master equation 260

Contents xvii

10.3 Moment analysis of the fragmentation-inactivation binary equations 261 10.3.1 General equations for the factorial moments and

cumulant moments 261 10.3.2 Moments of the multiplicity distribution at the

transition line 262 10.3.2.1 Brand - Schenzle fragmentation domain

(pF > l / 2 , a > - l ) 263 10.3.2.2 Marginal case : pF = 1/2 , a > - 1 265 10.3.2.3 Cayley fragmentation domain :

pF < 1/2, a > - 1 266 10.3.2.4 Evaporative fragmentation domain :

pF > 0 , a < - 1 268 10.3.3 Structure of higher-order cumulant correlations at

the transitional line 270 10.4 Binary cascading with scale-dependent inactivation

mechanism 271 10.4.1 First example : binary cascading with a = — 1 and

the Gaussian inactivation 272 10.4.2 Second example : binary fragmentation with

a = +1 and the Gaussian inactivation 275 10.4.3 A-scaling vs value of exponent r 275 10.4.4 Multiplicity fluctuations in different physical systems

and in the binary fragmentation . 276 10.5 Perturbative quantum chromodynamics including

inactivation mechanism 277 10.5.1 Multiplicity distributions in the dissipative

gluodynamics 280 10.6 Phenomenology of the multiplicity distributions in e+e~

reactions 281

Chapter 11 Fluctuations of the Order Parameter 285 11.1 Order parameter fluctuations in self-similar systems 286

11.1.1 The anomalous dimension 286 11.1.2 Critical cluster-size 288 11.1.3 Note about the correct order parameter 289

11.2 Example of the non-critical model 290 11.2.1 The weight functions 290

xviii Contents

11.2.2 Check of the linked pair approximation 290 11.2.3 Second scaling law 291 11.2.4 Note about the average size-distribution 292

11.3 Mean-field critical model: the Landau-Ginzburg model 293 11.3.1 Landau-Ginzburg free energy 293 11.3.2 Distribution of the extensive order parameter 293 11.3.3 First scaling at the pseudo-critical point 294 11.3.4 Gaussian first scaling in the disordered phase 295 11.3.5 Second scaling in the ordered phase 295 11.3.6 Correlation pattern in the Landau-Ginzburg theory . . 296

11.4 Example of the critical model: the potts model 297 11.4.1 Scaling laws for the order-parameter distribution . . . . 298

11.5 Reversible aggregation: example of the percolation model . . . 300 11.5.1 Order parameter in the percolation on the Bethe

lattice 301 11.5.2 The three-dimensional percolation model 304

11.5.2.1 Multiplicity distributions 304 11.5.2.2 Order-parameter distribution 304 11.5.2.3 Shifted order parameter 304 11.5.2.4 Outside of the critical point 305 11.5.2.5 Close to the critical point 307

11.6 Irreversible aggregation: example of the smoluchowski kinetic model 308 11.6.1 Basic behaviour of the order parameter 308 11.6.2 Scalings of the order-parameter distributions 310 11.6.3 Tails of the scaling functions 311 11.6.4 Scaling for the shifted order parameter 312 11.6.5 Origin of fluctuations in non-equilibrium aggregation . . 313

11.6.5.1 Argument of Van Kampen 313 11.6.5.2 Gelling systems 315 1T.6.5.3 Scaling of the second moments for gelling

systems 315 11.6.5.4 Non-gelling systems 317

11.7 Off-equilibrium fragmentation 317

Contents xix

Chapter 12 Universal Fluctuations in Nuclear and Particle Physics 321

12.1 Phenomenology of high energy collisions in the scaled factorial moments analysis 322 12.1.1 Nonsingular parts in the correlations 322 12.1.2 Choice of the variables 323 12.1.3 General phenomenology and experimental results . . . . 324 12.1.4 Self-similarity or self-affinity in multiparticle

production? 328 12.1.4.1 Self-affine analysis of n+/K+p data 329

12.2 A-scaling in pp collisions? 330 12.2.0.2 Aggregation scenario for pp and AA

collisions? 333 12.3 Universal fluctuations in excited nuclear matter 335

12.3.1 A-scaling in nucleus-nucleus collisions in the Fermi energy domain 337

Chapter 13 Final Remarks 345

Bibliography 349

Index 363

Chapter 1

Introduction

Ubiquitous Gaussian law is so spreaded over the Sciences that it seems unreasonable to seek for too many informations in it. Gauss derived its distribution in order to help astronomers to compute accurate asteroid trajectories. Quetelet, appreciating the universality of the normal function, used extensively this law to fit a large collection of data from all the areas of sciences (especially in biology and sociology) stressing the variance as an important parameter independent of the mean value [J. M. Keynes (1921)]. This law arises deeply in the number theory too.

To stay in Aristotelian logic : Gaussian form is the quality, and mean and variance are the quantities. But amount of information ends here for this law. So, much more interesting and recent applications should result in the cases where fluctuations of the investigated variable are not Gaussian.

Important part of this book will then be devoted to the strongly correlated systems with many degrees of freedom, such as those at the critical point of the second-order phase transition. In the statistical physics, this domain is closely associated with development of the renormalization group ideas and scaling theory of critical phenomena, as they are strongly linked to limit theorems. This approach allows for the mathematically rigorous treatment of renormalization group theory and, hence, the phase transitions, in the language of the probability theory. This particular point will be emphasized. It stems from the works of Bleher and Sinai (1973) (see also Baker Jr. (1972)) and further developed by Jona-Lasinio (1975). The basic idea is splitting the whole system into correlated blocks. Then the probability distribution of the investigated physical quantity is calculated for the blocks, and the renormalization procedure is expressing the distri-

1

2 Introduction

bution for large blocks in terms of the distributions for the smaller ones. These probability distributions dealing with larger and larger number of subunits, are assumed to be given by probabilistic limit laws, while the parameters of these distributions are properly renormalized at each step. When this scheme is realized, the renormalization procedures gives the universal behaviour of the infinite system.

Our presentation aims at developing such concepts and techniques, useful for attacking new problems in statistical physics of systems either at-equilibrium or out-of-equilibrium. These insights are exemplified by recent advances in nuclear and high energy physics, where scaling theory provided effective methods for analyzing suspected criticality. But similar reasoning and tools could be employed in any situation where a correlated system close to criticality is at stake. It is worth specifying here that this approach can afford new aids in the exciting novel field of dynamic critical phenomena.

We shall now proceed to outline the general plan : The monograph divides itself into three distinct (though connected)

parts : the mathematical tools (regrouped in the Chapters 2, 3, 5, 6 and 7), and their physical applications, either fundamental (Chapters 4, 9, 10 and 11) or applied (Chapters 8 and 12). The applied examples and data have been chosen in the areas of particle and nuclear physics, because they correspond to domains of research both presently very active and perfectly at the center of our subject. In particular, they deal with out-of-equilibrium finite systems expected to be close to some critical behaviour. And even if criticality is not proven in many cases, they are known to be strongly correlated at the microscopic level. Note however that in principle, no specialized background in these parts of Physics is needed to read and understand the corresponding sections. This is the same for the mathematical parts which do not require more than the undergraduate level in the mathematics of Probabilities. Even if the precise examples are rather specific to gain homogeneity, the approach and results developed in the present book are by no means attached to some definite areas of Physics. Applications exist in all other fields, such as turbulence, polymers, atomic physics, to cite just a few among the most evident.

Let us now review the chapters in greater details : Chapter 2.- The mathematical framework of the stable laws as limit

distributions of uncorrelated random variables and their fundamental scaling properties, is presented following the probability theory. There are no

Introduction 3

rigorous demonstrations here, but instead an attempt to give a synthetic and precise panorama of the main general results according to the point of view of the physicist. Nevertheless, one writes explicitly the precise assumptions required for the main theorems to hold. In addition, a large number of particular stable laws are given explicitly.

Chapter 3.- When the random variables are correlated, much less is known about the nature of the possible limit probability distributions. Some fundamental cases have been studied in mathematical physics and are discussed here. Most of them are connected to the critical phenomena. Emphasis is deliberately put on important physical implications, such as the renormalization group or the critical scaling theory.

Chapter 4.- This section is about immediate applications of the mathematical results presented in Chapters 2 and 3, to a general and important problem of Physics : diffusion, as modelled by random walks. Some more precise applied problems are treated in details within the framework of the probabilistic limit laws, as the linear polymers or diffusion in a random potential. This chapter is a collection of such standard examples of limit laws in the actual physics, as they have been deeply studied by mathematical methods. Its plan follows roughly the same logical structure as for Chapters 2 and 3.

Chapter 5.- Often, the experimental probability distributions are discrete. One studies in this chapter a general mathematical transform - the Poisson-transform -, allowing to pass naturally from a continuous distribution to a discrete one, while introducing minimum additional correlations. Some of these distributions are known to fit remarkably well some experimental data, even if no precise argument exists at present to explain such an agreement.

Chapters 6 and 7.- These chapters introduce important tools to extract correlations from experimental data. Essentially, they appear as combination of particular moments of probability distributions. Some of them are specific and focus on some particular characteristics of the distribution. Other are more general and serve to reveal global feature, such as intermit-tency. Real examples are given to illustrate how these tools should be used. With the Chapters 2, 3 and 5, these two chapters compose the mathematical core of the theory developed in this book, and the starting-point of the whole discussion for physical applications.

Chapter 8.- Currently, the Bose-Einstein interferometry is the centre of a very active domain of research in several areas of Physics. Here one

4 Introduction

discusses the successes and the shadows of this method in various examples of high-energy collisions. This requires use of the results detailed in the two preceding chapters about correlations. Some general trends in the current investigations on this problem are outlined.

Chapter 9.- The random multiplicative cascade model is presented here. It corresponds to a very general approach of strongly-fluctuating processes. The most important application is certainly turbulence, but this model stems too from various areas of physics dealing with multiplicative processes. This should be read as a precise introduction to the following Chapter 10. Note however that a complete discussion of the turbulence problem is beyond the scope of this section.

Chapter 10.- This Chapter concerns with a detailed contribution of the foregoing chapter to the problem of random successive fragmentation of an object. One deals here with dynamical, far-from-equilibrium systems, which are known to be difficult to broach. A particularly simple, though rich, model is presented with its critical and scaling properties, as well as more specific consequences in quantum chromodynamics theory.

Chapter 11.- The mean-field theory of the second order phase transition provides an elegant application of the ideas developed throughout this book. Even if this is a simple analytical model, it acts as a guide to discuss the other models of critical systems presented here. Some of them are standard at-equilibrium models (percolation, Potts, Ising models), but they are systematically treated with the tools developed in the chapters above. Some others are non-standard and far-from-equilibrium (Smoluchowski sol-gel transition, shattering transition), in order to exhibit the robust features of the theory of universal fluctuations.

Chapter 12.- This chapter is more specifically centered on analysis of recent experimental data from high-energy or nuclear Physics, within the framework of the universal fluctuations and the A-scaling. Emphasis is on conclusions which can be done, as well as ambiguities which cannot be solved, by use of these tools at the present time.

Acknowledgements

It is impossible to acknowledge the many influences on our present

Introduction 5

understanding and the ideas we have exposed from other theorists who contributed to our work via encouragement, discussion, opposition or alternative points of view. This book would never be completed without a stimulating interest and encouragement of the multiparticle production community. Here we want to acknowledge a particular role played by N.G. Antoniou, A. Bialas, B. Buschbeck, P. Carruthers, S. Hegyi, R. Hwa, A. Giovannini, W. Kittel, R. Peschanski.

Much of the discussion contained in this monograph stems from our collaboration with P. Bozek, whom we thank for many years of fruitful exchanges.

Finally, we would like to thank warmly X. Campi, A. Chbihi, K. Gudima, H. Krivine, J. Lang, R. Parades V. and A. Tucholski for many friendly exchanges.

Chapter 2

Central Limit Theorem and Stable Laws

Macroscopy is our natural human scale. But to gain in flexibility and freedom, Nature likes to play with a lot of tiny objects coupled together at the microscopic scale. This is why most of the interesting physical macroscopic variables are defined as sum of many microscopic random variables, and also why our only hope to understand such macroscopic behaviour is to seek for some universal limit behaviour. Without the universality, no investigation would be possible because of the quasi-infinite number of the parameters which should be taken into account. This is the land of statistical physics, but most of the other areas of modern physics are dealing with the same micro/macro problematic.

To be more explicit, let us consider a large number of real random variables Xi depending on the integer index i. This set is regrouped into different domains of N variables, and we are asking for conditions under which the global variable M^ = X\ + ... + XN defined for each of these domains, has a non-trivial limit probability distribution. In the following, Xi will be sometimes called the microscopic random variable, while the global variable MN will be called the macroscopic random variable.

Let us be still more precise. The indices i could be related to space location, and the space is divided into non-overlapping domains V; of size N. The random variable M/v attached to the macroscopic volume Vi will then correspond to the extensive physical quantity observed globally in this domain V;. Properly normalized, this variable will be considered as the averaged value of Xi over V;. We have then to ask the following question : how should we choose the shift and normalization factors, AN and BN in

7

8 Central Limit Theorem and Stable Laws

order that the random variable (the arithmetic mean) :

MN=^'f-A" ( 2 . j , has a smooth positive probability distribution when N goes to oo ? Answering this question will allow to define unambiguously the only way to define properly the averaging protocol (2.1) of Xj. This problem is connected to the relative magnitude of the fluctuations of random variables.

2.1 Central limit theorem for broad distributions

Under certain general assumptions, one can prove that the random variable Mjv, as defined in (2.1) approaches the Gaussian shape for large values of N. This important result is generically known as the Central Limit Theorem. It demonstrates in particular that many features of the Xj—distributions are removed by this averaging procedure, since they are no more present in the final shape of the limit distribution. We detail below the usual conditions of applicability of this theorem, and also what can happen when these conditions go to fail.

2.1.1 Central limit theorem for the sum of uncorrelated variables

Let us consider the random variable MJV as the sum of N statistically independent random continuous real variables Xj. Each variable Xj is assumed to be distributed according to the same probability density fx{x), defined as :

fx{x)dx = Prob[a; < Xj < x + dx] (2.2)

within the infinitely small range dx. The normalization of probability implies that :

/ fx(x)dx = 1

for the positive distribution fx- Integration is performed over the whole definition range of the random variable X.

To consider only microscopic variables of the same probability distributions is a very general practical case indeed, but this case is also needed if

Central Limit Theorem for Broad Distributions 9

one wants to find the general limit distribution. Otherwise, there is often one particular variable Xj with a distribution which dominates the other ones and hence the averaging is meaningless. Note however that, if all the Xj—distributions are close enough, even though not exactly identical, most of the Central Limit Theorem may hold true, as shown by Lindeberg's theorems [W. Feller (1971)]. These extended Central Limit Theorems allow, to a certain extent, for fluctuations of the parameters of the Xj—distributions. Even if conceptually important, these theorems will not be considered in the following, in order to keep formulation of both ideas and results as simple as possible.

The Central Limit Theorem states : If the common probability density of the Xj is such that :

f\ <i fx(x')dx' lim x2 / M > x , * ' , = 0 (2.3)

x~*°° J\x'\>xx '2fx{x')dx'

then the limit probability density of the variable MJV, properly shifted and normalized according to (2.1), approaches the normal distribution, that is to say :

1 f m2\ lim fMN(rn) = fnorm,ai{m) = —== exp —— . (2.4)

w - » ° ° ^/2-K \ 2 /

This is the Gaussian function with the second moment equal to 1 (see figure 2.1):

{m2)normal = / m2fnormai{m)dm :

When a mean and a variance of Xj are both finite, the term 'properly shifted and normalized' means that the normal distribution occurs for the reduced variable :

The moments of rank q are defined generally as :

{xq)= fxqfx{x)da


J nnrmnl' ^

Fig. 2.1 Plot of the normal distribution (2.4). For the Gaussian function with variance (m 2 ) , the two inflexion points where the derivative takes its extremal value ± l / ^ / 2 e 7 r ( m 2 ) 2 are at the abscissa m = ±-y/ (m 2 ) , and the corresponding ordinates

are ± l /y^2e7r(m 2 ) . These values can be used to calculate the variance of an experimental Gaussian distribution.

when these quantities are mathematically correctly defined. On the other hand, when the mean and/or the variance is infinite, the shift and the normalization are less obvious to characterize but they exist in some sense as long as (2.3) takes place. The complete shift and normalization will be given quite generally in Section 2.2.7.

The reciprocal of this theorem is also true, so all the distributions of random variables such as (2.1) and leading to the normal law (2.4) are characterized by the condition (2.3). The ensemble of all distribution functions verifying the condition (2.3) is called the domain of attraction of the normal law. This asymptotic condition is indeed very general, and explains a posteriori why the normal law is so widely spread in the Nature. In particular, it is the case for any bounded random variables Xj.

One comment about condition (2.3) is important to be noticed here : only the tail of the distribution fx(x), i.e. only the large-z behaviour, is needed to conclude about the appearance of the normal law. In this sense, the normal law depends only on the frequency of appearance of large - and

Central Limit Theorem for Broad Distributions 11

therefore rare -, values of the microscopic variables. Figure 2.2 shows for some values of N, the probability distributions of

the variables MN, defined as in (2.1) for all the Xj uniformly distributed over the interval [—\/3, v^3]. Hence, the mean value of the Xj equals 0 and

fy(m)

0.4

0.2

0.0

Fig. 2.2 Plots of the probability distribution functions of the MN random variables (2.1). The normal distribution is shown by the light continuous curve for comparison, a) for the Mi and M2 variables; b) for the M4 and Ms variables. The M4— and Ms—distributions have been computed using 108 times the pseudo-random generator described in the footnote of the page 12.

the variance is 1 i.e. (Xj) = 0 and (Xf) = 1. The uniform distribution corresponding to the Mi—variable, is shown in figure 2.2.a alongside the M2—distribution :

fM2{™) = (V6 — m)/6 when m > 0

fM2(jn) = (V6 + m)/6 when m < 0

In figure 2.2.b, M4— and Ms—distributions are compared with the normal law. The analytical expression for the successive distributions will be clear from the relation :


which is a consequence of a general recurrent relation discussed later in Section 2.2.1. There is a fine closed formula for the distributions /MN, which are the polynomials of order N — 1 [J. B. Uspensky (1937)] :

. (N+m^/N~/Z)l2 i

W"0 = 2^M £0 (-&(")(*-** + >

The following remarks help to understand the essential elements of the convergence to the normal law. First, note that the moments of the common Xj—distribution are :

(Xf+1) = 0

<**<> = ^ T

for any integer q, i.e., (M?) = 1, (Mf) = 1, (M?) = 9/5, etc. Higher moments than the fourth moment are indeed quite different from the values of the normal moments :

(M£) = M (2.5)

especially for higher orders. On the other hand, moments of Mjv are all given recursively by the following formula :

In particular, the successive values of the fourth moment of the Mjv are (Mpf) = 3(1 — 2/5N) and tend indeed to 3, which is the value for the fourth moment of the normal law, as given by (2.5). This convergence to the limit normal distribution is clear even for such modest values of the parameter N as shown in figure 2.2b. This is the basic for a 'quick and dirty' pseudo-random generator with the normal distribution, when accuracy is not essential *.

"Let us take half of the sum of four pseudo-random variables uniformly distributed over [—\/3, \/3]. The first three moments are the same as for the normal law and the fourth one is 10% off the normal value. This is the M4—variable of figure 2.2.

Stable Laws for Sum of Uncorrelated Variables 13

2.2 Stable laws for sum of uncorrelated variables

We return now to the additive unsealed case :

MN = Y,Xi • (2-6)

The quest for a statistical investigation of large objects corresponds to the characterization of the limiting distribution of such variable as M/v, with N the (large) number of subunits of the system. Logically, the problem can be described in two steps : (i) when the random variables Xj are all statistically independent, (ii) when there are correlations between the microscopic variables. In the latter case, results could depend on the correlations. Even the first step, when the variables are uncorrelated, is not trivial and needs some precise discussion.

One great achievement on the way to solve this fundamental question is that the corresponding problem for uncorrelated variables has been entirely solved : one knows all possible limit distributions of the random variables like (2.6) when N tends to infinity. On the other hand, the case when the variables are correlated, is not yet generally solved even if some precise results and tools are available.

2.2.1 The stability problem

Let us first discuss the probability distribution of the sum of N uncorrelated variables Xj. We will work mainly either with the probability density (2.2) which is generally defined as the probability for the random variable X to take values close to some definite value, or with the characteristic function of probability distribution which is nothing but the average of exp(—ikx) :

/

oo

e-ikxfx(x)dx . (2.7)

-oo

The latter function defines another complete description of the probability distribution. Following the general results on Fourier transforms, one actually knows that there is the same information in the probability distribution and in its characteristic function. For example, the formula (2.7) can be


inverted to recover the distribution fx(x) following :

1 f°° fx(x) = ± / eikx<j>x{k)dk .

So it is equivalent to work with any of the two descriptions. The characteristic function of the normal distribution (2.4) is the Gaus

sian law :

<j>normal{k) = e X p ( - P / 2 ) .

A useful general property is the development near k = 0 of the characteristic function as a series of cumulants. The first few terms write :

-lncl>x(k)=ik(X) + ¥-((X2)-(X)2) + ... (2.8)

The role of the variance appears here in a natural way. But the real importance of this function comes from a fundamental result : the convolution theorem. It tells t ha t : If X\ and X% are two independent random variables of respective characteristic function <pXi(k) and </>x2(fc), and C\ and c-i are two real positive numbers, then the characteristic function of the random variable M = C\X\ + C2X2 is :

<l>M(k) = 4>xAcik)<f>x2{c2k) . (2.9)

In particular, for any number of variables, the characteristic function of the sum of uncorrelated random variables is the product of the characteristic functions of each of them. This property can be used as well as the definition of statistical independence, i.e. the absence of correlation, of the random variables. Moreover, since the characteristic function is nothing else but the Fourier transform of its probability density, the density of the sum of two independent variables is the convolution product of their respective probability densities :

ftt{™) = /xj(a;i)/x3(a;2)<5(TO - {cxxx + c2x2))dxidx2 (2.10)

where 5 is the Dirac distribution. The key point to determine all possible limiting distributions of the

random variable (2.6) is the stability problem which can be stated as follows [P. Levy (1954)] :


What are the probability densities f(x), such that, if X\ and X2 are two independent random variables of the same probability density f{x) and C\ and C2 are two positive numbers, then M = C1X1 + C1X2 is a random variable with the same probability density f(fh) ?

Such a function is called a stable distribution. The importance of this problem is clear for physicists : if one knows its solution, then the distributions of the additive physical observables at the thermodynamic limit should be necessarily one of these solutions. Alternatively, its importance can be expressed by the following theorem [B. V. Gnedenko &z A. N. Kol-mogorov (1954)] :

The function f is a limit distribution for the random variable MN = (X\ +... + XN — AN)/BN, where all the variables Xj are independent and identically distributed, if and only if it is stable.

There exist simple illustrations of this stability. For example, suppose that X\ and X2 are two independent Gaussian random variables. Then, variable X\ has the density :

' * < •> -TsbH-^w 2 ) <"" and this is the same for X2—density with the parameters m-i and 02 • Then, it is simple to demonstrate that the variable M = c\Xi + c%Xi is yet another Gaussian variable with the parameters : a = cia\ + c^ai and a2 = c\o\ + c2(T2, i.e. the Gaussian distribution is stable. This can be shown either using the convolution property (2.10) or, more directly, using the characteristic functions (2.9) since the characteristic function of the Gaussian distribution (2.11) is just :

<j)Xl (k) = exp (iaxk - ^-fe2 J .

It is instructive to introduce now the concept of fixed-point of the stability condition, in order to get an idea of the sketch of the full solution of this problem. Suppose that all random variables Xj are statistically independent with a common probability law centered at 0, i.e. {X) = 0. Let us consider Mjv = YLJ=IXJ/BN- Then, because of equation (2.9) the characteristic function associated with MN is :

4>MNW = <f>x(J^) •


Let us also denote by TZN(<J)) the transformation over a set of characteristic functions :

KNmQ = <t>N(J^)

We are now seeking for the proper normalization Bjv leading to a positive limit distribution :

</>(k) = lim nN{^)){k) . N—Hx

Suppose that the characteristic function for the Xj—distribution is :

^(fc)=eXpi-53c#r

with some exponent /J,. Then the characteristic function for MN, provided that it exists, is :

( oo

-#£)<:,• (I A|/5JV)W

The choice BN = Nx^ leads then to the limit distribution :

0(fc)=exp(-ci|!fen • (2.12)

If one chooses BN increasing less than N1^, the limit probability distribution should differ from 0 only at the infinity. On the other hand, if BN increases with N more rapidly than N1^, the limit distribution collapses into a Dirac distribution. In this sense, BN = N1^ is the only choice leading to a non-trivial limit distribution of the stability problem for these distributions <t>x(k). In other words, the function (2.12) with BN = N1^ satisfies the relation :

nN(<l>) = <s>

which is the fixed-point relation of the transformation TZN-All these ideas can be generalized to solve completely the stability prob

lem for uncorrelated random variables, leading to a complete solution detailed in the following sections.


2.2.2 Complete solution of the stability problem for uncorrelated variables

2.2.2.1 The ensemble of one-dimensional stable distributions

All one-dimensional stable distributions P^p are given by the theorem of Gnedenko and Kolmogorov (1954) :

The stable distributions are the Gaussian law and those for which the logarithm of their characteristic function is given by one of the formulas :

ln^(Jfe) = -\k\» M +1/3 A tan(7r/x/2) J if fi ± 1 (2.13)

ln^(jfe) = -|fc| (l + i^-^-Ank) i f / i = l (2.14) V ^ \k\ J

with 0 and fi two real constants such that — 1 < 0 < 1, 0 < /i < 2. This theorem gives all stable distributions depending on one real vari

able, up to a constant shift (a; —> x + xo) and a constant dilation (x —> ex) of the whole distribution. Shift corresponds to adding a term i'yk in the formulas (2.13) and (2.14). On the other hand, dilatation changes the factor |A;|M into C\k\^ in equation (2.13).

Note that the stable laws with // ^ 1 are particular elements of the wider ensemble of the semi-stable laws for the index fi. These are the distributions that verify the scaling relation :

ln<£(Afc)=A'iln<?!>(fc)

for any real positive parameter A *.

2.2.2.2 Alternative formulas for the stable distributions

This two-parameters family of stable distributions can be written in various forms. Some forms are more interesting than others, according to their application. One of them, conveniently convergent, is :

Plii/3(x) = — - / sm(xk1/>1-kta.nilj + ip)e-kdk (2.15) 7VX COS %p JQ \ /

*For the particular case fi = 1, the same basic scaling occurs up to a shift of the whole distribution.


with the parameter ij} defined by :

tanip = f3 tan(7r/i/2) , with \ip\ < TTfl

if / x ^ l

and

Pit/3(x) = - / cos ( xk+ — kink j e fcctt if / / = 1 .

The case i = 1 is always particular because of the logarithmic terms involved*.

2.2.2.3 Range of values for fi

The values of \i larger than 2 are forbidden in the sense that the corresponding functions, even if mathematically well-defined, take negative values for some range of values of x and thus cannot be considered as the probability distributions. Such an example is shown in figure 2.3.

^5/2,0 (x)

Fig. 2.3 Plot of the function : P5/2 ,0i which takes negative values for \x\ > 3.761788 This function cannot be a probability distribution.

' T h e distributions with fj, = 1 are sometimes called theCauchy distributions, since they have been studied by Cauchy in an attempt to generalize the Gaussian law of errors [A. Cauchy (1853)].


2.2.2.4 Range of values for 8

The parameter 8 is related to the asymmetry of the curve. More precisely, it can be shown that Up. ^2 then :

1 + /J *i£> ^P^(x')dx'

Some particular cases are of importance: 3 vanishes for an even distribution and 8 = 1 (respectively : 8 = — 1) when the random variable Xj takes only positive (respectively : negative) values.

0.6

0.4

0.2

0.0 -5 0 5

x

Fig. 2.4 Plot of some symmetric (/? = 0) stable distributions : Pj/2,0) ^1,0 a n d /53/2,o-

Some typical symmetric stable distributions (8 = 0) for various values of \i with are drawn in the figure 2.4. We see in particular that the smaller is the value of \i the sharper is the central peak. Other analytical examples will be presented in Section 2.2.3.

Figure 2.5 shows examples of stable distributions for the largest possible value of B {i.e. 8 = 1). Note that the distributions for /x < 1 are positive defined only in the interval JO, oo) of the variable x.

For all values /J. < 2, positive 8 corresponds to a positive skewness and this distortion is more pronounced for small values of fi. Figure 2.6 shows


P*l<*>

Fig. 2.5 Plot of some asymmetric stable distributions (/3 = 1) : Pi/2,1, Pi,i and ^3/2,1-

the influence of the parameter /3 on the asymmetry of the curve for the intermediate value /x = 1.

2.2.2.5 Gaussian distribution as a stable law

The normal distribution corresponds to a limiting value fi = 2, and this is the only case where the variance of the random variable is finite. Notice that because of the definition (2.13), the distribution in this case is always given by the Gaussian curve, independently of the value of (3. In this sense, there is no asymmetric normal law.

2.2.2.6 Moments of the stable distributions

By far the most important parameter of this family is the characteristic exponent /x. We saw that it cannot be larger than 2 because P^^ must be non-negative (see the example in figure 2.3) and is larger than 0 to ensure the convergence at the origin.

The expansion (2.8) and the theorem giving the stable distributions allow to conclude about the finiteness of the first and the second moments


0.3

0.2

0.1

0.0 - 5 0 5

x

Fig. 2.6 Plot of the symmetric Cauchy distribution and asymmetric Cauchy distributions (/x = 1) : Pi,o, Pi,1/4, Pi,i/2> -Pi,3/4 a n d ^i,i- T n e larger is the value of /3, the more skewed is the curve.

of X distributed as a stable law, namely :

0 < M < 1 _ > (X) = oo and (X2) - (X)2 = oo

1 < M < 2 —• (X) < oo and (X2) - (X)2 = oo (2.16)

H = 2 —• (X) < oo and (X2) - (X)2 < oo

So in general, we cannot discuss the variance of the random variable distributed according to this kind of law. Instead, we have the theorem which says that :

/ / Xj are independent random variables whose probability distribution

is*

1 f°° PM*) = T" / eikX exp{-cn\kndk

2?r . / - O O

then the random variable MN = X),- Xj is distributed following the same

"The example below is written for /3 = 0 to keep the formulas as simple as possible.


probability law :

i r°°

p0(m) = ^- eik™exp(-CN\k\»)dk

with

Gjy = Cj + . . . + Cjy .

We recover clearly the Gaussian case for fj, = 2. Only in the case fj, = 2, Cj's are the variances of the variables Xj.

2.2.3 Explicit examples of stable distributions

2.2.3.1 Symmetric stable distributions ((3 = 0)

Many stable distributions can be expressed in terms of known analytical

functions. This is the case for the following symmetric stable distributions P»fi-

1 f°° P^o(x) = — e~W exp(ikx)dk . (2.17) 2^ J-oo

Hereafter are some closed results sorted by decreasing fj, values. • The Gaussian distribution fj, = 2, /3 = 0 :

P2fi{x) = 7^exv{-l) (2-18)

• The Cauchy distribution p. = 1,/? = 0 [A. Cauchy (1853)] :

IT 1 + Xz

• The Zolotarev distribution /i = 2/3, (3 = 0 [V. M. Zolotarev (1954); V. M. Zolotarev (1986)] :

^ A o ( z ) = V l £ e X P ( 2 ^ ) W~WI* ( 2 ^ )

where Wp,q is the Whittaker function [M. Abramowitz k. I. A. Stegun (1964)],


• The distribution p, = 1/2,/? = 0 [E. W. Montroll k J. T. Bendler (1984)] :

Pi/2,o(x) = y/2^ 2 \ / 2 d d ) 008(1/41*1)

+ I 2 S(y/2Z\x\ ) sin(l/4|d,)

with C and S the sine and cosine Fresnel integrals, respectively. Note that the first example P2lo(x) (equation 2.18) corresponds to the Gaussian law with a variance equal 2 and not exactly to the normal law. This is due to a particular normalization which is usually chosen for the characteristic function (2.13, 2.14).

Moreover, the following convergent series hold for all x > 0 :

for 1 < /x < 2

P«°M = £ E ( " ^ + 1 rffi Mn^/2) for 0 < M < 1

aft" (2J)!

j=i xi» T{j)

known as the Cauchy expansion and the Wintner expansion [A. Wintner (1941)], respectively.

The case /i —» 0 deserves a particular attention. Montroll and Bendler (1984) have shown that for the vanishing values of fx one obtains the following approximation :

w^-\-*m 1 f u2 log2 Id + 2 ^ G X P

(2.19)

with the small value

<r(l*) = IT 71" \ fj,

2"eJfWM) This means that the distribution is a sum of the broad long-tailed log-normal distribution and the narrow Gaussian peak at the origin. But strictly speaking, the case /i = 0 cannot be defined since the function is not normalizable.


2.2.3.2 Asymmetric stable distributions (@ — 1)

Several similar expressions are known for the asymmetric stable distributions (/? = 1). The corresponding distribution support is (—00,00) when 1 < fi < 2, but it is restricted to ]0,00) for the case \i < 1 . In the latter case, the stable distribution has a simple Laplace transform [C. P. Lindsey & G. D. Patterson (1980)] :

Jo

t» \ P„,l{x)e-tXdx = e X P r—r-r . ( 2 . 2 0 )

/o V cos(7T/x/2)y

Few known explicit expressions for the distributions PMii written for positive values of the variable x, are the following :

• The distribution for /x = 3/2, , 3 = 1 * :

x (1 4 2a;3 \ P 3 / 2 » = 2 l /3 35/ 2 7 r l /2C / [Q> 3 . I f J (2-21)

with U the second Kummer confluent hypergeometric function [M. Abramowitz & I. A. Stegun (1964)]. For negative values of x, the analytic continuation applies.

• The case /z = 1/2, /3 = 1 [P. Levy (1940)] :

P1/2il(x) = ^ f e~V2* (2-22)

• The case /i = 1/3, (3 = 1 [H. Scher & E. W. Montroll (1975)] :

23/2 / 25 / 2 \

P i A i W = 37/4^3/2^1/3 [^JI^J2)

where Ki/3 is the modified Bessel function of order 1/3. Moreover, the Pollard expansion [H. Pollard (1946)], which is the equivalent of Wintner expansion, is known in this case too :

p (x] _ /* y * (~1)J '+1 r 0 » sin(j7rM) M ' U ^ T T X ^ X* T{j) COSJ(TTfl/2) '

The great similarity of the Wintner expansion (for /3 = 0) and the Pollard expansion (for /3 = 1) is not a coincidence and there is a fine reciprocity

"To our knowledge, this distribution has not been known before. We outline the main points of its derivation in Section 2.2.4.


relation for stable distributions connecting them, as described in the next section.

2.2.4 The reciprocity relation for stable distributions

A first general relation between different sets of values of the parameters /x,/3 can be found readily from the general definition (2.15) :

P„,-p{-x) = P„AX) •

This allows to restrict the investigation of stable distributions into a positive range [0,1] of the skewness parameter (3.

A less obvious relation between stable distributions of indices fi and 1/fj, has been reported by Montroll and Bendler (1984) :

with the corresponding parameters and variables according to :

1 / n \ /nS\

- = tan {- j tan {j j

J-=tan(^)tan(^)

y = x' '* s m Sin ' ^ ——-

x sin ( -^— J > 0

where 6 is any real parameter for which the quantities above are all properly defined. One can reproduce several interesting results using these relations. For example, if one chooses 5 = l//x and \i > 2/3, then one can express •Pi//i,i/tan2(7r/2/j) a s a function of PMio- In particular, this results in the distribution Pi/2,1 a s a consequence of the Gaussian form of P2,o- In the same spirit, the Zolotarev distribution Pz/3,0 is connected to the asymmetric distribution £3/2,1 by taking S — 3/2. This provides a basis for the derivation of the expression (2.21).


2.2.5 The tail of stable distributions

Many interesting features of stable distributions for large or small values of the variable have been studied by Ibragimov and Lirmik (1971). The large—a; tail is a power law when /i < 2 :

P M ( l+/?)r( /x + l)Sin(7r/i/2) 1

This is the main reason for divergences of the moments of the random variable. In contrast, there is the essential singularity for small—a; in the case 0 < /z < 1, /3 0 :

^ ~ - p ( - ^ ) <**>

where

Urn a(u, P) = \ fi-tO

a ( l , / ? ) = 0

In all other cases, the value of the distribution at the origin is finite and equal to :

T(l//x) cos[arctan(/3tan(7r/^/2))//i] ~^M [l + )82taii2(7r/i/2)]V2M_ FfA°) = —— h , fl2*.„.2A,../o»l/2U ' (2-25)

A simple but important consequence of (2.23) is that the characteristic function of stable law is non-analytical at the origin k = 0, except for the Gaussian law (fi = 2) and the symmetric Cauchy law (fi = 1, /3 = 0) .

2.2.6 Moments of stable distributions

Because of the algebraic tail (2.23), the moments of the stable distributions, as defined by :

/

oo

\x\^P^{x)dx (2.26) •oo

exist only if either q is not too large or for q > - 1 when P^.,p(0) has a finite value. The latter case occurs because of equation (2.25) when /x > 1. In other words, the non-existence of some moments is a direct consequence of

Stable Laws for Sum of Uncorretated Variables 27

the non-analyticity of the characteristic function <j>(k) near k = 0. One can write more precisely :

(\x\q) is finite if <

— 1 < q < oo and /x = 2

- 1 <q< \i and 0 < fi < 1, \(3\ < 1 or 1 < /x < 2

—oo < q < (J, and 0 < /i < 1, |/3| = 1

There exist closed formulas, derived directly from equation (2.15), for the moments restricted to the ranges (—oo, 0] and [0, oo) :

/ ° ( - ^ p ^ W ^ - r<-«/"> <M«W2-*/")> OO

OO

I xq PIJ,tp(x)dx

fjF(—q) sin(7rg) cos^/^ -

r ( -g/ /x) sin(g(7r/2 + V>/M)) fiT(—q) sin(7rg) COS^/M ^

As a first consequence, one remarks that the case /j, < 1, for which tan(7r/z/2) is positive, and /3 = 1 yields :

/

o

for any value of q, since in this case tp = 7r/i/2. One concludes that the distribution vanishes in the whole range (—oo,0], as it has been asserted without demonstration on page 24. Moreover, the corresponding moments (2.26) are all given by the formula :

T(-q/fi) cos(ipq/n) " ' ' ^ ixT{-q)cos{-Kq/2){cos^/»

with tp denned in (2.15). This formula is correct also for [i = 2. This is another way to see that the average value of the variable is finite

only when // > 1, and that its variance is infinite for any value of /i < 2, as it has been written in page 21.

2.2.7 Asymptotically stable laws - domains of attraction

Let us recall that the stable distribution is the one which is exactly invariant under addition of independent random variables. This invariance happens


for the distribution of the sums of uncorrected microscopic random variables Xj when the corresponding sums are infinite. The essential question now is which probability distributions of microscopic variables Xj lead to the stable distribution P^p for the sums of Xj when the number of terms in the corresponding sums goes to infinity.

Let Xi,..., XH be a sequence of statistically independent variables, identically distributed with a common probability density f(x), and let M be the random variable :

MN = *I + --- + XN-AN

ON

where AN and BN are two real numbers chosen such that the probability of MN converges to the stable distribution P^p when JV tends to infinity. When the distribution f(x) for the variable X yields the limit density Pu,p{m) for M, then / is said to belong to the domain of attraction of P^tp-All domains of attraction are characterized by the following theorem [B. V. Gnedenko k, A. N. Kolmogorov (1954)] :

For all positive A, the ensemble of the probability densities f(x) such that :

L^i-i fW)dx' l i m {*'>WJ) ' ~ A" (2.28)

r* f(x')dx' i _ B lim J-°°JK ' i tL (o 29)

™ f~f(x>)dx' 1 + 0 [ ' constitutes the entire domain of attraction of the P^p.

The case \i > 2 in equation (2.28) defines the domain of attraction of the Gaussian distribution. This is also an alternative condition to (2.3) for the applicability of the Central Limit Theorem.

We see that in this approach only the tails of the distributions f(x) are necessary to characterize the domain of attraction to which those distributions belong. This remark has fundamental consequences in Physics due to the following theorem :

If f(x) ~ l/|a:|M+1 with 0 < fi < 2 for ±x —» oo, then f(x) belongs to the domain of attraction ofP^p with all the parameters AN, BN, /? explicitly known.


In particular, BN as appearing in (2.27) is such that BN ~ N1^ and :

AN = 0 if fi < 1

AN = N(X) if /* > 1

Logarithmic dependence appears in the marginal Cauchy case (j, = 1. In other words, when the tails of the distribution functions of the uncorrelated local variables X are algebraic and hence decreasing like l / |x | ' ' + 1 with 1 < n < 2, then the reduced global variable :

E f - i Xj - N{X) MN = ^ ~ 1

A rJ

1 / , (2-30)

has an asymptotically stable distribution of index fi when the number N of microscopic variables tend to the infinity. When 0 < \i < 1, this is the same with :

TN X-

These results for uncorrelated variables permit a complete classification of the ensemble of possible local distributions into a two-parameter family of domains. We shall see later that for strongly correlated variables, the same kind of classification is not yet known even though a concept of the fixed-point equation is used as a natural generalization of these ideas.

2.2.8 The concept of the A-scaling

The case 1 < \x < 2 deserves more attention since it corresponds to an infinite variance and, hence, to large fluctuations of a microscopic variable X with a finite mean value (X). This is indeed an important and general case in Physics.

Let MN be the sum variable. For /x in the range 1 < /x < 2 we get :

(MN)=N(X) .

The preceding characterization of the domain of attraction of the stable law writes generally :

Prob MN-NIX) _, '

m< j ^ <m + dm Pufiirnjdm


Introducing then the probability density f^ (m) of the variable Mjv, one obtains [R. Botet & M. Ploszajczak (2001)] :

as the limit distribution when the number N of subunits tends to infinity. This form of asymptotic scaling, discussed later in other contexts, will be called the A— scaling. In the present case, the scaling exponent is A = l//x. Note that in the considered case, the value of this scaling exponent is contained in between 1/2 and 1.

The case when 0 < fj, < 1 is more subtle because (MJV) cannot be defined equally well (see Section 2.2.2.6) and so the scaling relation like (2.32) cannot be written. However an approximate behaviour is expected in the following form :

N^fMjfn) ~ - P ^ ( ^ r ) (2.33)

without any connection between the average value of Mjy and N. Instead, we can show that N1^ is a typical value Mjy of Mjv for large values of N. The corresponding scaling law :

M*NfMN{^)-P^{^) (2-34)

resembles (2.32) with the exponent 1 instead of l / / i . In this form, the scaling relation (2.34) is perhaps less useful because M^ is not as precisely denned as the average value. Nevertheless, it shows that in the framework of the sum of independent random variables, it is pointless to search for relations such as (2.32) with the scaling exponent 1/fi larger than 1.

2.3 Limit theorems for more complicated combinations of uncorrelated variables

2.3.1 Product of uncorrelated variables

Most of the statistical physics is devoted to the analysis of additive quantities : relation between local extensive microscopic variables Xj to the global macroscopic observable MN (equation (2.6)). There exist exceptions from this predominance of additive quantities : some global physical quantities

Limit Theorems for More Complicated Combinations of Uncorrelated Variables 31

have a multiplicative structure or, uncommonly, they may have even more complicated forms. A few such combinations can be found in the literature, but the multiplicative case is by far most common in Physics. We shall see some specific applications in Section 3.3.4 about the renormaliza-tion group. The multiplicative structure appears when the success of a task is the consequence of a chain of successful independent random processes. A nice example is the analysis by Shockley (1959) of scientific publications of researchers. Publishing an article in a scientific journal is the result of several successive independent processes, each of them having its proper probability to be completed *. Hence, the publication probability of an article should be equal to the product P1P2P3P4P5P6P7P8 • • • of all probabilities of the success along the chain. Consequently, the Central Limit Theorem should hold approximately for the logarithm of the number of publications, leading to, say, Ms—distribution (as defined in page 11) for this logarithmic random variable. Shockley verified successfully this logarithmic-Gaussian shape on the publication output of 88 researchers of the Brookhaven National Laboratory. Note that this result depends on the structure of the process more than on the precise number of steps in the process or the proper distributions of the individual probabilities pj. One can for example consider some additional random steps in the chain without modifying the log-normal feature.

Let us be more formal and consider the random variable :

JV

WN = Y[Xj . (2.35) 3=1

The logarithm of WJV is additive and the Central Limit Theorem applies if the conditions for this theorem are realized for In Wpr. The resulting probability density for the reduced variable (the geometrical mean) :

W» = {%)U" (2'36) "One can name some of them : 1) finding an interesting problem to investigate, 2) having

competence to work on it, 3) being able to recognize a relevant result, 4) stopping the research at a convenient point, 5) writing clearly the results, 6) being able to profit from the comments of other persons involved in the same kind of problem, 7) finishing the manuscript and sending it to an editor, 8) responding to referee's criticisms positively, such that the referee finally recommends publication...


is the log-normal distribution [E. L. Crow k K. Shimizu (1988)] :

Jlog- ii(y>) = 1

exp ln2iu

if w > 0 wy/2n

= 0 if w < 0

Under this form, its moments are

(Wq) log-normal = exp(q2/2)

and this yields the following characteristic relations between moments :

(2.37)

(w)i \(w)2

which are verified for all values of q.

f (w) log-normal

Fig. 2.7 Plot of the log-normal distribution (2.37). The abscissa to = 1 corresponds to the median, i.e. the areas of the curve between 0 and 1, and between 1 and oo, are equal.

The shift parameter AN, as introduced in the geometrical mean (2.36),


corresponds to the median value :

/ f(w)dw = 1/2 Jo

and the scale parameter BN is connected to the variance as :

(fift) - (WNf = A2N (exp(2B%) - exp(B%)) .

Up to now the subject of this section may seem to be less important as compared to the Central Limit Theorem, even if the multiplicative processes may occur in various physical situations. But another point of view let this problem more interesting for our purpose. Imagine that the microscopic variables Xj can take the values 1/2 or 2 with equal probability. The q-th. moments of W^, the variables defined in (2.35), are equal to :

(N\ since I J is the probability to get the event X = 1/2 exactly I times

among N trials. Performing the summation in equation (2.38), one gets :

which diverges for any positive value of q in the limit N —> oo. The reason for those divergences is simple but worth to be mentioned here. The probability of the event {2 ,2 , . . . , 2} in N trials is exponentially small (1/2^), but the corresponding value of the random variable (2N) is exponentially large and, therefore, the contribution of this very rare event to the moments remains finite.

Several fundamental problems arise as a result of this feature. For example, one expects great discrepancies between small and large systems, because many rare states of a large system are not available in a small one, leading to a possible wrong determination of the moments. This is a serious limitation for numerical simulations of such multiplicative systems, since simulations of small samples provide mainly informations about the most probable value and not about the average value in these cases [B. Der-rida k H. Hilhorst (1981)]. Another point is that successive moments do


not scale simply with the order q *, meaning that each moment is governed by a different size-scale and this behaviour is connected to the multifractal structure of the fluctuations.

Another feature of multiplicative systems is their sensitivity to the short-range correlations [S. Redner (1988)]. Suppose that the sequence of random variable X is changed such that no immediate reversal is allowed. This means that all values for a rank 2p are exactly the same as for a rank 2p + 1, but the values of X2p+i are themselves uncorrelated. The beginning of the sequence is then one of the four possibilities : {2 ,2 ,2 ,2 , . . .} , {2,2 ,1/2 ,1/2 , - . .} , {1/2,1/2,2,2, . . .} or {1/2 ,1 /2 ,1 /2 ,1 /2 , . . .} . The above reasoning remains valid for the values 1/4 and 4, instead of 1/2 and 2, grouping the variables by two in N/2 'twin variables'. The average value of (Wff) is then equal to (y/IT/8)N instead of (5/4)^ for the uncorrelated case, and so, it is considerably larger. We shall see in Section 3.1.1, that such a problem with the short range correlations cannot arise for additive variables.

2.3.2 The Kesten variable

Another combination of importance for some physical problems, is the Kesten variable [H. Kesten (1973)] :

Xi + Xi x X2 + Xx x X2 x Xz + ... + Xi x X2 x . . . x XN . (2.39)

This variable appears in problems such as the diffusion in a medium with random energy barriers [B. Derrida & H. Hilhorst (1983)], or more generally in sleepy ladder problems : a point has to climb on top of a ladder, but there is an average time Tj to pass from j - t h rung to (j + l)-th rung. If it does not succeed to climb up, it goes down the ladder. The average time to reach the top is then the sum of typical times to reach the intermediate steps :

T = Ti+Ti XT2 + Ti XT2 X T3 + . .. + Tl XT2 X . .. X TN • (2 .40)

T has the same form as (2.39). If the times rq are distributed according to some common probability law fT, then the limiting distribution of the random variable T verifies the self-consistency condition [C. de Calan et al.

*For example, one does not get a formula like : (W^) ~ exp ((aq + b)N), as it is usual.


(1985)] :

fT(T) = Jo /T(T)/T ( J - l ) ^ • (2.41)

Moreover, following (2.40), the distributions of T and TX X (1 + T) should be equal for infinite N.

Some exact relations can be found from the equation (2.41). For example, the integer order moments of the T—distribution can be found recursively using the relation :

Assuming then large—T and the power law behaviour of the limiting distribution :

fr(T) ~ 1/T"+1

the self-consistency condition (2.41) yields :

This equation fixes the value of the exponent /i. This result is in general consistent with such algebraically large distribution of times T [H. Kesten et al. (1975)1.

2.3.3 The Gumbel distribution

The statistics of extreme values from an ensemble of random variables is of primary importance in Physics [E. J. Gumbel (1958); R. Rammal (1985)]. For example, the low-temperature thermodynamics of a system is often governed by the statistics of low-energy states.

The problem arises generally in the following way. Suppose that {X,} is an ensemble of N independent random variables and YN is the random variable whose value takes the minimum value in the ensemble {Xj} :

YN=Minj{Xj} .


The cumulative distribution of YN is :

TV

Piob[YN>y} = Tl I fXj(x)dx -n/ since if y denotes the minimum of the values of Xj, then all values of Xj must be larger than y. The probability density of the variable Yjv is just the opposite of the derivative of Prob[Yjv > y] with respect to y. Suppose now that all random variables X have the same probability distribution fx- In this case one obtains :

fYN(y) = Nfy(y)(l- f fx(x)dx\ . (2.42)

To see whether this probability distribution leads to some definite limiting distribution when the number N of variables tends to infinity, first one has to search for the most probable value of Yff [E. J. Gumbel (1958)]. This is achieved equating to 0 the first derivative of fyN(y)- This leads to the equation :

dfyjy) dy

/»oo

/ fx(x)dx = (N-l)fY(y) Jy

which admits a negative solution for the unimodal distributions fy- Let y* denotes this most probable value. Then one considers the scaled variable Wjv = (YN — y*)/-Bj\r. For a unique value of BN, one finds that (2.42) converges towards a limit distribution when N —> oo.

Three universality classes are known. The first one is when fx decreases faster than any power law for x —> — oo. A generic example from this class is the distribution :

with the positive parameters b and /3. The leading term of the most probable value of y is given by :

y* = (ln(iV)/6)1/'3

and for the new variable :

YN + wNyb)1"3

WN ln(JV)i//3-i/6i//3/3


the probability distribution of the random variable Wjv converges to the limit Gumbel law [J. Galambos (1987)] :

foumbeib") = ev (2.43)

whose characteristic function is T(l — ik). In particular, the average value of m for this distribution is :

(w) Gumbel = —C

where C = 0.577215... is the Euler constant [M. Abramowitz & I. A. Stegun (1964)]. Plot of this distribution is shown in figure 2.8.

fc^M

Fig. 2.8 Plot of the Gumbel distribution (2.43). The abscissa w = ln(ln(2)) = —0.3665... corresponds to the median. The negative—w tail is exponential while the positive—w tail decreases as the super-exponential : ~ exp(— exp(w)).

In the case when the distribution fx is bounded for negative values of the argument, the same kind of reasoning yields the Weibull distribution as the limit law :

fweibuii(w) = cwc le w

= 0

for w > 0

for w < 0 (2.44)


with the parameter c larger than 1. More precisely, suppose that the distribution fx{%) is exactly 0 for

x < XQ, and fx{x) ~ (a; — XQ)C~1 near the threshold. Then defining the variable :

YN -x0 WN (c/iV)Vc

one finds the limit distribution (2.44) when the number N tends to infinity. Moments of the Weibull distribution are all given by :

{wq)we%bull =T(q/c+l)

when the corresponding value if finite. The figure 2.9 shows typical Weibull distributions for a few values of the positive exponent c.

f (w)

Fig. 2.9 Plot of the Weibull distribution (2.44) for the values : c = 1 (the exponential distribution), c = 2 (the ), and c = 4. The abscissa m = ln x / c 2 corresponds to the median. The most probable value is at m = (1 — l/c)1 '*2.

2.3.4 The arc-sine law

Other combinations or more complicated expressions have been studied in the general framework of the probability theory, independently of any

Two Examples of Physical Applications 39

possible physical applications. One of it deserves to be mentioned as it could be related to some physical processes with a threshold.

Let Xj are independent random variables taking values +1 or —1 with equal probability 1/2. Consider now the variable MN :

M e(*i) + e(x1 + x2) + ... + e(x1 + ... + xN) MN =

where 0 is the step function equal to 1, if its argument is non-negative, and 0 otherwise. The variable MJV is just the normalized number of times the partial sums X\ + ... + Xj take non-negative values. Clearly, M^ takes values between 0 and 1. Levy (1948) demonstrated that MJV has the limit probability distribution, called the arc-sine law :

2 fArcSine(m) = - arcsin(v

/m) 7T

when N tends to infinity. This beautiful result is related to a more general problem of the convergence of experimental histograms to their proper probability distribution. More on this point important for applications can be found in the work of Gnedenko and Khintchin (1960).

These examples of non-additive independent variables give rise to rather complicated derivations. It is then not surprising that even less is known about such arrangement for correlated variables. We shall not discuss them any farther in this book.

2.4 Two examples of physical applications

2.4.1 The Holtsmark problem

To illustrate what has been presented up to now, we expound now two physical problems where the stable distributions arise naturally. The first very general problem arises when dealing with the force fluctuations. This problem has been called the Holtsmark problem [B. B. Mandelbrot (1982)], since Holtsmark studied it in the framework of atomic spectroscopy [J. Holtsmark (1919)]. We keep here this name even if this problem can be stated much more generally for any long range forces acting in a homogeneous medium. Chandrasekhar (1943) detailed its solution for the gravitational attraction of a star in a stellar cloud. It is also linked to the classical Olber's paradox. Here we will introduce this problem for the Coulomb forces, but it will be clear that the reasoning is not specific to this interaction.


Imagine a homogeneous plasma of identical classical charged particles contained in a sphere. The Coulomb force between any couple of particles at a distance f is au/r2, with u = f/r. The real number a is the coupling constant depending on the electric charges. The question is to compute fluctuations of the electric force exerted on a test charged particle which is placed at the center O of the sphere.

Let us divide the space surrounding the origin O in a large number of thin pencils of angular opening dQ. In each such pencil, the repulsive force exerted on the test particle by the particles inside of it will be in the direction of the pencil and of the magnitude : a/r2 + a/r2, + ..., with i~iir2i • • • the respective distances to the test particle. These distances are distributed according to : r2drdQ. This is the same for the opposite pencil, but forces will then act in the opposite direction. We can then rewrite the projection of the repulsive force on the pencil direction using standard notation X1+X2-I , with all random variables X, identically distributed. The common distribution is f{x) oc l/\x\^/2 since x oc 1/r2, and the sign of the Xj is random according to the proper direction of the pencil. Because the positive rj are uncorrelated (the homogeneity assumption), the force distribution must be the symmetric stable law PM)0 of index /x = 3/2, when the number of interacting particles in the system tends to oo. One deduces in particular that the average value of the force is finite (it should be 0 by the Gauss's theorem), but its variance is infinite, therefore the force fluctuations are quite large. The result holds for a mixture of positive and negative charges.

More generally, one can make a comment about the occurrence of these infinite fluctuations. Let us consider the two-point force a/rd'il depending only on the distance r in a d—dimensional space. The coefficient a has a constant modulus and is negative (respectively positive) with the probability (1 — /3)/(l + j3) (respectively 2/3/(1 + /3)). As above, this force can be rewritten as a random variable X distributed as : l / |x | '1 + 1 . If the interacting points are homogeneously distributed throughout a spherical volume, the total force acting on a central test particle would have the stable distribution Pfj,,p, at least when the value of /i is smaller than 2. One should recall that the value /x = 2 is a very particular one, limiting the domain of attraction of the normal law. In other words, if the exponent d/fj, of the force is smaller than d/2, the limit distribution will be the Gaussian law. This does not occur for gravitational or Coulomb forces since 2 is larger than 3/2, but if the forces decrease slower than l/rd / / 2 , then the


self-averaging implies a dramatic decrease of the fluctuations in the space distribution and we arrive at the mean-field limit.

2.4.2 The stretched-exponential relaxation

A second example of application of the stable distributions involves the time relaxation of strongly disordered complex systems such as glasses and polymeric systems. Let us begin by the Debye theory of relaxation [P. Debye (1929)] of a simple system with just one time mode r . In this case, any physical quantity, say v(t), is expected to relax exponentially to its equilibrium value. Let us be more precise : suppose that the system is set slightly out of its thermodynamic equilibrium at the initial time v(0) = VQ and the asymptotic equilibrium is characterized by t;(oo) = 0. Then one should get : v(t) = uoexp(—i/r), as the result of the assumption that the instantaneous decay is proportional to the net difference of its actual to its at-equilibrium value, i.e. : dv/dt = — V/T. This is in particular the case for the dielectric relaxation : the function v(t) describes then the temporal decay of polarization of a dielectric sample after a steady polarizing electric field has suddenly been removed [H. Frohlich (1949)]. The dielectric constant e(u/) is a simple function of the Fourier transform of v(t) :

«M-«OO= r j f f i , - ^ (2.45) Cst — Coo Jo "&

with est the static value, and e^ its high-frequency limit. In the cases of strongly disordered systems, a large number of various

independent relaxation times is expected to occur. By the way, all modes relax exponentially to their asymptotic values and the global relaxation law in disordered systems should be written quite generally as :

/»oo

v(t) =v0 e(T)e-^TdT (2.46) Jo

where 9(T) is the distribution of the relaxation times r . The decrease of v is then simply the Laplace transform of the distribution 0. In such complex materials, the relaxation typically has the stretched exponential form :

v(t)~voexp(-(t/T0)>*)


with a positive exponent n smaller than 1 [K. L. Ngai (1979); C. T. Moyni-han & P. G. Gupta (1978)]. This decrease occurs often experimentally on many time decades [G. Williams k, D. C. Watts (1970)]. Moreover an experimental conclusion, which is not yet fully understood, states that fi = 1/3 is a limit value for this exponent for most of these systems [I. A. Campbell et al. (1988)], If the theory of a simultaneous independent relaxation of the modes is correct, one sees that according to relation (2.20) the behaviour of v{t) expressed as the Laplace transform (2.46), is identical to the Laplace transform of the asymmetric stable distributions P ^ i . Hence, the relaxation time distribution should be related to this stable distribution :

ni \ ^0 p / Tp

T* cos

This shows that the stable asymmetric distribution of the modes should occur in disordered systems, but this does not explain why this is so.

For the particular case of the dielectric relaxation, the dielectric constant is usually written in terms of the normalized dielectric parameter ep :

e(w) - Coo _ , , s • a t \ ^st too

with e'D and e , the real and (opposite of the) imaginary parts of the normalized dielectric function. Comparing to the definition (2.17), one sees that the equation (2.45) yields [E. W. Montroll k J. T. Bendler (1984)] :

£D( W ) = 7rg0WT0i^,o(wTo) (2-47)

making a connexion between the dielectric loss and the stable symmetric distribution of index \i. Figure 2.10 shows a fit of such dielectric loss for the polyvinylacetate polymer with the formula (2.47) for fi — 0.56. This fit is found to be correct over five frequency decades.


log(co/rad.s') Fig. 2.10 Experimental primary normalized dielectric loss d'D as a function of the decimal logarithm of the frequency w, for the polyvinylacetate at T = 62.5°C (after [Y. Ishida & K. Yamafugi (1961)]). The continuous curve represents : nvouiToPp,o(wTo), according to the equation (2.47) with the parameters fj, = 0.56, vo = 19.58 and TQ = 5 x 1 0 - 3 s.

Chapter 3

Stable Laws for Correlated Variables

The case of the macroscopic variable MM which is the sum of a large number of correlated microscopic variables Xj plays a crucial role in Physics. Unfortunately, no general theory exists in this case which would permit to characterize the stable law in the problems like phase transitions. Consequently, our goals should be more modest than in the previous chapter. In fact, several well developed mathematical tools exist to characterize the proper normalization of the random variables leading to an existence of limiting distributions. On the other hand, the exact shape of these distributions is generally difficult to find rigorously. One can say, that for weakly correlated variables the Central Limit Theorem gives rise to the universality of the limit law which is independent of the details of the local variables, while for strongly correlated variables this universality exists in a weaker form of the universality of some scaling and its characteristic exponents, but not of the whole limit law.

To achieve this study, we have to introduce the notion of covariance as the fundamental correlation between different random variables. This needs explicitly that the index j labelling random variables Xj is associated to some metrics. Hence, in the following, this index j will be considered as the location in an appropriate metric space whose dimensionality is equal to the dimensionality of the index. This is the general case and the underlying metrics is not difficult to find in any particular application *.

*For example, this could be an ordinary space for the renormalization group or the time for random tossing of a coin.

45

46 Stable Laws for Correlated Variables

3.1 Weakly and strongly correlated random variables

The previous chapter was devoted to the discussion of an asymptotic stability of the distribution of the macroscopic global variable which is given by the sum of uncorrelated variables. In the present chapter, we are going to investigate what happens to the distribution of the global variable if the underlying microscopic variables are correlated? Are these correlations going to change the asymptotic stability of Mjv?

Let us start with a simpler case : Xj are random variables of the same finite average (X) and the same finite variance (X2) — (X)2. If there are no correlations between random variables, we expect that the sum MN has asymptotically the normal law distribution with the average N{X) and the variance N((X2) — (X)2). But now, let us consider that the variables Xj are correlated, and more precisely that the covariance function :

Ci = {XjXj+i} - {Xj){Xj+l}

is non-vanishing and is a function of the distance I only (the stationary process). This is a common case in Physics when the number N of microscopic variables tends to infinity. The variance of the global variable MN is :

N

(M2N)-{MN)2 = N({X2)-(X)2) + 2J2(N-1)C1 . (3.1)

i=i

In the following, we will search for the condition which determines that the sum appearing on the right-hand side of equation (3.1) provides a dominant term when N —> oo.

Suppose that the correlation function decreases as a power law :

Q ~ i /r

with a positive exponent a. If a > 1, then the sum is of order N and the variance of M^ remains proportional to N. The correlations decrease too fast to be relevant. In this case, the variables are said to be weakly correlated *. On the contrary, if a < 1 then the variance of MN behaves

*Note that this reasoning holds for random variables Xj attached to a one-dimensional index. For variables Xjli..,jd in a d—dimensional space, the condition for irrelevance of correlations writes : a > d . For example, this could be the case for the problem of spins in a d—dimensional lattice. We now come back to the one-dimensional case for simplicity, but reader should be aware that some constraints involving covariance behaviour may depend explicitly on the space-dimensionality.

Weakly and Strongly Correlated Random Variables 47

like N2~a. Consequently, fluctuations of this variable are enhanced by the correlations and the variables are said to be strongly correlated. In this case, the resulting asymptotic distribution can be changed, as we shall see in the following section for the correlated Gaussian random processes. This is the same for the marginal case a = 1 which leads to :

(M2N) - (MN)2 ~ JVln(iV) .

This argument gives a precise prescription to decide whether variables are weakly or strongly correlated. In the latter case, when the correlations are relevant, the limit probability distribution is still to be determined. This task has not been solved in the general case. An important step towards finding solution of this problem was achieved in the case of a correlated random Gaussian process. This process will be briefly described in the following section.

3.1.1 Correlated random Gaussian processes

The case of correlated variables X, is commonly quite complicate and, hence, the limit theorems in this case are not numerous. Nevertheless, an excellent starting point to attack this problem is provided by the observation that, generally speaking, j-point correlation functions ((Xi — (X\))---(Xj — (Xj))) decrease quickly with j . Hence, one can expect that the average value and the covariance functions carry most important informations about the random process. This is the reason why one introduces the following Gaussian process as the first approximation.

A stochastic process is Gaussian if the probability distribution of random variables X\,..., -Xjv is iV-dimensional Gaussian distribution :

N

/x i xN(xi,. • •,xN) JJdxi x = l

= Prob[a;i < X\ < x\ + dx\, • • •, x^ < X^ < £;v + dx^\ N N

= 5/vexp « 2 J til amm'(xm ~ {Xm)){xm> - (Xm>)) 2

TTI=1 m ' = l

(3.2)

with the real normalization factor B^. The matrix A = (ammi)mtm< is symmetric positive definite. This is a natural generalization of the Gaussian


distribution for the correlated variables. A basic feature of the matrix A is that the covariance of Xj —variables

is given by the inverse matrix * A~l :

({Xm-{Xm)){Xm,-((Xm,))) = (A-1)rntm, . (3.3)

In this case A is diagonal. It is easy to see that the probability distribution of such a process depends only on the average value and the covariance of various random variables and not on more complicated higher order moments. This should be clear with the following property.

Nice features of the Gaussian process concerns the average correlation functions of Xj—variables. A fundamental relation is the connection between the j—point correlation function and the covariance function :

{(X1-((X1))...{Xj-(Xj))) = 0

for odd j and :

(({X1-((X1))--(Xj-(Xj)))

= E Y[{^m-{{xm)){xml-{{xml))) J2 n ( i _ 1 w»' pairing pairs pairing pairs

for even j . In the above expression, the sum runs over all the different ways in which one can divide the set of j points into j/2 pairs.

The Gaussian process has also remarkable stability properties. If X\,...,Xff is a Gaussian process, then any linear combination of these random variables :

N

Yi = J2c^Xi > l = l,...,N . 7 = 1

is also a Gaussian process. This property is true for any linear transformation of the Gaussian process, such as e.g. the integral transform of a continuous Gaussian process.

"This is the counterpart of the uncorrelated case for which the inverse of the coefficient in the exponential normal law is just the variance of the random variable.

Weakly and Strongly Correlated Random Variables 49

In particular, we shall see in the following section that the Central Limit Theorem holds for the variable :

MN = Y/Xj/N

and the Gaussian process Xj with correlations :

(XjXj+l)-{Xj)(Xj+l)~i/l<*

provided that a > 1.

3.1.2 Taqqu's reduction theorem

Let us suppose that the microscopic variables Xj with a one-dimensional index j represent a Gaussian random process, i.e. we assume that the two-point correlation functions define entirely the correlations. In addition, the covariance function of Xj is supposed to decrease as a power law at large distances : Ci ~ l/la. Let us write now MM more generally as :

with / a regular real-valued function, such that the variance of f{Xj) is finite. As usual, the coefficients BJV will be defined later since they depend on the conditions both on the function / and on the value of a.

The function / is developed as a series of Hermite polynomials * according to :

oo

fix) = Y, CjHiW/jl .

Taqqu introduced an index m*, and called it the Hermite rank of the function f. This is the index of lowest rank for which the coefficient Cj is not vanishing.

The main general result [M. S. Taqqu (1975b)] is the following :

'These are the polynomials [M. Abramowitz & I. A. Stegun (1964)] : Ho(x) = 1, H\{x) = x, H2(x) = x 2 - 1, . . . , Hj+i(x) = xHj(x) - jHj-i(x).


If a < 1/m*, then the limit distribution for MN exists and, moreover, this limit probability distribution is the same as the distribution of the modified random variable :

U BN

with

BN = Nl~am'l2 .

In addition, the non-normal limit distributions are explicitly known as the Wiener-Ito multiple integrals [R. L. Dobrushin & P. Major (1979)]. The Gaussian case is recovered when m* = 1.

A simple parameter proving that the limiting process is not Gaussian is the third standardized cumulant :

(M3N}-3{Mjf)(MN)+2(MN)3

((M*N) - (Mw)2)3/2 • W

For the Gaussian variable, this parameter vanishes. Davison and Cox (1989) showed how to compute efficiently the limiting value of this cumulant in the cases investigated by Taqqu. This allows to sort the cases into the 'non-Gaussian distributions' (for which the third cumulant does not vanish) and the 'possibly-Gaussian distributions' (for which cumulant (3.4) is zero). Of course, in the case of vanishing third order cumulant, the higher order cumulants should be studied as well to prove that the distribution is indeed the Gaussian distribution. In particular, the limiting distributions are expected to be Gaussian when a > 1/m* [P. Breuer & P. Major (1983)].

3.1.3 Rosenblatt's model

Simple examples are useful to understand different possible ways of investigating these difficult problems. One of them has been given in 60-th by Rosenblatt (1961) and remained unique until the preceding theorem was proved, fifteen years later.

Suppose that the covariance function for a correlated Gaussian process Yj decreases as :

Q = l / (1 + Z2)a/2 , 0 < a < l / 2

Dyson's Hierarchical Model 51

and the variables Yj have the mean value equal to 0. With these variables, one can build the new random variables by the non-linear transform :

X, = Yf

which correspond to a non-Gaussian process. The sequence

MN = 2^ Nl-a j=0

has a limiting probability distribution defined explicitly by its characteristic function :

(f>(k) = exp 3=2 J

where the coefficients Cj for j > 2 are known as the multiple integrals :

-16 d,Xi(\xi - X2||o;2 - I31 •»iir

and the Xj—integrations are over the intervals [0,1]. This characteristic function does not coincide with any stable distribution known for uncorrec ted variables since it is analytical near the origin without being either of the Gaussian form or of the Cauchy form *. This historical example is now an illustration of the Taqqu's reduction theorem for the case when m* = 2 and a < 1/m*.

3.2 Dyson's hierarchical model

Correlations between random variables in Physics are generally given by a Hamiltonian ri rather than by explicit probability distributions P or covari-ance functions C;. On the other hand, according to the Boltzmann-Gibbs statistical mechanics, P and ~H are related at the equilibrium according to the formula :

ri/T -Soc -\nP

where S is the entropy at a temperature T.

*One should keep in mind the remark in the bottom of Section 2.2.5


Let us consider a finite system AJV to which is attached some macroscopic random variable M\N. This system Ajv is the sum of two disjoint parts Ajv and AJV», with the mutual interaction given by a Hamiltonian /H{ANII AJV") which depends on the actual states of the subsystems AJV<

and AN" . The random variable MAN is itself a normalized average of the corresponding variables for the subsystems : M\N — {MAN, + M\N„)/B, with B > 0. The probability distribution of M\N attached to the whole system is then given by :

/AJV(TOAT) = a-N dmNldmN»exp(-(3'H(AN>, AN"))

x fAN,{mN>)fAN„(mNI,)5 (mN N' B

N" j (3.5)

with (Zj\r a positive normalization factor. In this equation, m^r and mjv" represent the respective values taken by the random variables M\N, and MAN„. We recognize here that, except for the statistical correlation factor exp[—/3%(A;v, A J V ) ] , which comes from the explicit interaction between the subsystems A'N and Ajy, the form of the equation (3.5) is quite similar to the equation (2.10) for independent random variables. Equation (3.5) is the basis for the renormalization group theory which will be discussed in the following section in more details. Here we shall focus on a simpler though non trivial example of the hierarchical Dyson model.

This particular model has been introduced by Dyson (1969) as an intermediate step to study rigorously the one-dimensional Ising model with ferromagnetic long-ranged interactions decreasing as 1/r2. Following the works of Anderson and Yuval (1969), this model was known to be relevant for an important physical problem occurring in magnetism of materials with magnetic impurities. This particular example of magnetic interactions is also important in itself, since it is at the borderline between the two well-defined behaviours. For one-dimensional interactions ~ l/rd with d < 2, the phase transition is not possible, generalizing the classical result of Ising. However for d > 2, a second-order critical phenomenon occurs at a finite temperature [D. Ruelle (1968)]. Thouless (1969) suggested that the case d = 2 behaves singularly. Below, we shall follow more closely the analysis of Bleher and Sinai (1975), as summarized by Cassandro and Jona-Lasinio (1978), which concentrates more on a problem of the stability of the distributions than on the proper structure of the critical behaviour.

In the Dyson model, the interactions are introduced in a hierarchical

Dyson's Hierarchical Model 53

way. At the level p, the system is of size N = 2P as the union of two subsystems of sizes TV' = TV" = 2 P _ 1 . The random variable MM is defined as the arithmetic average of the corresponding random variables at the preceding level : MN = (Mjv + MNn)/2. Moreover, the two subsystems interact according to the Hamiltonian :

Hp = -c>>(mN'+2mN''Y

with the parameter 1 < c < 2. Within these rules, the equation (3.5) becomes :

fAN(mN) = aNexp((3c?mN) / / dmNldmN»

x fhN,{rnNI)fhN„(mNn)5 [mN ^——— j

Since the parameter c is larger than 1, the exponential factor diverges with increasing p and there is no finite solution of this equation. We have then to consider more sophisticated normalization of M^. Let us take the following variable :

YN = V>/2MN .

The preceding equation is now :

fAN(yN) = aNexp[/3(c/b)?y2N}

I /A- {-&) fK«- \—vr~] dm' If b < c, then for the same reason as above there is no positive solution. However if b > c, then the exponential factor tends to 1 as p —>• oo and we recover the same equation as the equation (2.10). The solution is a normal distribution in this case. The situation b = c is then the most interesting since with this particular normalization the limit distribution is given by the equation :

f(y) = Aexptfy2) J J dy'dy" f(y')f{y")5 (y - yf^Af^j • (3-6)

Bleher (1975) showed that this equation has always the normal distribution as the solution. However if *J2 < c < 2, then there exists a particular value B = Bcr which does not depend on the iteration p and for which another


solution exists whenever f3 > f3cr. This is the sign of a phase transition with pcr as the critical temperature, and the relevant limiting probability distribution is therefore non-Gaussian.

The large—y behaviour of the non-Gaussian limit distribution deserves particular attention since it will be discussed later in the general context of the critical phenomena. Let us define the distribution f(y) as the product of the Gaussian solution fdy) and of the non-Gaussian part <j>(y) [G. A. Baker Jr. (1972); G. A. Baker Jr & G. R. Golner (1973)]. Inserting in equation (3.6), one can show that <f> should satisfy the functional relation :

^ ) o c ^ ( ^ )

for any value of y. The solution of this equation behaves as :

<Ky)~exp(-a2/5 + 1) (3.7)

with the exponent :

which is always larger than 2. The relation (3.7) indicates that the limit distribution f(y) decreases faster than a Gaussian distribution for large arguments.

3.3 The renormalization group

A concept of the renormalization group is an important application of the preceding ideas upon correlated random variables in statistical physics [K. G. Wilson & J. Kogut (1974)]. It comes from the consideration of an ensemble of a large number of random variables coupled by interactions. In principle, the number of degrees of freedom is too large for the system to be studied directly. So, one tries to reduce its complexity and to extract by a series of variable transformations only the dominant behaviour, removing at the same time the irrelevant features. Several renormalization group approaches have been developed according to the specific physical problem under investigation. These comprise Gell-Mann - Low multiplicative renormalization group transformations [M. Gell-Mann & F. E. Low (1954)], Wilson transformations [K. J. Wilson & J. Kogut (1975); S. K. Ma (1973)] and others [F. J. Wegner (1974)]. Jona-Lasinio and his

The Renormalization Group 55

colleagues [G. Jona-Lasinio (1975); G. Benettin et al. (1976)] gave proofs of the mathematical equivalence of these approaches.

3.3.1 The renormalization group and the stability problem

If the coherence length £ can be properly defined *, then one can divide the space in subsystems of size £. Inside such coherent subsystem, one can suspect the ensemble of local random variables to behave like a single block random variable. These block random variables are only weakly coupled since they are spaced by distances larger than £. Hence, and after a suitable renormalization of the random variable and of the lengths, one deals with a system of new independent random variables.

More interesting is the situation when £ = oo. In this case, the system is correlated at any distance and we have to deal explicitly with all correlations. The stability problem can then be reformulated as follows : a block random variable with some stable distribution can be separated into the sum of several block random variables having the same distribution.

One can formulate this idea more rigorously following similar lines as in the preceding section and opting for Kadanoff presentation [L. P. Kadanoff (1966)]. Suppose that a d-dimensional, possibly infinite initial system is divided into blocks V^p\ Then each block V^ is divided itself into N smaller blocks, say Vj , where p is the index of the renormalization procedure and the ratio of the typical size lp/lp+i is set equal to some constant value p > 1. By the way, N ~ pd. The global random variable for a block V(p) is called M^ and is just a normalized average of the similar random variables defined for the subsystems :

BN

with a positive parameter 5JV yet to be determined. The at-equilibrium probability density for the variable M^ can then be written with the densities of Mj as :

Ax,vf" (m ) = / e _ / 3 W ( { y / }) I J dmjfyip+D (mj)6 I m - — ^ mj J

(3 .8)

•This could be the typical length appearing in the exponential decrease of the covariance function C; ~ exp(—I/O-


where the effective Hamiltonian %{{V^P '}) depends both on the actual states of the blocks at the level p +1 and on the interactions between them. The problem now is to find the value of BN for which the asymptotic solutions of equation (3.8) are the common distributions / . Equivalently, one seeks for the fixed-point solution of the transformation :

/*(*) = |nd m; e xpH^(W)]rK)* (™ - ^ E "

3.3.2 Scaling features

Let us now calculate the covariance at the level p. One obtains :

{M^M^X = Jmimi, IJdm,-/vo,)C rrij) .

After inserting the distribution fv(P) (equation (3.8)) and inverting the 3

sequence of integrations, one gets :

p+i

The notation j G V^ means that the summation holds over all subsystems of level p + 1 contained in the system V>-p'. We renormalize the lengths in such a way that the system at the level p is geometrically similar to the system at the level p+1. Within this change of length-scale, if one supposes that the distance I between systems i and i' is large, then the distance between a subsystem m belonging to the system i and a subsystem m1

belonging to the system i' should be approximately equal to pi. Then the preceding relation can be written more clearly as :

This scaling relation leads to non-trivial stable solution of the covariance only if the factor BN is chosen such that :

BN = Nl-a/2d


with

Ci ~ l/la

independently of the level p. In particular, the equation for the limit distribution function of the random variable writes in this case :

/*(m) = /"Udm,-e- ' J W«T O '» / *K)<5 [ m - N"^'1 ^ m , J . (3.9)

This corresponds to a fixed-point equation after a renormalization of both the lengths :

1(P) = jy-l/e^p+l) p_ 1 0 j

and the random variables :

M ( P ) = jV«/2dM(p+1> . (3.11)

Note however, that this equation requires an integration over an infinite number of variables which usually cannot be realized. Some appropriate additional mathematical tools are needed to overcome this basic difficulty.

3.3.3 e-expansion

Some known distributions correspond to particular asymptotic solutions of the equation (3.8). We shall see now one such a family. As discussed before, Gaussian processes are defined only by their covariance function. If one considers these covariance functions as the Fourier transform Ci on the integer one-dimensional lattice :

Ci = dk <j)(k) exp(2iirkl)

with the particular choice :

<6(fc) = |l-exp(2"rfc)|20(fc)

where 0 is the periodic function :


then the Gaussian distribution is completely determined. The particular factor |1 — exp(2i7rfc)|2 appears in <f>{k) to cancel the summation of the Fourier weight exp(2i7rfcZ) when passing from the distribution function to the covariance function. Such a structure will be seen later in Section 3.4.4. The important point here is that all physics is hidden in the function 7.

Denned as above, the distribution function fGaussian whose covariance function is Ci, is translationally invariant on the lattice. If, in addition, 7 is the homogeneous function of the coordinates :

7(Afc) = A3-a7(fc)

then C; ~ l/la. Under these assumptions, the stability equation (3.8) can be rewritten as a self-consistent equation with the unknown function 7 [Ya. G. Sinai (1976)].

Now the problem is to define the linear stability of this solution, i.e. to investigate whether the recurrence process (3.8) for the distribution fGaussian which is perturbed by a small amount leads again to /Gaussian or to another, possibly non-Gaussian, solution. The general scheme is known. Let us take :

J = J Gaussian * y*- ~r "*)

as a trial function for the stability equation (3.8), and let us rewrite this equation in a more compact fixed-point form :

f = nN(f) • (3.12)

Since h is expected to take small values, one can linearize the fixed-point equation (3.12) and expand the solution in terms of the eigenvectors of the corresponding linear equation. The general solution of this eigenvector problem is :

nN{fGaussianhk) = N ^ ^ h k .

The eigenfunctions hk are expressed as the Wick products. For symmetric functions, only the even values of k are present. The linear stability can be discussed now looking to the sign of 1 - ka/2 for different modes k. For example, one sees that the Gaussian solution k = 2 is linearly stable as long as a > 1 . We recover here the rule for the relevance of long-range correlations in a context of the Gaussian process, already discussed on page 46. If the mode k = 2 does not exist, i.e. the distribution does


not belong to the ensemble of Gaussian distributions, the mode k — 4 is unstable as soon as a < 1/2.

In fact, this is the general case. If a > 1/2, the limit distribution is unique and Gaussian, while non-Gaussian limit distributions are allowed if a < 1/2. In the latter case, the rigorous answer to this problem must then be discussed with the complete fixed-point equation, including the Hamiltonian of the system, and the full limit distribution for a < 1/2 depends indeed on the details of the physical interactions. Nevertheless, the first approximation in this case is :

/ ~ /Gaussian X (1 + e/l4)

with e = 1 — 2a. The recurrence equation (3.8) allows to expand self-consistently the limit solution in powers of the parameter e.

These results can be generalized straightforwardly to the d-dimensional space. The structure of Gaussian distribution is quite similar and one finds that the Gaussian solution is stable whenever a > d. The limit distribution is Gaussian if a > d/2. Non-Gaussian solution of the fixed-point equation (3.8) may exist if a < d/2. In addition, some general partially proved results are known which define more precisely the domain of attraction. For example, if d > 4 then the short-range interactions lead to the Gaussian limit distributions with the covariance exponent a = d—2. This means that the correlations are irrelevant for those space-dimensionalities as soon as a > d—2. The scheme developed above for a general d, can be written here as a method to find the limit stable distributions as expansions in terms of powers of e = 4 — d.

3.3.4 Multiplicative structure of the renormalization group

As previously, one defines 7t^r as the transformation changing the parameters of one system into the parameters of the same system with renormalized lengths and random variables. The lengths are changed according to :

l^N1/dl .

We have the composition law :

Ti-Nxii-Ni — T^NiNi.


to which the terminology of 'group' refers. When close to the fixed point, the equation :

/ = nN{f)

can be linearized and one defines in this way the linearized part TV^ of the whole transformation "R-N- This leads formally to the relation between these commutative linear operators :

£(Kn)£(Kn) = £(K»k _ ( 3 1 3 )

As explained in the previous section, linear stability analysis requires study of the eigenvalues of these operators. Equation (3.13) means that for any eigenvalue Xj (N) of it^ , one should get :

Xj(N1)Xj(N2) = Xj(N1N2) . (3.14)

This shows the multiplicative structure associated to the renormalization-group TIN- This can be also interpreted in terms of a functional equation for correlation functions [M. Cassandro & G. Jona-Lasinio (1978)]. Equation (3.14) implies in particular that all eigenvalues are power laws of the system size :

Xj{N) ~ N"'

and the corresponding eigenvectors are called the scaling fields [S. K. Ma (1973)]. These fields are relevant or irrelevant according to the positive or negative sign of their exponent Uj, respectively. If the renormalization procedure is performed many times, the weight of the irrelevant scaling fields vanishes as an inverse power law of the system size, independently of their initial values. On the contrary, if the initial relevant scaling field is different from 0, its weight will diverge exponentially with the order of the renormalization and the fixed-point is unstable. This is an illustration of the symmetry breaking during the second-order phase transition. To be able to observe criticality in such a case, one has to impose that the initial relevant field is identically zero. This gives one criticality condition per relevant field. Generally, the number of such fields is small. For simplest magnetic systems, the two scaling fields are the field conjugated to the order parameter and the distance to the critical temperature.

Self-Similar Probability Distributions 61

3.4 Self-similar probability distributions

The basic property of distributions arising from the renormalization argument is the scale-invariance under the space dilation. This idea can be stated as the starting point as well, and allows to study these kind of random processes in an elegant way. This is the concept of fractal. Most of these ideas were developed in a well developed mathematical framework long before an introduction of fractals [A. N. Kolmogorov (1940)], but it was Mandelbrot (1977, 1982) who emphasized this notion in many scientific areas, contributing effectively to a diffusion of this concept.

3.4.1 Self-similar processes

Let us consider a random process Xj depending on the one-dimensional index j . Xj is said to be a self-similar process if the probability distributions of X\j and of XHXj coincide for all values of A for some common value of the parameter H. The exponent H, which appears to be characteristic of the process even though it does not fix the exact form of the probability distribution, is called the Hurst exponent. All moments of the distribution of Xj can be estimated straightforwardly because of the scale invariance. For example the covariance (XXJX\J>) is equal to \2H (XjXj>), what implies Ci ~ l/l2H after taking j = 0, j ' = I and A = 1/Z as a particular case. Strong correlations correspond to H < 1/2, according to the discussions on pages 46 and 58. Of course, these definition and properties can be extended to processes depending on a d-dimensional variable. In this case, one should define d different Hurst exponents. Such processes are then called self-affine processes and the coordinate corresponding to the smaller value of the Hurst exponent becomes the dominant one asymptotically.

Let us take an example. In Section 3.3.1 about the renormalization group, the random variable M^ was associated to the length l&h The stability equation (3.9) with the renormalization rules (3.10) and (3.11) for /(p) and M^ respectively, can then be reinterpreted as the asymptotic equality of the probability distributions of M(Xl) and of Aa/2M(Z), where A = Nl/d. So, this random process is self-similar, at least asymptotically when p -¥ oo, with the Hurst exponent H = a/2. We recover in particular that the covariance is : Q ~ l/la.

The limit non-Gaussian distributions arising in the Taqqu's theorem (Section 3.1.2) are another examples of self-similar processes.


3.4.2 Euler theorem

The self-similarity implies strong constraints on different moments of the distribution. One of the simplest, though leading to numerous non-trivial conclusions in Physics, is the Euler theorem.

If Xj is a self-similar process with a Hurst exponent H, then :

as a direct consequence of the self-similarity. This identity should hold for any value of A as soon as the moments of order q are defined. This is a particular case of the homogeneity property which can be written for a general function / of a variable x as :

f(Xx) = Xaf(x) .

If / satisfies this identity, then / is said to be the homogeneous function with the homogeneity index a. Differentiating once this function with respect to x and taking A = 1, one finds the identity :

"dT = af{x)

which constitutes the Euler theorem. This identity can be extended to the case where the function / depends on d variables and is homogeneous following :

f(\a*x1,...,\a«xd) = \a°f(x1,...,xd) . (3.15)

This is the case when / is a moment of random self-affine process. Therefore exponents ai,...,ad are possibly different. Moreover, if / is a homogeneous function then any derivative of / is a homogeneous function as well. Homogeneous functions are closely related to appearance of algebraic singularities near the origin.

When equation (3.15) holds, then Euler theorem states that :

^2ctjXj — (xi,---,xd) = aQf(xi,...,xd) .

This theorem plays a central role in some fundamental theorems in Physics, such as the Virial Theorem between average kinetic energy and average potential energy, or the behaviour of thermodynamic functions near a second-


order transition point, as exemplified by the Widom's hypothesis for the thermodynamic potential, introduced in equation (3.21) below.

3.4.3 Self-similarity of fractals in the renormalization group approach

Fractals are objects invariant by any change of length scale. Formally, one can introduce them in a spirit similar to the renormalization group, as both are based on the same idea. This has been discovered by Suzuki (1983) at the very beginning of the introduction of the fractal concept by Mandelbrot.

Let us consider a geometric fractal figure at a resolution scale a\. This means that, typically, fine structures of lengths smaller than a\ are not seen and the resulting object T\ is an approximant of the total object at this resolution. When the fractal is looked at a smaller resolution scale a^ = ai/b, then finer details of the morphology of the figure appear. This defines a new object T2 which can be written as the image by some transformation % of the first approximant T\. In this way one defines a sequence of more and more detailed objects :

Tj+i = TbFj . (3.16)

The fractal object is just the limit of this sequence when j —» 00, i.e. when the resolution becomes infinite. With the words of the renormalization group, the limit fractal T* is the solution of the fixed-point equation :

T* =thT* . (3.17)

Following the renormalization group approach, one considers the sequence of transformations (3.16). However, after each transformation % one modifies the length scale by the factor b in order to be able to superpose the fractal approximant Tj onto a part of the next approximant Fj+i. This renormalization of the length scale is itself a geometrical transformation which will be denoted by St,. Let us consider now any quantity Q{J-) related to a metric, as for example average moment of the volume of a part of the fractal (this could be some length or some surface as well), then the fractal dimension is the exponent Df defined by the relation :

Q(SbTj+1) = bD'Q(Fj)

which should be valid for j large enough. The relation (3.17) yields for the


limit fractal (j = oo) :

Q(SbF*)=bD'Q(F*) . (3.18)

This is the basic self-similarity law for fractals and this is also a direct consequence of the existence of the fixed-point in equation (3.17).

A general set of fractals plays a central role in the theory. They are called mono-fractals, and are characterized by the same value of £>/ for any definition of a geometrical quantity in terms of moments. For example, the volume of a part of a random object can be defined as (yq)1/q for any value of q, V being the ordinary volume and the average is taken for different parts with the same characteristics. Under these conditions, the relation (3.18) can be interpreted as the self-similarity of the distribution function of Q, namely the equality of the distribution of Q for the object SbT* and of bDfQ for the object T*. The fractal dimension is then analogous to the Hurst exponent defined for the self-similar process.

3.4.4 The power spectral density function

A practical equivalent specification of the two-point correlation function of a given random process is the power spectral density function, which is just the Fourier transform p(k) of the covariance function Q . In particular, the strong correlations in the random process correspond to a power spectrum with a singularity at the origin, while p(k = 0) is finite when long-range correlations are irrelevant. Another useful feature is that when the process is self-similar with the Hurst exponent H, then the power spectrum behaves as the power law :

p{k)~k2H-1 .

This constitutes an easy way to compute or measure the Hurst exponent. The search for the stable distributions by the self-similarity constraint

is far from being achieved. However, one important result is the following. Consider a random one-dimensional Gaussian process. The self-similarity of this process depends only on the form of its covariance function. A theorem of Sinai (1976) states that :

Random one-dimensional Gaussian process is self-similar if and only if


its spectral density writes :

oo 1

p(fc)=A|l-exp(2nrfc) |2 £ « = —OO ' '

and f/iis corresponds to the decrease :

Ci ~ l/Z"

0/ i/&e covariance function. The exponent a is constrained in between 0 and 1. The complete equiv

alent classes for the d-dimensional space (d > 1) are not known but one should note that we quoted one of them in Section 3.3.3, as it occurs naturally in the linear stability analysis of the fixed-point equation.

3.4.5 A-scaling framework

It has recently been argued that some particular scaling of the probability distribution of the random order parameter should hold at the second-order critical point [R. Botet &; M. Ploszajczak (2000)]. This A-scaling writes :

(X)*fx(x) = *{zA) , *A = ^ T (3 '1 9)

with a real exponent A contained in between 0 and 1. The A-scaling is defined as the asymptotic behaviour of {X)Afx(x) when both x, and (X) tend to infinity but value of the scaling variable z& is fixed. The mean value (X) plays here a role of a scale parameter. A typical application of the A-scaling is for finite systems. In this case, (X) varies as a function of the system size and a comparison of the distributions for different sizes can be used as a check of this scaling relation. We have seen already such a form in Section 2.2.8 about the stable laws. We recall that the value A = 1/2 corresponded to the case of the normal law.

In the case of the second-order critical phenomena, it can be shown that this scaling (3.19) holds too for the value A = 1, providing some possible test of criticality for finite systems of different sizes. Moreover, if XL is the global order parameter corresponding to the d-dimensional system of size L, then the average of this random variable scales at the transition as :

(XL) ~ Lgd .


The parameter g is called the anomalous exponent and its value is between 1/2 and 1. Writing the A-scaling for variable X\L and for variable XHXL,

one finds that the random process corresponding to the A-scaling is self-similar with the Hurst exponent [R. Botet &: M. Ploszajczak (2001)] :

H = gA .

It is interesting to note that the reverse is also true. Suppose that Xz, is a self-similar random process with Hurst exponent H. Then, for any value of A, the variables X\L and XHXL have the same probability distributions. This means in particular that :

(XXL) = *"{XL) •

Putting A = 1/L, one gets :

(XL)~LH .

Now the probability density of \HXL is given by the renormalized probability density / of XL as : X~H f{xi,). Choosing A such that A = LA, one finds simply the A-scaling (3.19). Of course, the value of A is not fixed by this procedure and only the general form of the scaling is constrained. The Hurst exponent gives then the value of the anomalous exponent. These ideas will be discussed in more details in the following sections.

3.5 Critical systems

As suggested in the Section 3.3 about the renormalization group approach, the thermodynamic system at a second-order critical point is a typical example of a large set of strongly correlated random variables. The limit distribution of the order parameter or of any critical observable is then expected to depend on the appropriate correlation functions, hence on some precise features of the physical interactions. Nevertheless, a line of thought can help to get some informations about these distributions and their stability. This is the concept of the scale-invariance, or the scaling theory [M. E. Fisher (1973)].

Critical Systems 67

3.5.1 Anomalous dimension

The finite-size scaling analysis allows to link the finite-system behaviour to its thermodynamic limit, that is when the system size N is infinite (i.e. the total number of sites is infinite). Near the critical point, the coherence length £ can be defined, for example as the typical length on which the decay of the covariance function for the order parameter is significant. But we know that at the critical point, £ must be infinite - since one cannot define any physical length scale in the system and, hence, the covariance function should decrease like a power law of the distance. One puts first : £ ~ l/ev, with the positive critical exponent v. The quantity £d defines a coherence volume in the d-dimensional Euclidean space. Let us denote by M^ the order parameter of the system. Because of the infinite-ranged collective interactions, this quantity behaves singularly with e near the critical point. The fundamental finite-size scaling close to e = 0 for the same quantity in a system of size N is :

MN(e)^e>3f(N/f) .

Since N is finite, there is no collective effect involving an infinite number of sites and the quantity Mjv is not singular at the transition. In order to remove the singular behaviour with e, the only consistent choice for the function / close to the origin is a power law with the exponent —f3/vd . This leads to a power law for MJV vs the size of the system :

MN~\/Nf)/'"1 .

The anomalous exponent g for the order parameter is introduced as :

In the case of the second-order critical phenomenon, one obtains in this way :

g = 1 - (3/vd

and this exponent is always smaller than 1. The same kind of arguments can be been extended to any non-singular

quantity. For example, if c depends smoothly on e near the critical point at the thermodynamic limit N = co, then this quantity cannot depend on the size N of the system when e = 0 and the system is finite.


3.5.2 First scaling

Let us consider now some thermodynamic system of interacting microscopic random variable Xp. Index r*is a d-dimensional vector related to a site location in the Euclidean space. The standard terminology of "thermodynamic system", means that the system is large, it is at the thermodynamic equilibrium, and the variables Xp are correlated. Suppose, in addition, that the temperature, or more generally the driving parameter, is chosen such that the system is close to the second-order criticality. This domain is characterized by the non-analytic behaviour of the asymptotic order parameter M^, defined as the limit for N —¥ oo, of the finite-size observable averaged over a finite volume VN of size N : MN = J2rzvN Xj/N. The distance of the driving parameter to its critical value is denoted by e and this parameter is assumed to be small. At the critical point e = 0, the actual state of the system is strongly fluctuating and this induces the large fluctuations of the order parameter. The goal is to characterize the probability distribution of the random variable Mjv-

Let N be a fixed (large) size of such a system. The state of the system is generally characterized by the thermodynamic potential per site :

G = F-hMN .

In this expression, F is the free energy per site and h is the field conjugate to the order parameter. The free energy is related to the free-field Hamiltonian for the system and to its entropy. The probability distribution of a variable Mjv for a temperature T is :

1 / NG(e,h)\

with the proper normalization by the partition function Z. This leads directly to the relation between the moments of the variable Mjv and the thermodynamic potential :

WM)>4(^)7M*. (3,0, where q is integer.

The fundamental similarity-law assumption states that for such a second-order critical system, the singular part of the thermodynamic potential G(e, h) should scale near the critical point as a homogeneous function

Critical Systems 69

[B. Widom (1965); F. J. Wegner (1976)] :

G(Ae, A2-Q-"/i) = A2-aG(e, h) (3.21)

for any positive A. We consider here systems with two relevant scaling fields, as it is generally the case, so the two positive critical exponents a and (3 characterize completely the critical class. Within this assumption, the q-th. moment (3.20) of the variable Mjv is found to scale as :

(MN(Xe, \2-a-?h)) = X^(MN(e,h))

and this relation can be extended also to the case where q takes the non-integer values. Such a scaling has deep consequences. For instance, taking h = 0 and A = 1/e one finds that all normalized moments (M^)/(Mjv)9

are independent of e near the zero-field transition. Following the finite-size scaling analysis, these moments cannot depend on the size N of the system when it is finite . Then, the characteristic function of the probability distribution P( e 0)( r n) is :

/ • O O

(3.22)

Hence, the characteristic function does not depend of k, N and e separately but only on the reduced variable : fc(M/v(e,0)). Taking an inverse Fourier transform of the function <j>, one obtains the following relation :

771

\MN)

near the critical point. In contrast to (Mff), the scaling function $ does not depend on the size N or on the driving parameter e. The law (3.23), which has been called the first scaling law [R. Botet et al. (1997)], is characteristic of large fluctuations :

( M ^ ) - ( M J V ) 2 ~ (Mjv)2 .

The scaling limit in relation (3.23) is defined by the asymptotic behaviour of (Mjv)P(»n) when both m and (Mjv) tend to infinity, but the value of the scaling variable z\ = m/{Mn) is fixed. {MM) plays a role of a scale

J — oo

oo

£< q=0

P(«

. (-•q

i,0)(m)e %

-ik{MN(e

<?!

kmdm

. o )»«


parameter, replacing N in many phenomenological applications. The normalization conditions :

/ $(zi)dzi = 1 , / z^z-Cjdz-i. = 1 (3.24) Jo Jo

imply that moments of $(zi) are independent of (Mjy). In equation (3.23), the function $ is the inverse Fourier transform of the

function <f>, which is usually not exactly known except for very simple cases. Nevertheless, it has been demonstrated that the tail of the distribution function for the large values of the order parameter is related to the critical exponent 5 characterizing the singular behaviour of the order parameter with the small field [R. Botet & M. Ploszajczak (2000)] :

(Af0O)~A1/* .

The general form in the limit z\ —> oo and m —> oo is :

$(zi) ~ exp(-czf+ 1) ~ exp(-c 'm 5 + 1 ) . (3.25)

c and c' are some positive constants. It is important to note here a particular feature of the large-zi tail of

the scaling function $, since we shall mention this feature many times in next sections. The basic thermodynamics implies that the exponent S has to verify the Buckingham-Gunton inequality [M.E. Fisher (1969)] :

2-7? < ^ - 1

d - 5 + 1

for any space-dimensionality d. The exponent 77 characterizes the divergence of the generalized isothermal susceptibility and is always smaller than 2. One deduces then that the value of 6 must be larger than 1 and the scaling function $ decreases faster than a Gaussian law *.

3.5.3 Second scaling

Let us suppose that an investigated system is not at the critical point or that one studies a random variable MM whose average value is not singular

*The magnitude of critical exponent 5 is such that : 8 + 1 = 4 in mean-field theory of the second-order phase transitions, 8 + 1 ~ 6 in the three-dimensional Ising model, and <$ + 1 = 16 in the two-dimensional Ising model.

Critical Systems 71

at the critical point. One can then write the free energy per site as a regular function of the value m of this variable :

F ~ a ( m - m * ) 2 .

The thermodynamic equilibrium condition dF/dm = 0, means that m* is the most probable value of M^. Or equivalently, since one has :

F oc - l n ( P )

therefore the probability distribution of MM is a Gaussian function and the average value of M;v is proportional to m*. The distribution P = PMN can be written under the scaling form :

V ^ > P M * ( m ) = * ( z 1 / 2 ) , z 1 / 2 = m ^ ^ ) - (3.26)

with $ a Gaussian function. This form is called the second scaling law [R. Botet et al. (1997)] and is characterized by small fluctuations :

(MN) - (MN)2 ~ (MN) .

3.5.4 A-scaling

One may ask what will happen if the observable quantity is not the order parameter but some N—dependent function of the order parameter M^ :

MN = Nai - Na2MN (3.27)

with

ax > g + a2 - 1 . (3.28)

The condition (3.28) assures that the order parameter does not characterize a leading term of Mjv-behaviour, and for large N :

(m) ~ i V a i .

Writing (3.23) with Mj\r instead of MJV and taking into account that :

PMJV (m)dm = P ^ (m)dm

one finds the A-scaling law :

( m ) A P ^ ( m ) ^ $ ( 2 A ) , A ^ g + " 2 - l < 1 ( 3 2 9 )


with the A-scaling variable :

rh — rh* ZA =

(rh) ™ \ A

The A-scaling is defined as the asymptotic behaviour of (rh) APMN (rh) when both rh, rh* and (rh) tend to infinity but the scaling variable ZA is fixed. The mean value (rh) plays a role of the scale parameter. The normalization of the probability distribution Pj^ (rh) and the definition of the average value of rh provide two constraints :

/

oo $ ( Z A ) ^ A = 1

- ( m ) i - ^ (3.30)

rOO

lim /

oo z&$(z&)dz& = 0

• < m ) 1 - A

which are consistent with A < 1, because the scaling function $ is positive. The scaling function 3>(ZA) defined in (3.29) has an identical form as $(zi) except for the inversion of abscissa axis. In particular, its tail for large ZA has the same form :

$(zA) ~ exp ( - c " 4 + 1 )

as given in (3.25). One should mention in passing that if

ai < g + a2 - 1

in (3.27), then :

(MN)~Na*(ri)

and A = 1. As an important example, one may see from equations (3.27) and (3.29),

that the A-scaling in the extensive variable* :

MN = N(l - MN)

can be used to determine the anomalous dimension since in this case : A = g. For this reason , N(l — M^) is a very useful variable in the

"One supposes here that 1 is the upper limit value for the order parameter variable Mp/.

Critical Systems 73

phenomenological studies. At the phase transition :

(N(1-MN)) ~ i V

but the finite-size corrections are algebraic.

3.5.5 Studies of criticality in finite systems

Several features of finite systems are important if one wants to study either the criticality of the corresponding infinite system or the distance to the critical point. One should name here :

• the A-scaling (this includes the first scaling law as well) • the form of the tail of the scaling function 3>(.ZA)

• the anomalous exponent.

All these features are closely related with properties of the scaling function which characterizes a finite system at the equilibrium. According to the preceding discussion (c.f. Sections 3.5.2 and 3.5.4), a logarithm of the scaling function $ ( Z A ) :

\n$(zA) = -f3Tf(azA,c) , fiT =-—

is related to the non-critical free energy f, in either ordered (c > 0) or disordered (c < 0) phase.

The discussion of A-scaling refers to the scaling of probability distributions as a function of the system size. Actually in many phenomenological application, the precise meaning of the system size is not necessary and one can use for this purpose the average value of the observable, i. e. the A-scaling of the normalized probability distribution P^r(m) of the variable m can be written as :

<m)AP (m)(m) = $(zA) , 0 < A < 1

(3.31)

ZA = (jn — m*)/(m)

where (m) and m* are the average and the most probable values of m respectively, and the system size N is assumed to be a monotonous single-valued function of (m). On the contrary, an exact meaning of the system size is


important if we want to find the relation between the scaling parameter A and the anomalous dimension.

In phenomenological applications, it is often difficult to get the probability distribution with a sufficient accuracy for the values of scaling variable which are far from the most probable value. It is then more judicious to work with moments of the distribution instead of the distribution itself. For example, when the system undergoes A-scaling, then properly normalized cumulant moments (Sections 6.1.2, 6.1.4, equations (6.3), (6.10)) :

are independent of the size of the system. An important consequence of (3.32) is that the generating function of

m—distribution :

M{u) = ^ e m u P ( m ) = F({m)Au)

is a function of the reduced variable (m)Au only. This provides a generalization of the remark, following equation (3.22) for the generating function in the case of the first scaling.

Chapter 4

Diffusion Problems

All notions of asymptotically stable distributions are present in some real or realistic situations in Physics. We describe below some of them in a context of the particle diffusion, to illustrate the general discussion contained in the preceding chapters. In order to emphasize consequences of general theorems cited there, the discussion in the following sections of this chapter is presented as close as possible in accordance with the presentation in the corresponding sections of Chapters 2 and 3.

Even if at the first sight, one can think of this kind of diffusion problems as somewhat specific, one should keep in mind that the fluctuating motion of a mode in a macroscopic dynamical system with large number of degrees of freedom is a particular realization of the diffusion problem.

4.1 Brownian motion

Direct applications of previous ideas connected to the Central Limit Theorem are the random walks. These are standard modelling of the Brownian motion of particles, as sequences of random rectilinear paths. We recall here three complementary theoretical approaches to the Brownian motion, before coming up to the precise discussion of the random walks.

4.1.1 Fick's representation

Let us consider a medium containing large number of Brownian particles, and n(f,t) is the particle density, i.e. the number of particles per unit volume. Fick's law gives the flux j*of particles crossing the unit surface per

75

76 Diffusion Problems

unit of time :

3=-VVn

with T> the diffusion constant which is a characteristic of the medium. This flux induces a change in the particle density according to :

^ = - V J = VAn (4.1)

which is the diffusion equation for n. Let us consider now the low-density limit of this problem. In this case, equation (4.1) is the result for non-interacting diffusing particles and we are just dealing with the motion of individual independent particles. Hence, the particle density is :

n{r,t) = Jn(r0,to)P((r,t)\(r0,t0))dr0 (4.2)

where P((r*, t)\(ro, to)) is the transition probability that the considered particle will be in r*at t, knowing that it was in ro at to- Since the equation (4.1) must be satisfied for the form of n(f, t) given in (4.2) with any initial condition n(fo,to), one finds that the probability distribution P should verify the diffusion equation :

^ P ( ( r , t ) l ( f o , t 0 ) ) = p A p ( ( r ^ ) | ( f o ! t o ) ) _ ( 4 < 3 )

The normalized probability distribution, which is the solution of equation (4.3) in the d-dimensional Euclidean space with the particular initial condition :

P((f , t) | (fo, to))=<5(f-rb)

is given by :

P(C.t)IW,M) = (4,p(( I ^ e x p ( - J ^ f ) ) • (4.4)

This is the Gaussian law for the probability distribution of the location of the Brownian particle. A consequence of (4.4) is the law for the displacement in time :

((f-f0)2)=2dV(t-t0) • (4.5)

Brovmian Motion 77

Note also that we have the Markovian relation :

P((f,t)l(n),to)) = yP((r,t)l(* !i,*i))P((* !i>ti)|(rt.,to))dr1 (4.6)

indicating that the history of the particle in the time interval [ti,t] is independent on what happened to it in the preceding interval [to,^i]-

4.1.2 Ornstein-Uhlenbeck representation

Another, more precise approach consists of following the motion of a particle of mass M subject to a random force R(t). We suppose in addition that R(t) is a white noise Gaussian process, i.e. (R(t)) = 0 and {R(t)R(t')) = (R )8{t — t'), with (R2) = est . The Langevin equation governing evolution of its velocity v is then [G. E. Uhlenbeck & L. S. Ornstein (1930)] :

dv -* M— = -Mjv + R(t) (4.7)

at with the positive constant friction coefficient 7. The general solution of this equation with the initial condition : v = VQ for t = to, is :

v(t) = v0 exp[- 7 ( i -t0)] + j dfexp[-7(* - 0 ] ^ • (4-8)

Within an assumption that R(t) is a random process with (R(t)) = 0, the integral appearing in the above equation corresponds to the fluctuating term, while the remaining term is the average value of the velocity for time t. In addition, since R(t) has been assumed to be a Gaussian process therefore v(t), which is a linear transformation of R (c.f. Section 3.1.1), is a Gaussian process as well. By the way, distribution of R(t) is entirely characterized by its covariance functions :

((v(t) - (v(t))) (v(f) - (v(t')})) = - ^ L e x p [ - 7 ( * ' - t)] 2M 7

(8* IM2

x (1 - exp[-27(t - *0)]) (4-9)

The same occurs for the position r, which is also a Gaussian process


with :

W 7 7 l M (4.10)

M V V 7 following equations (4.7) and (4.9). Then, defining the diffusion constant as :

m) - (m)f) = ^ (t -10 - i ( i - e-^-<°))

V (R2)

(2rfMV) one recovers equation (4.4) in a form :

1 - exp[-7(i - *o)] P((f,t) |(f0 , to)) = A-KV i - 1 0

x exp

7

( r - f o ) 2

-d/2

42)( t_ t 0 _lz«Bb2i*z*2}l) (4.11)

We have then explicitly two regimes according to the value of the time increment t — to as compared to the correlation time 1/7. For example, relation (4.10) for small times (t — i0 4CI/7) writes :

( ( f ( t ) - ( f ( t ) ) ) 2 ) ~ D 7 ( t - t o ) 2 •

The meaning of this short-time relation is that the Brownian particle undergoes some approximately rectilinear path at a finite average velocity \fD^f. This defines a typical length I = yVh and we can call it the mean free path. After travelling this length, diffusing particle looses memory of the initial conditions f*o, VQ at time to.

One can also analyze these two behaviours in terms of the Markovian condition. Instead of equation (5.21), one finds :

P((r,t'Wo,t0)) = |p((r , t ) | ( r i , t i ) )P((r '1 , t1) | ( fb, io))df '1

with the time t' given by the implicit equation :

7 ( t ' - t0) + exp[- 7 ( t ' - to)] = 7 ( t - t0) - 1 + exp[- 7 ( t - h)]

+ e x p [ - 7 ( t i - t 0 ) ] . (4.12)

Brownian Motion 79

In this sense, this is no more a Markovian process and there is a memory of initial conditions over the delay time I /7 . However if t-to > 1/7, then the Markovian condition (5.21) is well satisfied since in this case equation (4.12) gives again t' — t. In this model, the finite correlation time 1/7 separates the non-Markovian process (short times) from the Markovian process (long times). For long times, equation (4.4) is indeed recovered, but one should notice that the probability distribution is Gaussian at all times.

4.1.3 Fokker-Planck representation

As a last approach to the Brownian motion, we shall discuss briefly the Fokker-Planck equation [H. Risken (1984)], which is verified by the distribution function P(v,t) for a particle to get a velocity v at a time t *. We introduce the distribution function f(v,t) for a given configuration R(t), i.e. for a given particle, and the distribution P(v,t) is the average value of f(v,t) over a large ensemble of such random configurations R(t).

Conservation of the volume in the velocity-space ensures that :

dmt)+vJd4mt))=0 (4.13)

at v\dt

where V# is the gradient operator with respect to the velocity. Let us then consider the evolution equation (4.7) for the vectorial field v(t). Integrating equation (4.13) over a time interval St which is large as compared to the correlation time I / 7 but small as compared to the temporal variations of / , one gets :

ft+5t - (dv \ f(€,t + 8t)-mt) = -J Vfff-(ti)/(i;, ti)J dt! .

This equation can be solved recursively and the general solution is [R. Kubo et al. (1985)] :

rt+st -. /dv \ f(v, t + 6t) = f(v, t) - jf V* ( -^(h)f(v,*)J dh

*(vo,to) = (0,0) will be supposed implicitly to simplify the notation.


This solution yields the time-shifted distribution f(v,t + St) as a transform of the distribution f(v, t) by the linear operator. Replacing now dv/dt by equation (4.7) and averaging over different configurations R(t), one can write the Fokker-Planck equation for the distribution T?(v, t) :

%™-M«+% (v, t) = V* ( 7<J + £ a V* P(i7, t) (4.14)

where (4.9) and (R(t)R(t')) = (R2)5(t - if) have been used. The solution of this equation is the Gaussian distribution analog to (4.11). One recovers indeed a simple diffusion equation (4.3) when 7 = 0.

The main importance of this approach comes from the ability to work directly with the probability distribution of the variable. Randomness has yet been included in the equation through an averaging process prior to solving the equation. Such a derivation can be done as soon as the Langevin evolution equation :

dv/dt = ?(v,t;R(t))

holds for the diffusing quantity v with the short-correlated random noise R(t). If in addition the noise is a Gaussian process, then the corresponding Fokker-Planck equation involves only second order derivatives.

Sometimes, one has to deal with a nonlinear evolution equation instead of the Langevin equation, because the noise itself depends on the diffusing variable. In this case, two standard scenarios are known : the Stratonovich interpretation [R. Stratonovich (1966)] and the Ito-McKean interpretation [K. Ito & H. McKean Jr (1965)]. They lead to slightly different equations, though final solutions are essentially the same.

It should be noticed that these Fokker-Planck equations can also be derived starting from the master equation [R. Kubo et al. (1985)] which can be written quite generally as :

— (v,t) = - wziSP(v,t)du+ I wv-atsP(v- u,t)du . (4.15)

The first term on the right-hand side is the probability of the event : (v) + (u) —• (v + u). This loss term corresponds then to a disappearance of particles with the velocity v. The second term is the gain term and expresses an appearance of particles with the velocity v : (v — u) + (u) —¥ (v).

Random Walks 81

Developing P(v- u,t) as a linear operator acting on P(v,t) :

V{v-u,t) = e-^V{v,t)

one recovers the usual Fokker-Planck equation. This alternative scenario is less used since it requires knowledge of the full transition probabilities, what is rather unusual. Nevertheless, we shall discuss such an approach for some fragmentation models in Chapter 10.

4.2 Random walks

4.2.1 Gaussian random walks and Gaussian Levy flights

We showed in Section 4.1.2 that Brownian particles at a short-time scales move on roughly straight trajectory of typical length £ which is the mean-free path. We can then imagine modelling of this diffusion by a random walk in the Euclidean space. Particle trajectory in this case is a sequence of random segments of constant length £, each one passed in a constant time r . The randomness comes from the random directions of segments. Location of the particle after N steps £j is given by the sum : rjv = 2 j = i tj- All random vectors £j of common length £, are supposed to be independent one from another. This sort of statistical independence in this context is referred to as the random walk without memory.

For a notation convenience, we shall discuss here the one-dimensional case of the random walk. Note will be added when some additional behaviour occurs for higher space-dimensionalities. As a matter of fact, all features discussed below are known for a d-dimensional case as well but are just more complicate to write down.

Let us suppose that the diffusing particle starting at time t = 0 from the origin, undergoes successive independent jumps of equal length £ on a line. Its spatial coordinate after N jumps will be denoted by the random variable MNl. The "left" (respectively : "right") direction will denote a jump to the negative (respectively : positive) values of the coordinate. Left (respectively : right) jump occurs with a probability p_ (respectively : p+) at every time increment T. The normalization yields : p_ +p+ — 1, and the unbiased random walk corresponds to the symmetric case : p_ = p+ = 1/2. For a particle diffusing according to such a random walk, its location at time tN = NT will be : r = fh£, with integer m. The probability PJV(MJV = m)


to get the particle at the point fht at time t^, verifies the master equation which is formally equivalent to the equation (4.15) :

PN(MN = fh) =p_Pjv_i(Mjv-i = T O + 1 ) + P + P J V - I ( M J V - I = m - 1) .

Its solution is the binomial distribution. For an unbiased random walk, one obtains :

V^- = ^ = w{(N + m)/2) (4-16)

with \fh\ < N. The values of rh and N in (4.16) are of the same parity, otherwise :

PN(MN = m) = 0 .

Note that the average value of TO2 is equal to N. The leading behavior of the higher rank moments is :

(m2g) = (2q - l)\\Ng + 0(Nq-1) .

This result, for the moment of rank 2 can also be written as :

(r2) = 2Vt (4.17)

with r = fht, t = NT, and the diffusion coefficient V = £2/2r. This is the one-dimensional analogue of equation (4.5) or (4.10) if the correlation time is neglected. The asymptotic solution of equation (4.16) for large values of N is the Gaussian distribution :

whose moments of even order 2q are given exactly by :

{m2q) = (2q - l)\\Nq .

Derivation of equation (4.18) requires using Stirling approximation for the factorial terms in the equation (4.16)

Another way to derive this asymptotic formula is by using the Central Limit Theorem. In fact, the value of the random variable Mjv is given by the sum over iV jumps of the discrete values Xj — ±1 defined for each jump. Since the variables Xj are all bounded and : (Xj) = 0, {X2} - {Xj)2 = 1, therefore the Central Limit Theorem (see page 9) states that the random

Random Walks 83

variable MJV = $21=1 Xj/yN has the normal limit distribution, leading to (4.18) *.

The latter analysis can be generalized, since we referred only to a small part of the Central Limit Theorem. For example, one can imagine the length-distribution of the uncorrelated mean-free paths. More precisely, one takes the length i fixed and a distribution oiXjji in the form : f(x) oc 1/x^1, with an exponent /x > 2 which is the same for each independent variable Xj. This is called the Levy flight of index /i. Then, the conditions of the Central Limit Theorem hold implying that the random variable Mjv = £?=i Xj/y/N has again the Gaussian shape.

In addition, a simple argument can help to understand better the limiting value fi = 2. Since the variables Xj are statistically independent, therefore the average value of <(Dj=iXj)2) is equal to J2j=i(Xj)- Knowing the common probability distribution for all Xj : f(x) oc l/x^+1, the average squared diffusion distance is :

/

CO / flOO

x2f{x)dx I I f{x)dx (4.19)

where e is the smallest possible length of the jump. If e is positive, then the preceding equation gives a finite value for (($^.7=1 Xj)2) only for /i > 2, namely :

(4.20)

Notice the failure of this formula in the limiting case /x = 2. If the succession of jumps occur at regular times NT, the random walk obeys equation (4.17) with the diffusion coefficient :

M e2

V = 2( /x-2) r

We recover the standard diffusion coefficient T> = £2/2T in the limit fi —> 00. In all other cases (2 < fi < 00), the Central Limit Theorem tells that the

* Since the variable m takes only integer values of the same parity, two successive values of m are 2 units apart and the proper density distribution corresponding to (4.18) is just : PJV(MJV =rh)/2.


distribution of the diffusion distances remains the Gaussian function, but its variance depends explicitly on the value of n, as seen in equation (4.20).

4.2.2 St. Petersburg paradox

In the preceding section, values of jj. must range from 0 to oo to ensure that a function / is normalizable. We have studied the case fi > 2, but the remaining range of /i—values (0 < fi < 2) is also interesting. In this case, the second moments (4.19) are infinite. This seems at first look paradoxical because one does not expect the divergence after a finite number of steps for such a well-defined Levy flight. This kind of behaviour is sometimes called the St. Petersburg paradox. It has been introduced in the different context of a game martingale by Bernoulli in 1714 and edited in the Commentary of the St. Petersburg Academy. The Bernoulli's game goes as follows : you have to flip one coin until a head appears. When the head is there, you win some amount of money and you quit the game. The quantity of money given by the bank is G = 1 if head appears at the first flip, it is G = 2 if head appears at second flip (tail being at the first flip), more generally, it is G = 2J if head appears for the first time on the (J + l)-th flip. The probability to win at the (j + l)-th flip is 1/2J, so that the average gain is :

(G) = 1 x (1/2) + 2 x (1/4) + . . . + 2j x (1/2 J + 1) + . . . = oo .

But this result is obviously misleading since the median gain is only 1 *, and one has to flip the coin an infinity of times to get an infinite gain. In other words, one can say that very rare events correspond to very large gains and these improbable events contribute to a finite amount of the total gain.

It is interesting at this point to see the problem from another point of view. Let us write the Bernoulli problem as a sort of asymmetric random walk : a walker starts from the origin of a one-dimensional axis. With the probability 1/2 it stays at 0 and becomes inactive, i.e. walker is not allowed to move any more in the following steps, or with probability 1/2 it jumps to the site 1. This is the step 1. At the step j , let us suppose that the walker is active in site Xj. Then, it has the possibility to become inactive in Xj with probability 1/2, or to jump to the adjacent site Xj + 2j~1 to its right with the probability 1/2. For allowed jumps, the probability to get a

•The probability of the gain 1 is 1/2, so the probability to get gain larger than 1 is 1/2 too.

Random Walks 85

jump of length £ is then 1/21 In this sense, this problem corresponds to a particular Levy walk with \i = 1. Hence, the walker undergoes successive jumps on sites : 1, 3 , . . . , 2J — 1 . . . until it becomes inactive by chance. Similarly to the original Bernoulli problem, the average distance passed by the walker is :

1 x (1/4) + 3 x (1/8) + . . . + {2j - 1) x ( l / 2 ' + 1 ) + . . . = oo .

But it is more relevant to ask what is the average distance (.^ for a finite number JV of steps. In this case one finds that :

(£N) = N/2-l/2 + l/2N+1 .

Moreover, higher order moments (q > 1) of £jv behave quite differently, even though all are infinite at the limit N = oo since :

2(g-l)(iV+l)

(Or) - 2q _ 2 • One can write this result in a different form :

<^r) ^ 2 ^ 2 = ? < ^ ) ' _ 1 ioTq>1

<^-m (4.21)

The logarithm appearing in the last equation is a reminiscence of the marginal Cauchy case /x = 1. There is a limit distribution to this process and one can define it by its characteristic function :

M f c ) = e ^ g e x p | ^ (422)

This function (see figure 4.1) belongs to a class of functions studied by Weierstrass at the time when the principal quest of mathematicians was to know if the continuous function can be non-derivable. One can see immediately that the function <j>\ (k) in equation (4.22) exists for all values of k, is bounded (its modulus is smaller than 1), but all its derivatives at k = 0 are infinite since all moments of £N are infinite. This is one quirk amongst many other particularities for these kind of functions which gave the first deterministic examples of fractal curves.


1.0

0.5 ' ' ' ' ^ ' ' • --0.5 0.0 0.5 -0.5 0.0 0.5

Fig. 4.1 Plot of the real (on the left) and imaginary (on the right) parts of the characteristic function (4.22). The fractal irregularities are clear. For comparison, the characteristic function of the normal law (not shown on the figure) has the real part ~ exp(—k2) and the imaginary part is identically equal to 0.

4.2.3 Non-Gaussian Levy flights

Random walks for Levy flights exhibits features which are analogous to the St. Petersburg paradox. In this section, we shall discuss some consequences of this problem.

4.2.3.1 Anomalous diffusion

Let us consider the Levy flight of index / i a s a random walk for which the step-lengths are distributed according to the probability law : f(x) oc l/x^1, and the exponent \i is contained in between 0 and 2. As an illustration, figures 4.2 show random Levy flights for various exponents fi *. The definition (4.19) of the average squared diffusion distance, is no more

"The Levy's distributions of index fj, can be simulated using the following algorithm : let X be a continuous (pseudo-)random variable uniformly distributed over [0,1]. Then, the random variable Y = {X"1/^ — l)/a with a = 21/ f J - 1 is distributed according to SY{V) = a / i / ( l + oy) M + 1 , i.e. in the Levy form when the argument y is large enough. Note also that the distribution tends to the exponential function : fy (y) = ln(2)/2y

when fj. —> oo.

Random Walks 87

Fig. 4.2 Plot of the Levy flights with exponents fj. = 1/2, 3/2, and 5/2. The upper figures visualize flights between points, while in the lower figures only points belonging to the Levy walks are shown. Strictly speaking, only the latter figures represent t he Levy ensembles. Length scales in the figure are in the ratios 1000/2/1, respectively. This gives an idea of the radius of each ensemble. Note that the third figure (/i = 5/2) is statistically equivalent to a Gaussian random walk, the few long jumps being irrelevant in this case.

correct in these cases (the term on the right-hand side of equation (4.19) is infinite) and should be replaced by the anomalous diffusion law :

( ( ? * ' • ) ) * ^ • (423)

This can be guessed by the following argument. Let <j>x{k) and <J>MN (fc) are the characteristic functions of the Xj—distribution and MJV~distribution (MN = Ylj=i Xj)> respectively. Then from the convolution theorem (2.9) one has :

to*(*) = *£(*) •


Since Xj —distribution is normalizable, this relation can be written as :

4>aN(*) = ( l + J[exp(-ikx) - l]fx(x)dx\ (4.24)

even if all moments (Xq) are infinite for q > 0. For small values of k, the characteristic function is approximately :

<I>MN (*) - exp \N f[exp(-ikx) - l}fx(x)dx\ (4.25)

and according to the relation (2.8), can be formally put in the form of a series :

P Y / _ , . N f c {{M%) - (MN)2)k2

1! 2!

Choosing fx(x) ~ l/x^1 for large x, equation (4.25) becomes :

4>aN(k) ~ exp (-NkP /"[expC-iTi-1/") - l}e(ku1/fi)du\ . (4.26)

The function 6 in the the above expression is defined as :

fx{l/x) fi9(x)

X>J,+I

Since the distribution function fx is such that : fx{x) ~ l /xM + 1 , at least for the large x, the auxiliary function 0(x) has a finite value 6(0) at origin. Because of this property, one can replace the function 6(ku1^11) in (4.26) by the constant 0(0) when the value of k is small enough, and this leads to the important conclusion that <j>MN(k) is a function of the variable Nl/^k instead of being a function of N and k separately. One has in particular for the variance : (M&) - (MN)2 ~ N2/^ when 0 < \i < 2. When fi = 2 or li=l, the logarithmic terms appear in the variance.

To complete this discussion, one should notice that the moments (Xj) and (X2) for \i > 2 are finite and the leading term on the right-hand side of equation (4.24) is : (1 + i(Xj)k - (X2)k2/2)N, implying the Gaussian behaviour of M^ with the variance of MJV increasing proportionally to N.

Random Walks 89

4.2.3.2 Continuous Levy flights

The Levy flight in a random medium can alternatively be defined by the continuous Langevin equation :

-=F(r)+R(t) (4.27)

where F is generally the force field depending only on the location of the diffusing particle, and R(t) is the uncorrected random noise with distribution fii(R)dR = dR/\R\,l+l which should at least be true for large values of \R\. The main difference with the previous description is that now the time variable is continuous instead of being discrete. One can derive then the corresponding Fokker-Planck equation for the probability distribution ¥{r, t) to get a particle at a distance Fa t a time t if the particle was at the origin at t = 0 :

^ ^ - = -W[P{f)V{f,t)\+V^V{f,t) . (4.28)

Equation (4.28) is the analogue of equation (4.14) in this particular case. The non-local fractional operator VM is defined for any function / as :

^f& = -^j *'***&& and the ordinary Laplacian operator is recovered whenever /x = 2.

We focus hereafter on the one-dimensional free-space case, i.e. in absence of any force field. The multi-dimensional quenched random force has been treated by Fogedby (1993).

The solution of the Langevin equation is immediate when the force field vanishes, if one supposes that the diffusing particle starts from the origin at the time t = 0 :

r(t) = I R(t')dt' . Jo

Then, the probability distribution of a random variable r is given by P(r,t) = (<5(r — r(t))), with 5 the Dirac distribution which also can be written as :

1 r°° 6(x) = -J^Jikxdk . (4.29)


The distribution P is then:

P(r>t) = i - j°° /exp (-ik f R(t')dA\ eikr dk .

The quantity / 0 R(t')dt' corresponds to a continuous version of the sum of independent random numbers. The common probability distribution of variables R is : f(R) ~ l / | f l | ' J + 1 , and this law is known (c.f. Section 2.2.7) to belong to the domain of attraction of the stable laws of index /x, provided the sum is normalized by the factor t1^, as explained in (2.30) and (2.31). One deduces that the Fourier transform (exp(—ik J0 R(t')dt')) is just the characteristic function of the symmetric stable distribution of index /i :

'exp (-ik I R{t')dA \ = exp (-2>/1(|*|*1/'*)'1)

with some positive coefficient V^. The probability distribution P is then :

1 f°° P(r,t) ~ — / exp(ikr-Vll\k\llt)dk 2* J-oo

for large times t. In particular, the first scaling form appears :

t^P(r,t) ~ Ppfiir/t1'")

which is consistent with (2.32) and (2.33). Moreover, the mean squared distance is given by :

<r2) = 1 {V.tf"

making more explicit the anomalous super-diffusion law (4.23).

4.2.3.3 Return to the origin of the random walk

A simple example can help to understand the logic of this study on the asymptotically stable distributions. Let us consider a symmetric one-dimensional random walk with discrete jumps of length 1. This means that the walker starts from the origin 0 and at each integer time step it has a probability 1/2 to jump one site to the right and 1/2 to jump one site to the left. We record now the increasing sequence of times t0 — 0, ti,..., tj,... when the walker returns to the origin 0. The differences Xj = tj — tj-i form a sequence of even integer random variables. They are statistically independent since when returning to the origin 0, the walker has forgotten

Random Walks 91

all preceding steps and this is equivalent to starting a new process. The probability for a one-dimensional variable Xj to be equal to 2£ is given by :

V^ = = w{1)^ri • (4'30)

Note first that the characteristic function of the V\ distribution is known :

<l>xn(k) = 1 - \/l - exp(-ifc)

with y/l — exp(-ik) a series of odd powers of \/ik. One concludes that this characteristic function <j>Xj{k) is not analytical at fc = 0 and all positive order moments of the distribution V\ are then infinite.

Considering now the random variable (this is the scaled iV-th return time) :

N

MN = tN/N2 = YJXj/N2

3 = 1

one obtains in the limit of infinite N :

ln<pMN(k) -> -y/ik

which following relations (2.13) shows, that the limit distribution of Mjy is the stable law with (i = 1/2 and 0 = 1.

One can recover this result with a following more general argument. Applying the Stirling approximation to the formula (4.30), one obtains asymptotically (£ —> oo) :

P1(Xj=2£)~l/£3'2 . (4.31)

Using the argument leading to (4.26) and since the sum of all Xj for j between 1 and N is equal to i^, one derives for positive fc :

4>tN/N2(k) ~ exp (~\Vk f [exp(- iu- a) - l] 6(ku2/N2)di

When N —> oo, one can replace the function 6(ku2/N2) by its value at 0 (0(0) = 1/y/ir), and the integral / ^ o ( e x p ( - i u - 2 ) - l)du by -\/27r(l + i). Performing a Fourier transform one finds :

Vi{MN<nt/2) = —== I x-3/2exp(-l/2x)dx . (4.32) V27T Jo


Note the normalization factor N1^ = N2 for £/v, which is needed to get a non-trivial asymptotic distribution law.

This result can also be seen as a simple application of results discussed in Section 2.2.2. Since the probability density of Xj is a power law ~ l/a;'J+1

with the exponent /i = 1/2 (c.f. (4.31)), the normalized random variable tu/N2 has the limit probability distribution (2.22) Pi/2,i- This is the same result as (4.32).

This kind of probability problems is important in Physics, because the probability of the return to the origin is the measure of self-interaction of the diffusion process. Some exact results are known. For example, summing Vi(Xj = 21) (equation (4.30)) over all possible values of £ one gets :

oo

e=i

which means that the random walker will surely return to the origin. Polya (1921) proved that this is the same for a two-dimensional square lattice. But for a hypercubic lattice in the space of Euclidean dimensionality d > 2, the return probability is always smaller than 1 because of an incomplete filling of the space by such a random walk. In particular, for the usual case d = 3, the exact value is derived [C. Domb (1954); W. H. McCrea & F. J. Whipple (1940)] :

< 1 5 / 3 ( ; i ' 4 ' 4 ) = 1 ~ V6r[l/24]r[5/24]r[7/24]r[ll/24]

where T is the Euler Gamma function. As a consequence, the probability for a decent random walker to return at least once to the origin of the three-dimensional cubic lattice is only about 1/3.

4.2.4 Random walk in a random environment

Few examples of stable laws in the marginal case fi = 0 are known, even though they are forbidden in principle by the normalization condition. The following example exhibits such a case and in addition, is important for the dynamics of disordered media.

Let us consider a diffusing particle randomly walking in a random potential. Neglecting inertial effects, its movement should be driven by an

Random Walks 93

equation :

0 = -M-yv+ F(r)+R(t)

instead of the Langevin equation (4.27). The force field F is a function of the local potential U : F — —VU. This force is random, uncorrected, static (the quenched disorder) and depends only on the location r. After averaging over the spatial disorder, its mean value and variance are given by (F) — 0 and (F2) = a2S(f—f'). As usual, R corresponds to an uncorrelated thermal noise and does not depend on the location of the diffusing particle. Since there are two sources of disorder, therefore there are also two different physical problems : first one corresponds to the temporal evolution of a given system, and the second one is to ask what are the features at a given time but averaged over various independent random systems. Generally, these questions lead to different results. We shall exemplify here the one-dimensional case [J.-P. Bouchaud et al. (1990)].

Let us consider first the case of a fixed random spatial configuration. To move, the particle has to cross successive potential barriers by thermal activation. The time required to do this is, in the average, given by the Arrhenius formula :

—(€?) with ro the characteristic trial time. In the one-dimension space, the potential is such that : F = —dU/dx, and can be considered as a random walk. Its variance is (U2) = (F2)x *. Combining this result with the Arrhenius formula, suggests an anomalously slow diffusion behaviour :

x2(t)<x\n4t (4.33)

where x denotes time average of the distance for a given spatial disorder [K. Golosov (1983, 1984) ]. Moreover, the time-averaged variance of the distance x can also be derived. Surprisingly, this variance is finite, say x2, for the infinite time. This means that the all diffusing particles have almost the same history in this quenched environment. As a consequence, there is no limit distribution in the sense that the distribution probability

*This follows from the equation (4.9) applied to the parameters : t—tx,v-+U,R-* F , 7 - > 0 ,M = 1.


follows a static law, shifted by x(t). In other words, the front diffusion is a non-dispersive travelling wave.

Quite different is the case of the sample average, as studied in details by Sinai (1981). Since the random potential can be considered as a spatial random walk, one can ask what is the probability to reach a random barrier of magnitude U at a distance x. This probability is [W. Feller (1971)] :

and the time to cross it is :

t ~ T0 exp(U/2kBT) .

This simple argument leads essentially to the exact result, as derived by Kesten (1986) for the corresponding discrete-time model :

X0 \n2(t/T0)(P{x,t)) = fKesten ( ^ )

\XQ\O. (t/To)J

with the scaling function :

fKesten(z)=8-±^-e-^^^ (4.34)

in the variable : x

Z = T • x0\n (*/To)

This scaling function is shown in figure 4.3. fKesten{z) is not a stable distribution since the tail is exponential for large arguments. In particular, the anomalous sub-diffusion (4.33) is recovered in this case, independently of the strength of the quenched disorder [Ya. G. Sinai (1981)].

A following heuristic reasoning can capture the essential correspondence with the stable laws [M. V. Feigel'man k V. M. Vinokur (1988)]. One can define trapping times r in domains of width x as the typical times spent in a potential well of extension x before the particle is able to reach another domain by the thermal activation. The distribution of these times is such that fT(r)dr = fu{AU)dAU, with the potential variation AU distributed as a random walk on the distances x :

/[/(A(7) oc exp(-a(A[/) 2 /z) •

Random Walks 95

f (Z)

Fig. 4.3 Plot of the Kesten distribution (4.34). It has an essential singularity at z = 0, which is responsible for the flat peak and the exponential tails.

Since the thermal activation yields the relation :

T oc exp \2kBTj

one deduces that the T—distribution should be

fAr) exp(—a In2 T/X)

It seems that for large enough x, the distribution of trapping times is essentially ~ 1/T. This corresponds to a limiting case fj, = 0 of the general theory of stable distributions. A cut-off of the distribution fT(r) at Tcut ~ exp(v^) ensures the normalization of this distribution. Note however that contrary to stable distributions Pn,p, there is no such universality for /i = 0 and the proper form of this scaling function, which is found to be PQ in the case discussed above (c.f. equation (4.34)), depends in fact on the cut-off needed for its normalization.

All derivatives of the limit distribution PQ vanish in 0 [G. H. Hardy & J. E. Littlewood (1918)] and z = 0 is an essential singularity of this distribution. Moreover, its integer rank z—moments are all rational numbers :


(z) = 5/24, (z2) = 61/1440, etc. *. The corresponding biased problem, where a drift force (F) is present,

has also been discussed in details [J.-P. Bouchaud & A. Georges (1990)]. In this case, one recovers approximately the stable Levy distributions.

4.2.5 Sinai billiard

The case /x = 2 is marginal in the sense that it separates the Gaussian domain (/i > 2) from the domains of other stable laws. In addition, the logarithmic corrections are expected in this case. We give now an example of such realization.

In the classical billiard imagined by Sinai (1970), single point particles move in a two-dimensional plane filled in with a regular square array of fixed circular scatterers centered at the nodes of a square lattice. The point mass particles are reflected elastically when they collide with such disks and they move freely in between the disks. Since these disks are assumed to be rigorously fixed, their mass is considered as infinite and the velocity of the moving particles is conserved and can always be set equal to unity. The trajectories of the particles are sequences of rectilinear paths.

This kind of diffusion process is indeed fully deterministic and the proper trajectory of a particle depends only on its starting point and a starting velocity. Nevertheless, the movement is chaotic and nearby trajectories diverge exponentially in time. This means that, in spite of its very simple definition and deterministic character, any long-time prediction concerning a single trajectory is impossible and the system can only be studied from a statistical point of view.

Because of this chaoticity, trajectories of the particles are like random walks with some distribution of the step lengths. Here two cases should be discussed according to the value of the distance L between centers of neighboring disks, as compared to the disk diameter 2R.

When L < 2/2, then the disks are overlapping and every diffusing particle is trapped inside of a finite domain. In this case, diffusion at large distances is forbidden, i.e. diffusing particles see the closed horizon.

More interesting is the case when the distance L is larger than the di-

*More generally, the moments of the distribution (4.34) can be expressed by :

/-He \z\9~1Po(z)dz = 2«+iL{2q + 1 ) / T T 2 « + 1 , with the Dirichlet function L(q) =

Y/%0(-iy/(2j + l)".

Random Walks with Memory 97

ameter of the disks. Now, arbitrarily long straight trajectories appear and diffusing particles see the open horizon. Simple geometrical considerations allow to derive the solid angle (~ L(L - 2R)/£2) in which there is no obstacles between a wandering particle and a disk at a distance £. One concludes that the distribution of random lengths £ of the diffusion paths behaves as (L - 2R)/£3 for large I *. Taking now the chaos assumption as a serious ground, i.e. neglecting the complicated correlations which occur during such a diffusing process, one can apply the general results about the Levy flights (c.f. Section 2.2.2). This means that the asymptotic distribution of distances passed by a diffusing particle is the Gaussian function, but since the exponent \x of the path-lengths takes the marginal value 2, the diffusion is weakly anomalous [A. Zacherl et al. (1986)] : (£2) ~ tint. This result holds for other scatterer shapes than disks, provided they are convex.

4.3 Random walks with memory

4.3.1 Random walks with Gaussian memory

We go now to the correlated random variables in the diffusion processes. An important particular case of correlated variables Xj is when the probability distribution of the j-th. step depends only on the value of the variable at the step j — 1. To be more precise, let us write the conditional probability to get the value Xj = Xj if the variable Xj-\ took the value Xj-\ :

p(xj\xj-i) = (xj-ilAlxj) (4.35)

The matrix A is analogous to the evolution matrix. It is then implicitly assumed that the set of all possible values of the random variable Xj is finite and takes, say, I different values. Since

53 (XJ-I\A\XJ) = 1 *C j — 11 3

*One should mention that the simple argument reproduced above and leading to the dependence : f(£) ~ 1/t3, neglects a number of features. For example, the short paths are characterized by a distribution f(£) ~ l/£2 for square scatterers. This result is not universal in the sense that the short length paths distribution depends on the form of scatterers. For instance, it is f(£) ~ l / ^ 5 / 2 for disks as shown by Bouchaud and Le Doussal (1985). In addition, several complex features are evidently due to the regular structure of the lattice.


the matrix A is a norm-conserving probability matrix whose largest eigenvalue is equal to 1.

Let us consider now periodic boundary conditions for convenience. This means that the variable X/v is correlated to X\ by some relation as (4.35). This additional condition is expected to be irrelevant in the limit of large number of variables. One can then write the probability distribution of

N

P(MN=m)= £ {xi\A\x2)---{xN\A\x1)6\Yf-^-m\ (4.36) {XI,. . . ,XJV} \ i = l /

where the first sum runs over all possible configurations of the set of the N random variables Xj. Writing the Dirac (^-distribution as (4.29), the equation (4.36) becomes :

1 f°°

P(MN = m) = —J 5 ^ <zi|^(fc)|s2>-"<xjv|^(*0l*i>

(4.37) {x i , . ..,xN]

x exp(—ikm)dk

with the new matrix A{k) which is defined by :

p(xj\xj-i)exp(ikxj-i/VN) = {xj-i\A(k)\xj) .

The sum appearing in (4.37) is also equal to the trace of AN :

'£{x1\AN(k)\x1)

{*i}

and therefore :

1 f°° l

P(MJV = m) = — / V A f {k)e-ikmdk (4.38) 2 7 r J-<*>j=1

where Xj{k) are the eigenvalues of the matrix A(k). Knowing that the largest eigenvalue of A(0) is 1 and expanding the perturbative term : exp(ikxj-i/y/N) for small values of k/y/N, Schneider (1985) was able to prove that the largest eigenvalue of A is :

1 - ak2/N + 0( iV-3 /2) ~ exp{-ak2/N)


with a a positive coefficient which depends on the precise values taken by the p(xj\xj~i). The Gaussian structure of the probability distribution of MM follows after replacing the sum Ylj=i ^ W m equation (4.38) by the largest dominant term (ma,Xj{\j})N. However the variance of this random variable depends explicitly on the correlation rules p(xj\xj-i) even if it behaves proportionally to N.

This was a particular example of the Gaussian processes discussed in Section 3.1.1. The argument can be extended for finite range correlations as well. But when the correlations are long ranged, things become more complicated. In this case, the above derivation can be formally rewritten until one discusses the trace of A(k). Since the matrix is dense, the spacing between consecutive eigenvalues can be quite small and the sum Xw=i -\^ W should be replaced by an integral over j . The AT—dependence of the dominant k—term is then less obvious to derive as it depends strongly on the eigenvalue distribution. This can lead to non-Gaussian limit distributions.

4.3.2 Fractional Brownian motion

Long before fractals have been introduced in the scientific community, the concept of Brownian motion was extended by Mandelbrot and Wallis (1968) to take into account some additional long range temporal correlations. Formally, the one-dimensional temporal fractional Brownian motion rnit) is a single valued function such that its increments rji(t') — rn(t) have a Gaussian distribution with a variance :

(\rH(t')-rH(t)\2) oc \t'-t\2H . (4.39)

The value of the exponent H is in between 0 and 1. Averaging in (4.39) is considered over many independent samples. A discrete realization of this motion can be achieved considering the recurrent definition [B. B. Mandelbrot k J. W. Van Ness (1968)] :

M - l

XN+1 =XN+RN+Y, [U + 1 ) " - 1 / 2 " 3H-1/2} Rif-j (4-40) 3=1

where Rj are uncorrected random numbers having a Gaussian distribution with zero mean and a common finite variance (-R2). The parameter M stands here for the finite memory of the process. In principle, M should be chosen as infinite, but in practice, values of M larger than the needed


number N of steps, or alternatively of order of few thousands, are often sufficient to get correct features. Notice however, that pure fractional Brownian motions are quite difficult to simulate and specific algorithms, such as the mid-point displacement [M. F. Barnsley et al. (1988)] or wavelet synthesis [G. W. Wornell (1993)], have been studied to provide rapid and accurate numerical results.

When M = 0 or H = 1/2, one recovers indeed the standard random walk. On the contrary, H ^ 1/2 implies long range correlations of range M, which can be quantified by the covariance function :

{XN,XN) = i ( i?2) (N'2H + N2H - \N' - N\2H) .

All these features resemble the standard Brownian motion with the addition of long-ranged memory. One can be more precise about such an idea. From the definition (4.40) of the discrete fractional Brownian motion, one deduces that :

(|XJV-Xo|2) = X ) ^ - 1 { ^ _ , ) 3 = 1

M-N

+ £ ((N+j)H~1/2 - JH-1/2) <#.,) • 3=0

In particular, when Rj are all identically distributed, one recovers the fundamental relation (4.39) in the form :

(\XN-X0\2) ~ {R2)N2H . (4.41)

Still more generally, the relation (4.41) is true for the fractional Brownian motion in any space-dimensionality. One can show in this case that there exists an effective diffusion constant [J. R. Hunter et al. (1993)] VJJ, depending only on (R2) and on the dimensionality of the Euclidean space, for which the probability distribution of the fractional Brownian motion :

P((f,0|(r-o,*o)) = {4nmt'to)2H)d/2«P ( - 4 4 V - t ) 2 g )

is formally identical to the expression (4.4) after replacing the time t by the temporal variable t2H. Not surprisingly, it has been shown that this distribution is the solution of the effective Fokker-Planck equation [K. G.


Wang & C. W. Lung (1990)] :

^(f,t) = 2HV*Ht™-lVf Q ( £ 2 } V ? ) P(f, i) (4.42)

which generalizes equation (4.14) in the frictionless limit : 7 = 0. It is worth mentioning that one recovers the results quoted for the Levy flight of index fi, as described in Section 4.2.3, with the formal equivalence H = l/n, and the diffusion constants being related according to :

V*H = (2H)l/2HV1/H .

Even if these results look similar, the reason for the anomalous diffusion is different for the fractional Brownian motion and for the Levy flights. Whereas in the first case, this is the effect of long range correlations between steps, in the second case this is due to the power law form of the distribution of the length of the steps.

Some examples of two-dimensional fractional Brownian motion are shown in figure 4.4. Repulsive or attractive correlations are clearly seen depending on whether H is larger or smaller than 1/2.

^ H=3/4 H=l/2 H=l/4

Fig. 4.4 Examples of the two-dimensional fractional Brownian motions (N — 4096 steps) with Hurst exponents H = 3 /4 ,1/2 (the Brownian case), and 1/4. These plots have been obtained using equation (4.40) for both the x— and y—increments. The length scales are in ratio 10:1:1 for H =3 /4 , 1/2 and 1/4, respectively. Note that, even if self-crossings are allowed, the long-range repulsive correlations tend to stretch the trail in the H > 1/2 case. On the contrary, the trail collapses in the H < 1/2 case.


4.3.3 Flory's approach for linear polymers

Linear polymers consists of long flexible chains of molecules in a fluid solvent. The polymer of TV + 1 monomers could be seen as a special case of a random walk of N steps, the step length corresponding to the monomer extension. If there were no interactions between monomers, a model should be the pure random walk. This has been shown to be correct for large space dimensionalities (d > 4). However, in the ordinary three-dimensional space and for smaller space dimensionalities, the interactions cannot be neglected and they are not short ranged. Since the polymer is rather flexible, even monomers far apart in the polymeric chain can interact strongly. These interactions depend also on the quality of the solvent. Schematically, one defines three types of solvent : the "good solvent", the "0-point" and the "poor solvent". In good solvents (high temperature), there is short-range repulsive interaction between monomers and the most probable configuration is a swollen, elongated polymer. In poor solvents (low temperature), effective monomer interaction has an attractive tail and the equilibrium state corresponds to a compact coiled configuration. Both behaviours occur at different temperatures for a given solvent and the limit between them is at a definite temperature, called the #-point. At this temperature, repulsive and attractive parts of the effective interaction between monomers balance exactly, and the equilibrium configuration is expected to be intermediate between the swollen and the compact structure.

We shall discuss here the problem of polymers in a good solvent. Under this condition, the equilibrium state is defined by a balance between randomness of steps and volumic constraints since two monomers cannot occupy the same place at the same time. The physical origin of this exclusion comes from the preferential solvation of monomers by the solvent molecules rather than by other monomers. This volume exclusion constraint is the basic characteristic of the self-avoiding walk model. It is identical to the random walk model, except that walks cannot overlap at any time *. Successful, though approximate, analytical treatment in the dilute regime has been done by Flory (1953). It is based on the following argument : the probability distribution PSAW to get the end-to-end distance f for such a polymer with N monomers, is a product of the corresponding probability

*More precisely, in the continuous space the end point of each step is excluded from the spheres of radius a centered at the end points of all other steps. The value of a is usually taken equal to the mean length of the step.


PRW for the random walk and the probability Ps for the random walk to get no intersections over the N steps [L. Pietronero & L. Peliti (1985)] :

PsAwi?,N) = PRW{r,N) x P s ( r , N ) . (4.43)

The first probability distribution on the right-hand side is given by the equation (4.4) with t - t0 replaced by N. Moreover, all distributions have spherical symmetry, i.e. they depend only on | f\.

The probability of no self-intersection can be evaluated approximately as follows. The walk of end-to-end distance r extends over the average volume proportional to rd in the d-dimensional Euclidean space. So, the polymer fills in this volume with an average density equal to a,dN/rd, where aj. is a geometrical factor. One deduces that at each successive step, the walk has the probability 1 - a.dN/rd to avoid any of the N preceding steps. The probability for the random walk to avoid self-intersection after N steps is then :

P s ( f , 7 v ) ~ e x p ( - ^ ) (4.44)

with some prefactors which may depend on r. Putting this expression into (4.43), one finds the leading behaviour :

P W r , A 0 - x P ( - ^ - ^ ) . (4.45)

This implies the Gaussian tail for the probability distribution of the end-to-end distance. Nevertheless, its average value is anomalous. One can see this feature if one notice that the maximum of the probability (4.45) occurs for the value :

r* oc AT3/(2+d) ( 4 . 4 6 )

with the constant proportionality factor dajV/2. Developing PSAW as a Gaussian near its maximum, one can show that :

(rq) ~ JV3"/(2+d)

for any positive q and large N. This result should be compared to the corresponding moments for the random walk : (rq) ~ Nq/2. This means that for space dimensionalities d < 4, self-avoiding walks are stretched as compared to the random walk. Figure 4.5 shows an example of such self-avoiding walk.


RW

Fig. 4.5 Comparison between the two-dimensional random walk (on the left) and the two-dimensional self-avoiding walk (on the right). The step length is 1. Both walks are of length N = 52 in the continuous Euclidean plane and are plotted at the same length-scale. The self-avoiding walk is clearly stretched as compared to the pure random walk because of the excluded-volume constraint.

This stretching of the walk shape is the main consequence of the excluded-volume constraint. In other words, this constraint implies long-range interactions between the steps of the walk (between the monomers) and this results in a fundamental modification of both the limit probability law and behaviour of moments.

The formula (4.44) is indeed a poor approximation but, anyway, the final result is rather good. One can compare this formula to real linear polymers, to computer simulations of self-avoiding walks, and to exact formulation *. The exponent 3/(2 + d) which is characteristic of the moments of the end-to-end distance is exact for space dimensionalities d = 1, d = 2 [B. Nienhuis (1982)], d = 4, and it deviates by no more than 1% from the correct value for d — 3. One can note also how the probability (4.44) is crippling for the numerical simulations. If one wants to build a self-avoiding walk from

*de Gennes (1979) showed that the equilibrium statistics of linear polymers in a good solvent was equivalent to a critical ferromagnetic Heisenberg model with n-dimensional spins, in the limit n —> 0.


a collection of N—steps random walks, the probability to find one proper SAW is about exp(—N) *. This explains why some specific algorithms have been developed to study this problem numerically [K. Kremer & J. W. Lyklema (1985)].

The reason for such an excellent agreement with the Flory argument is not trivial. For example, Fisher and Essam (1961) proved rigorously for the self-avoiding walks on a regular lattice of dimensionality d > 1, that the generating function of PSAW '•

oo

M(f,u) = £ ^SAw(r, N)e~uN (4.47) iV=l

decreases exponentially with the end-to-end distance r ~ \r\. Making then the assumption that PSAW (F*> •W") has asymptotically the scaling form

PsAw(r,N)^(r)-df(^j (4.48)

with (r) ~ N", f(z) = fo(z) exp(—czs+1), /o a slowly varying function, and c a positive constant. The saddle-point method applied to the integrand of (4.47) leads to :

M(r, u) ~ exp ( - r y v ^ + W W + D + i ] )

One concludes that the two exponents v and 6 are linked by the simple relation :

5 = i//(l - v)

and the tail of the probability distribution P behaves as :

P ^ - e x p ^ - c r 1 ^ 1 - " ) ) . (4.49)

The assumptions done by Fisher and Essam seem to be very reasonable. Some of them, like (4.48), formally identical to the first scaling (3.23) (c.f. Section (3.5.2)), have been proved by the renormalization group technique [J. des Cloizeaux (1980)]. So, one can trust safely that the tail of the distribution function P S A W is essentially non-Gaussian, contrary to the mean-field assumption (4.45). Yet taking the Flory approximate values for the exponent v, i.e. v = 3/(2 + d) in accordance with the equation (4.46),

*For a walk of 100 steps, this probability should be about 10


one sees that this distribution function decreases faster than a Gaussian for large distances r when d = 2 or 3, for which one has respectively : 1/(1 — v)= 4 or 5/2. Not surprisingly, these results presented here for the linear polymer problem in good solvent, are consistent with the approach developed in Section 3.5.2 about the limiting distributions in the critical systems.

4.4 Random walk as a critical phenomenon

4.4.1 Criticality of the Brownian motion

For the space dimensionalities d > 2, the Brownian movement of one particle is a limit case of the second-order critical process . This can be seen on the example of the correlation function which is defined as the probability to find a diffusing particle at a distance r from the origin at least once during a time interval t, knowing that the particle was at r — 0 at a time t = 0. Integrating equation (4.4) with respect to the time, one finds :

0(r) = - L_ f°° u(d-4)/2e-udu

The function 9(r) is nothing else but the two-point correlation function for the Brownian motion. It behaves as a Gaussian function :

1 exp(-r2/4TH) » « - n{4nV)d/2-Hd/^ / (4-50)

for finite t and large values of r such that r/\f4Vt —¥ oo, and becomes an algebraic function :

0(r) ~ l/rd~2

for r/y/4Vt -4- 0, i.e. in the limit where t —> oo and r is finite. Defining the driving parameter as e = 1/t, the latter case is typical of the critical behaviour and can be discussed in terms of e.

At the point e = 0, one cannot define a correlation length and using the common notation for these critical phenomena, one obtains :

6(r) ~ r 2-d-

Random Walk as a Critical Phenomenon 107

with the critical exponent : rj = 0. When e > 0 is small and positive, the correlation length can be denned as :

£<xi/yi

following (4.50). Hence, this correlation length diverges with e with a critical exponent v = 1/2. In the same spirit, the susceptibility is defined as the integral of the function 6{r) over the whole Euclidean space. The susceptibility diverges as 1/e and this fixes its critical exponent : 7 = 1. In this way, all critical exponents may be defined along the same lines. One finds here the standard critical exponents for the mean-field second-order critical behaviour.

Nevertheless, one has to note that even if all critical behaviours can be properly defined, it is only a limit case in the sense that the "critical point" is at t = 00. This is somewhat analogous to the case of a one-dimensional Ising model, for which the "critical temperature" is exactly at 0.

4.4.2 Criticality of the Levy flight

There are random walks for which the Central Limit Theorem is not valid. This is in particular the case when the distribution of the independent step variables Xj is very broad : f(x) oc l/a^"1"1, with the exponent fj, smaller than 2. We have discussed such processes in Section 4.2.3 under the generic name : the Levy flights. For these particular kinds of random walks, some jumps are of considerable extension and the diffusion is anomalous in the sense that the relation (4.17) fails. Instead, one gets the superdiffusive behaviour (2//i > 1) (c.f. relation (4.23)) :

In addition, the limit probability law is a stable distribution P^Q(X) as defined by (2.17) and not a Gaussian distribution. This can be written as :

Prob

with some positive constant c. The same analysis as in Section 4.4.1 can be done here too, and leads in particular to a coherence length diverging as £ ~ N1/*1 for N —> 00. In terms of the critical phenomena, it means that the critical exponent v is equal to l//x, or in terms of fractals, that the

N

r <y^Xj <r + dr (, cr

3=1 JVV/* /J ,° VjVi/M

dr (4.51)


fractal dimension of the trail of the Levy flight is equal to /x [B. D. Hughes et al. (1981)] :

These particular results hold for any space dimensionalities. On the contrary, the value of the critical exponent rj depends on the space dimensionality. For example, integrating (4.51) over N steps one gets the two-point correlation function whose behaviour at N —> oo gives the value : t] = 2 — fj, in the one-dimensional case. Note that the critical exponents for a pure Brownian motion are recovered for the marginal value fi = 2 and not for the value fi = oo, as one could expect naively.

4.4.3 Criticality of the self-avoiding walk

The equivalence between the problem of the polymer in good solvent and the limit Heisenberg ferromagnetic model [P. G. de Gennes (1979)] allows to study directly the self-avoiding walk problem as a critical phenomenon. In this case, one finds analytically the precise result for the scaling function f(z) occurring in the probability distribution (4.48) :

f(z) ~ z ^ - 1 ) / " for z -» 0

(4.52)

f(z) ~ z(i-7+"*-*/2)/(i-.0 e x p ( _ z V ( i - ) ) for z ^ oo

with the positive critical exponent 7 = v(2 — rj), which is also related to the total number Af of self-avoiding walks of given size N. More precisely one puts :

Af~eNN'i-1 .

Q is the effective connectivity constant which, unlike critical exponents, is lattice dependent. For example, it is conjectured to be equal to y 2 + \/2 [I. G. Enting & A. J. Guttmann (1989)] for the hexagonal two-dimensional lattice while it is close to 2.64 for the square lattice [A. R. Conway k A. J. Guttmann (1996)]. Even if the limit behaviours of f(z), both for large and small values of z, have been found here qualitatively similar to the Levy stable laws, they are definitely not equal to any P^p laws since the respective exponents 1/(1 — v) and (7 — l)/ i / cannot fit the formulas 1/(1 - l//x) and (—(j, — 1) valid for the Levy laws.

Random Walk as a Self-Similar Process 109

In two-dimensions, one finds : / (z) ~ exp(—cz4) *, whereas in three-dimensions : / (z) ~ exp(—cz25). These shapes agree closer to the numerical results concerning the third standardized cumulant (3.4), which is known to deviate strongly from 0 for d = 2 and d = 3 self-avoiding walk numerical simulations [M. E. Fisher & B. J. Hiley (1961)]. For d = 4, one recovers exactly the Gaussian shape since v = 1/2 and 7 = 1. One recovers then the same results as for the random walk.

4.5 Random walk as a self-similar process

4.5.1 Self-similarity of the Brownian motion

The criticality of the Brownian motion can also be analyzed with the help of the fractal concept. For this approach, one has to imagine the trail of a Brownian particle in the Euclidean space. This trail is an object for which one has to study the average mass M(-R) of a part contained inside a sphere of radius R, centered in a point of the random trail. This mass is given by the integral of the two-point correlation function 8{r) (as defined in Section 4.4.1) over the sphere. For the infinite time (t —> 00), one gets : M(R) ~ R2, for all Euclidean dimensions d > 2. This is the fundamental scaling relation characterizing a fractal object of fractal dimension Df — 2. Such an object is known to have infinite correlation length and to be geometrically self-similar, what is a typical feature of critical phenomena. This approach is convenient to understand a role of the space dimensionality. For d < 2, the object is compact in the sense that Df = d, or equivalently, it fills the whole available space. These are also the cases where the walker probability to return to the origin is equal to 1, as discussed on page 92. More generally, these features are true for the trails of the Levy flights of index fi. The fractal dimension is then equal to Df = n for diffusion in the Euclidean space of dimension d> fi.

From the probabilistic definition of the self-similarity (as discussed in Section 3.4.1), the Brownian motion is self-similar because f\t has the same probability distribution as A1/,2r*t for all parameter A. This is a simple consequence of equation (4.4). In fact, writing the probability distribution

•More precisely : z5/8 exp(—cz4), but the exponential term gives the leading behaviour.


at a time t :

P(XHr,t){XHr)d-1d{XHr) = V{r,t)rd-ldr

and using equation (4.4), one obtains :

1 / (XHr)2 \ P(A*M) = {^vx^tynexp \-wWt) •

This distribution is identical to P(r, Xt), with the Hurst exponent equal to H = 1/2.

The same reasoning holds identically for the fractional Brownian motion, as denned in Section 4.3.2, and H is then the Hurst exponent following the general definition of the self-similar processes. This holds also for the Levy nights of index n, as studied in Section 4.2.3, with H = l//x.

4.5.2 Anomalous diffusion in the fractal space

Up to now, we discussed only the particle diffusion in the Euclidean space. One can generalize this process to a random motion on disordered fractal structures. One example is the random diffusion on the self-avoiding walk. This could be related to the statistical description of the evolution of the excitation which is free to move on a given polymer. Since the self-avoiding walk is topologically linear, the diffusion of the Brownian particle should be Gaussian in the curvilinear distance t along the chain :

where it has been supposed that each self-avoiding step is of unit length. The equation (4.49) writes then :

P M ) o c e x p ^ - c ( - ) J . (4.54)

Combining equations (4.53) and (4.54), one gets the probability distribution of the movement of the random walker :

/ • O O

P ( r , t ) = / F(r,HyP(e,t)dt Jo

( / r 2 x l / ( 2 - ^ oc exp I —c'

Random Walk as a Self-Similar Process 111

by the steepest-descent approximation. Using the Flory value for the exponent v, one obtains :

/ r (2d+4)/(2d+l)

P(r,t) oc exp \-d' f3/(2ct+1)

for the tail of the distribution as long as d < 4. When d = 1, one recovers indeed the Gaussian shape since the self-avoiding walk is nothing but a portion of a line in this case. But for any d > 1, the tail is anomalous and decreases always more slowly than the Gaussian.

It has been suggested [R. A. Guyer (1984); S. Havlin et al. (1985)] that the following scaling should occur in this problem of diffusion on disordered geometries :

with the dynamic exponent vw characterizing the anomalous diffusion, and the static exponent v characterizing fractality of the substrate *. A more precise derivation, using the prefactors in equations (4.53) and (4.54), allows to put (4.55) in this form [S. Havlin & and D. Ben-Avraham (1987)] with :

f(z) ocexp (-21+53 j

at large—z. Such a non-Gaussian behaviour with a characteristic exponent depending on the values of v and vw is expected to hold generally for diffusion on disordered fractal structures [S. Havlin & and D. Ben-Avraham (1987)]. Note that this scaling law (4.56) can be written also in the form :

(r)dP(r,t) = / ( r / ( r »

with

(r) ~ tvw .

This scenario seems clear for loopless fractal structures . It is not so simple when the loops are relevant and/or the fractal morphology is more complicated (e.g. in case of the multifractals), in which case, an infinity

*i/ is just the inverse of the fractal dimension of the structure on which the particle diffuses.

(4.55)


of exponents vw could be needed to describe the diffusion [C. Evertsz & J. W. Lyklema (1987)]. When this occurs, the scaling equation analogous to equation (4.56) is not known.

Chapter 5

Poisson-Transform Distributions

Commonly, one observes discrete distributions. This may result from an unavoidable discretization in small systems of the "true" continuous probability distribution which is the principal object of studies. Prom this point of view, an observed distribution is the distribution composed of the true probability distribution and the distribution responsible for the discretization. The basic assumption in this approach is that the transition from continuous distribution to its discrete realization is known a priori and hence the continuous distribution can be reconstructed experimentally from large number of measurements. It is the continuous distribution, extracted in this way, which is then analyzed using mathematical tools of the Central Limit Theorem for continuous variables and continuous distributions. Hence, the idea is to "prepare" the result of experiment in such a way that the asymptotic results of Mathematics for continuous variables can be applied.

The second approach does not aim at the separation of the effects of the discretization and at the determination of an asymptotic continuum distribution. In this approach one aims at best "experimental" determination of the discrete distribution from a broad ensemble of experimental events for different system sizes and then using the concept of the A-scaling, which describes how finite-size distributions collapse into a limit distribution, one tries to disclose the trace of the limit law in small system discrete distribution. In this approach, the A-scaling is a fundamental conceptual tool which allows to stay within the realm of discrete distributions and, at the same time, allows to investigate the observed phenomenon in relation to its continuum generalization.

113

114 Poisson- Transform Distributions

We shall illustrate the above mentioned problematic by discussing the phenomenology of both the particle production in ultrarelativistic collisions of leptons, hadrons and nuclei, and the fragment production in nucleus -nucleus collisions at intermediate energies. In both these domains, precise experimental data and relevant theoretical analysis are currently available. It is one aim of the present book to give an overview of most recent results and developments concerning the analysis of discrete distributions in these domains of Physics. The observables in the field of particle and nuclear collisions (usually the number or the mass of the particles/nuclear fragments) are easy to apprehend, even for non-specialists, and we hope that the methods discussed in this book will be accessible to physicists of other areas. By the way, the reader should keep in mind that the present framework and methodology are obviously of much broader application, and is not limited to strong interaction physics.

The natural approach towards the phenomenological understanding of the particle/nuclear reaction data is to analyze the statistical features of the observables which are hopefully dependent only on a few relevant key parameters, but weakly on the particular dynamical process involved. This brings us to the study of the probability distributions and the correlations of various kind. When comparing in this way the data for different experimental parameters, the hope is that the data exhibit certain universal aspect and hence the search for scaling behaviour is sensible.

5.1 The class of Poisson transforms

The general problem we want to address is the change from the continuous distribution to its discrete realization, and the influence of this transition on the statistical fluctuations. Various ways are available in order to model such a process, but by far the most important method involves the Poisson distribution which is not a source of additional correlations in the system. For that reason, we begin discussing a discrete version of continuous distribution which belong to the following class of Poisson transforms :

P ( n ) 5 P M ( n ) = r ^ ) n g f H n ) x ) / ( g ) < f a ( j U )

Jo n-

The Class of Poisson Transforms 115

satisfying :

oo oo

£ P ( n ) = l , X ) n P ( n ) = (n> (5.2) n=0 n=0

where f(x) is the (positive) continuous weight. The corresponding positive discrete random variable n will be called the multiplicity to follow the name used in nuclear physics and denoting asymptotic number of particles/fragments in a given reaction event. This scenario is valid as far as the discretization does not introduce the additional correlations , e.g. due to some conservation law. This is a normal approach when the basic process is Poissonian and the average value of the random variable, i.e. x(n) in accordance with equation (5.1) is distributed according to the distribution law / . The discrete relations (5.2) correspond then to the two normalization conditions :

/•OO /.OO

/ f(x) dx = 1 , x f(x) dx = 1 Jo Jo

(5.3)

This type of distributions is also used in P - representation of quantum optics [R. J. Glauber (1963); E. C. G. Sudarshan (1963)], and they are known to describe the photoelectron counting experiments, the galaxy distributions [P. J. E. Peebles (1980)] and multiparticle momentum distributions. We must emphasize that the choice of the class of Poisson transform (5.1) to pass from continuous to discrete distribution, even though natural, is by no means a unique and other distributions, e.g. Bose-Einstein distribution, Pascal distribution, binomial distribution , etc. yield behaviours which are qualitatively similar to those described below for Poisson transform. We shall give brief examples in Section 5.4.

^From the mathematical point of view, the idea underlying such a functional transform is as follows. The scaled Poisson distribution :

{n)fpoisson{x) = < „ > M £ ) I e - < » > * TV.

is positive, has a maximum of order y/(n) at x = n/(n), and its integral over [0,oo) is equal to 1. Let us then denote z = n/(n). One deduces that (n)fpoisson behaves as the Dirac distribution when n becomes infinite, or more precisely :

(n)fpoisson(x) ~> 8(X - z) (5.4)


when n —> oo, leaving z constant. Still in other words, when the scaled multiplicity stays fixed, (n)P(n)

tends to the weight function as the limit distribution :

(n)P(n) -»• f(z) (5.5)

when n —> oo. From this explanation, it is clear that other choices for (n) fPoisson could lead to the same feature for the corresponding transform, provided asymptotic equation such as (5.4) is satisfied.

The two important conditions : n —> oo and n/(n) fixed, will be seen in the following as the general and central requirements for this scenario to apply.

5.1.1 General functional relations for the Poisson transforms

Several interesting relations about these transforms are worth to be mentioned here. First, let us notice that the discrete distribution (5.1) depends explicitly on the mean value (n). In many experimental realizations, this parameter is a function of time or initial conditions, e.g. the total mass, the total energy etc. From this point of view, the distribution P n depends on two variables : n and (n), which do not play a symmetric role in this problem. However, there exist fine relations between n—variations and (n)—variations of the discrete distribution. This can be seen directly by calculating the first derivative of the equation (5.1) with respect to (n). One finds :

( n ) ^ ^ = - ( n + l)P(n + l ) + n P ( n ) . (5.6)

This equation is formally identical to the master equation (4.15) written for the probability distribution P(n).

In the same spirit, it is fundamental to remark that the whole distribution P(n) can be entirely calculated if one knows alone the value of the void probability P(0) as a function of (n). This appears thanks to the formula :

v ' n! d(n)n K '

Both P(0) and P(n) in this expression are functions of the average value (n). This relation can be particularly interesting for analytical applications


Table 5.1 Examples of some useful Poisson transform distributions with their associated weights. The shape parameter k must be positive. Kv is the modified Bessel function of the second kind [M. Abramowitz & I. A. Stegun (1964)] with index u. It should be noted that in the generalized Gamma distribution, which is leading to the Stacy distribution after Poisson transform, 7 > 0 is the scaling exponent and k — ( r ( ( A + l ) / 7 ) / r ( A / 7 ) ) 7 > 0 by the normalization conditions. Hx'x is the Fox H-function [C. Fox (1961); A. M. Mathai & R. K. Saxena (1978)].

Distribution P(n) /(*)

S(x - 1)

exp(—x)

Poisson W W « P ( - < " » n!

Bose - Einste in ,. , ) (,„_LH

Pasca l

Bessel-K 2£&££^Kn-k-1(2^) jfc e x p ( - f c / * ) / x * + 2

(„+fc-l)l «n)/fc)" n!(fc-l)! (l+(n)/fc)n+»'

kk

r(fc) — x f e 1 e x p ( - / c x )

Stacy (0<7<1) ^rWT)HU

Stacv f-Y=l) r(n+A>! «">/fc> oxacy ^7 1; n, r (A) (1+<„ ) /A)

r((x+i)/y) (n)r(A/7)

Stacy (7>1) ^ 3 ^ <">r(A/-y) r«A+i) / 7 )

( l - n , l ) (V7,l /7)

(1 - A / 7 , 1 / 7 ) ( n , l )

?7-)kx/'>xx-xexp(-kx-»)

r(A/7)

i ^ x A 1exp(-fcx)

fo^s^extf-fcxT) r(A/7)

because, following relation (5.1), P(0) is just the Laplace transform of f(x).

5.1.2 Examples of Poisson transforms

Various examples of useful distributions and their corresponding Poisson weights f(x) are listed in Table 5.1. Plot of some Bessel-if distributions are shown in figure 5.1. One should note in particular the convergence of the scaled discrete distributions towards the limit distribution, in accordance with (5.5). Two particularly important cases will be detailed in the following sections : the Pascal distribution* in Section 5.2 and the Stacy distribution in Section 5.3.

'Pascal distribution is called frequently the negative binomial distribution.


0.05

0.00

<n>Pn

1.5

1.0

0.5

0.0

Fig. 5.1 Plots of the discrete probability distribution corresponding to the Poisson transform of the weight : exp(— l/x)/x3. These are the Bessel-K" distributions for the shape factor k = 1, as detailed in Table 5.1. On the left side, there are three unsealed examples with (n) = 16, 64 and 256. On the right side, the same distributions scaled according to : (n)P(n) vs the scaled variable n/(n), are shown. For large values of (n), one recovers correctiy the weight distribution (the continuous curve) in accordance with (5.5).

Some other Poisson transforms can be expressed by less elementary functions. For example, Poisson transforms corresponding to the Gaussian weights xk exp(—ax2) are related to the parabolic cylinder functions of negative order D-n-k-i [M. Abramowitz & I. A. Stegun (1964)]. An important particular case is the Poisson transform of the half-Gaussian which can be expressed as :

P(n) {-z)n dn

n ! dz - (e*2/2Erfc(z/V2)) < - • f(x) *-*l*

with the value z = (n) y/ix/2 of the current variable. Erfc is the error function [M. Abramowitz &; I. A. Stegun (1964)]. This kind of formula can be derived using the general relation (5.7) connecting P(n) to the value of P(0), which is valid for any Poisson transform.


Another useful general relation for the Poisson transform distributions is the following series :

wr(«>-£:(-;')&~/M-i>.ff which is valid as long as both infinite sums exist.

5.1.3 Generating function for the Poisson transforms

Generating functions of probability distribution provide exactly the same information as the distribution itself, but some features are sometimes more accessible using these alternative tools. We discussed previously the particular example of the characteristic function in the general framework of the probability theory in Section 2.2.1.

Another example is the factorial moment generating function for the discrete distribution P(n), as defined by :

oo

g(u) = J2unp(n) • (5-8) n=0

Its main property concerns the factorial moments (n(n — 1) • • • (n — q +1)), which are given by the successive derivatives d9Q/duq, calculated at the value u = 1 *.

The corresponding generating function for the Poisson distribution of mean value (n) is the exponential function :

Qpoisson(u) = e ^ 1 - " ) ^ . (5.9)

Hence for the functions in the class of Poisson transforms (5.1), the moment generating function is the Laplace transform :

/ •OO

QP-T(U)= / e-^-u^n>uf(u)du Jo

which is formally identical to the void probability P(0), with the change of the mean value : (n) -> (1 — u){n). As an example, for the Pascal

'Sometimes, the moment generating function is more conveniently written as : Q(u) = ^ ° t _ . ( l — u)"P(n). It is obviously identical to (5.8) after a substitution u —• 1 — u.


distribution (c.f. Table 5.1) one gets :

Qpascaliu) = TTTTi \i \ /;ifc • (5.10) [1 + (1 -u){n)/k\k

Similarly, for the Bessel-K distribution :

2 QBessei-K{u) = f ^ ^ ) (fc(l - u)(n)f+1)/2 Kk+1(2y/k(l - u)(n)) .

The positive Poisson weight f{x) can be regarded as a probability distribution on the interval 0 < x < oo. It is then interesting to compare the n—moments of P(n) with the a;—moments of f(x) :

(n(n - l) • •. ( n - g + 1)) _ 1 fl«gp_r {n)i {n)i dui lu==1

/ Jo

xqf(x)dx = (xq) (5.11)

for any positive integer value of the index q. These relations connect the moments of the continuous distribution f(x) to the factorial moments of the discrete distribution P(n). One advantage of the factorial moments is that they automatically eliminate the shot noise.

This particular property of Poisson transform distributions implies other useful relations between moments of f(x) and P(n). Some of them will be seen in the following sections. Notice that the relations (5.11) are specific of the Poisson transforms. Conversely, if they hold true for any positive integer q, then GP-T is the moment generating function of a Poisson transform of weight / .

5.2 Pascal distribution

Pascal distribution plays a prominent role in phenomenology of high-energy multihadron production [A. Giovannini (1972)], photoelectron counting distributions due to thermal light from k sources of equal intensity [L. Man-del (1959)] and count statistics for galaxies [P. Carruthers & D.-V. Minh (1983)]. Anscombe (1950) has stressed possible relevance of this distribution in biology. We shall discuss now some properties of this important distribution belonging to the class of the Poisson transform distributions.

Pascal Distribution 121

5.2.1 Definition and moments of the Pascal distribution

The Pascal distribution of index k :

n > \ k-1 J (1 + (n>/fc)»+* ( 5-12)

can be identified with the generalized Bose-Binstein distribution for A; cells having equal average occupancy (n)/k [M. Planck (1932)]. Formally, the Pascal probability is the n—th term in the expansion of unity, following :

with the correspondence :

1 , (n)/k l + (n)/k ' * (l + (n)/k)

i.e., (l-p)/p= (n)/k. The normalized factorial moments of the Pascal distribution (5.12) are

given explicitly by :

(n(n - 1) • • • (n - q + 1)) _ r(fe + q) (n)i ~~ T(k)kfl '

Figure 5.2 shows examples, for various values of k, of the weight functions which are needed to get the Pascal distribution by Poisson transform. The tail for the large argument is always exponential.

5.2.2 Recurrence relations for the Pascal distribution

A necessary and sufficient condition for a multiplicity distribution to be the Pascal distribution is that it satisfies the recurrence relation :

P(„) _ « + / * (» -1 ) n ^ 1 ( 5 1 3 )

P(n - 1) n

where a and (3 (/? < 1) are positive constants. Moreover, the index k of the Pascal distribution is : k = a//3, and the mean value (n) is : (n) — a / ( l — (3). This relation yields the Pascal distribution if supplemented by the normalization condition.


fPasJX)

Fig. 5.2 Plot of the limit scaling function kkxk_1 exp(—kx)/V(k) of the Pascal distribution for : k = 1/2, 1, 2, and 4. When k —> oo, the function approaches the Gaussian shape centered at x = 1 and of variance 1/fc.

General solution of the recurrence relation (5.13) [C. Bender k, S. Orszag (1978)] :

n - l

P(n)=aP(0)n J'=I

/ 3 j + a (5.14)

can be convenient for some applications. In this relation, P(0) must be determined by the probability normalization. It is immediate from the equation (5.14), that the large—n behaviour of the Pascal distribution is given by :

P(„) „ ^ 1 / f ^ - i

except for the limiting values of the parameters a or 8 which will be detailed below. This behaviour is reminiscent of its Poisson weight function since a/B = k.


5.2.3 Limit cases of the Poisson distribution

Some particular examples of the Pascal distribution merit a remark here as they behave somewhat differently from the standard case.

• In the limit /3 —>• 0, the recurrence relation (5.13) gives the Poisson distribution, with (n) = a.

• If a —> 0 at a constant 0 < /? < 1, the Pascal distribution (5.14) reduces to the distribution :

(5.15)

for n > 1

with the coefficient :

• The case a = j3 corresponds to the geometric distribution of ratio (n)/(l + (n)), also called the Bose-Einstein distribution :

n n ) (l + (n))»+i •

• For a > 0,/3 < 0 and the integer ratio a/(3 = —K, one obtains the positive binomial distribution :

P W = ( T ^ ( : ) H3)» (5-16)

with the average value :

/ \ a

5.2.4 Stability of the Pascal distribution

The section is devoted to the stability of certain distributions P(n) satisfying the A-scaling (3.29), as introduced in the Section 3.5.4 (c.f. Sections 2.2.8 and 3.4.5). We have seen in Sections 3.5.2 and 3.5.3 that this sort of distributions appear naturally in critical systems.

P(0) = l + a l n ( l - / 3 )

on

V{n)=a^- ,


The generating function of a distribution P(n) satisfying (3.29) has the form * :

° ° /.OO

g(u) = y " u " P ( n ) ~ / $(z)u<n>+*<"> dz .

Computing dQ/du and dQ/d{n), one derives the differential equation :

eg d(n)

= lnu £ . £ + (l-A)B (n) au

The general solution of this equation is :

g(u) = A((n)Alnu)uW

(5.17)

(5.18)

with A an arbitrary function reflecting the initial conditions and subject to the constraints : A(0) = 1 (the normalization of probability), A (0) = 0 (the definition of (n)).

Investigation of P(n) for n » 1 is equivalent to the study of Q for u —> l - . Solution of (5.18) for z —> 1~ can be written generally as :

g{u) ~ A{(n)A(u - 1)) x exp( - ( l - u)(n)) .

The generating function g appears then to be a product of the function A and of the generating function (5.9) of the discrete Poisson distribution . Consequently, the corresponding probability distribution which verifies the A-scaling is the convolution of a distribution characterizing the initial conditions, and of the Poisson distribution with parameter (n). In the case of first scaling (A = 1), the generating function is a function of a single reduced argument (n)(l — u).

Interesting feature appears when the initial condition £o(u) is itself the Pascal distribution (equation (5.10)) with multiplicity (n)o- In this case, the solution of the differential equation (5.17) with the normalization constraints :

0(1) Q'iX) = (»>

9{u) = i + (n)0 / (n)

k \(n)o

-\ -k

( !"«)

* A-scaling variable in this case is : z = (n — (n))/(n)A .


x exp[-( l - u)«n) - (n)A(n)J-A)] . (5.19)

The solution (5.19) means that if at t = 0 we have (n) = (n)o and the initial distribution P(n) is given by the Pascal distribution with index k, then at later times for which (n) becomes larger than (n)o, the probability distribution P(n) is the convolution of the Pascal distribution with the same index k and the Poisson distribution. In this sense, any distribution which behaves as a Pascal distribution for large values of the multiplicity n and which verifies the A-scaling has a stable tail. In the case of the first scaling, one obtains immediately that the Pascal distribution is time-invariant, since the Poisson generating function vanishes in the solution (5.19) and so :

G = Go •

5.2.5 Origins of the Pascal distribution

There are many ways to arrive at the Pascal distribution and their appeal depends on the considered problem. As mentioned before, Planck (1932) showed that the Pascal distribution is obtained by dropping Bose-Einstein particles into k cells of equal average occupancy (n)/k. This explanation is perhaps somewhat artificial but the Pascal distribution is not artificial in any respect. Pascal distribution is indeed a ubiquitous distribution in nature, found in many unrelated fields. In this respect, it is similar to the Maxwell distribution of molecules in thermal equilibrium which does not depend on any details of the intermolecular forces.

In the previous sections we have already given some of the physical mechanisms leading to the appearance of the Pascal distribution. One is given by averaging of Poissonian distributions with the weight function given by the Gamma distribution :

fGamma{x) = ^xk-1e-kx . (5.20) 1(A)

We shall see another such mechanism in Section 6.5.2 about the clan model, representing the Pascal distribution by the convolution of a particular Pascal and Poisson distributions. The realization of self-similar Cantor set structures as an origin of the Pascal distribution was put forward by Car-ruthers and Shih (1987). Giovannini (1979) has shown that jets in quantum


chromodynamics (QCD) in the leading-log approximation (LLA approximation ) have the structure of the Markov branching process of Pascal distribution type *. Below, we shall detail the Pascal distribution as a solution of certain stochastic differential equations.

All those numerous physical mechanisms are much too detailed to explain convincingly a ubiquity of the Pascal distribution in Nature. To understand the importance of this distribution one may notice, however, that the Pascal distribution derive its origin from the leading singularity of the generating function Q. Writing this singularity as :

one recovers the Pascal distribution with index k and mean value :

(n)=k/(ak-l) .

Hence, whenever the generating function of the studied physical problem is dominated by a single pole, the probability distribution P(n) is closely approximated by the Pascal distribution. From this point of view, the ubiquity of the Pascal distribution is a statement about the ubiquity of generating functions with single-pole dominance. The above remark can help in approximating complicated probability distribution by the Pascal distribution.

5.2.6 Stochastic differential equation leading to the Pascal distribution

Markovian processes play a prominent role in many domains of the statistical physics. We discussed some of them in the framework of diffusion (Section 4.1.2). The probabilities in this theory obey the differential-difference equations referring only to the present time, i.e. the earlier history of the system is irrelevant to the prediction of the future once the appropriate boundary conditions are given. As a simple example, let us consider the evolution equation :

^ f a * ) = A P ( n - l , t ) - A P ( n , t ) . (5.21) at

*The perturbative quantum chromodynamics in the higher order, next to next to leading logarithm approximation (NNLL approximation) does not however lead to the Pascal distribution.

Stacy Distribution 127

If we have P(n) = Sn>0 (5 is here the Kronecker symbol) at t = 0, then the future evolution of the probability predicted by the equation (5.21) is given by the Poisson distribution :

P(n,i) = ^pexp(-Ai)

with the average multiplicity : (n) = Xt. Equation (5.21) can be rewritten as :

( n > ^ M = n P ( n ) - ( n + l )P(n + l)

which is nothing else but the master equation (5.6). The average multiplicity (n) becomes a new evolution parameter in this equation. The simple transformation : t -> (n), allows to see that one can examine the evolution of P(n) in two ways. For a given collision energy (the size of the system), we can consider actual time-evolution of P(n) up to a certain cut-off time tCut, e.g. the hadronization time in the multiparticle production, or freeze-out time in the nuclear multifragmentation. Alternatively, one can vary the collision energy by changing (n) and study how the distribution P(n) changes. Since we expect that tcut increases with (n), therefore dP(n)/d(n) gives indirectly information about dP{n)/dtcut.

One should mention that the validity of Markovian processes in the context of high energy physics is questionable. For physical processes one expects that many degrees of freedom are involved in the evolution process and, moreover, that the phase relations are important. Examples of that kind are provided by the system of coupled bosons or the forced quantum oscillator [P. Carruthers & K. S. Dy (1966)]. Both aspects are absent in equation (5.21). The validity of above equations require the chaotization of many degrees of freedom which will contribute with random phases, and the appearance of collectivity which will reduce the number of effective degrees of freedom.

5.3 Stacy distribution

In Section 5.2 about the Pascal distribution, we took the point of view of the discrete random variable. But we know from the relation (5.5) for Poisson transforms, that it is equivalent to describe the discrete distribution or the continuous weight factor, the latter one being just the limit scaling

128 Poisson- Transform, Distributions

function of the former one. For the Stacy distribution, it is easier to discuss the continuous generalized Gamma distribution , because its analytical expression is quite simple.

5.3.1 The generalized Gamma distribution and its moments

The Gamma distribution is given by the formula :

Mx) TO x^e-^ (5.22)

This is a three-parameter family of distributions. Figure 5.3 shows some examples of such distributions corresponding to the particular case k = 1, for various values of 7 and A.

/ (x) Gamma

L M

Fig. 5.3 Plot of some generalized Gamma distributions with k = 1. (i) For the parameter A < 1, the maximum of the curve is at x = 0. (ii) For the parameter A > 1, the maximum occurs for a finite value of the argument x* = ( ( A - l ) / f c 7 ) ) 1 / 7 .

In particular, one remarks that the most probable value of this distri-


bution is 0 if and only if A < 1. More precisely :

x~ = <

0 for A < 1 (5.23)

((A-l)/fc7)1 / 7 for A > 1

This feature has been found to be related to a kind of critical phenomenon in some physical applications. This will be described briefly in the next section.

Moments of the generalized Gamma distribution are easy to calculate :

,q) = r((A + g)/7)

These expressions correspond also exactly to the normalized factorial moments {n(n — 1) • • • (n — g + l ) ) / (n) 9 of the discrete Stacy distributions, according to the general result (5.11) about the Poisson transforms.

The generalized Gamma distribution is quite common in Physics. One of the reasons is that it behaves like a power law ~ x A _ 1 for small arguments and as an anomalous exponential law ~ exp(—kxJ) for large arguments. For the values of 7 smaller than 1, this corresponds to the stretched exponential, while for 7 > 1 this is a sharper cut-off than an exponential cut-off. This kind of behavior appears naturally in many areas, such as the stable distributions of index 0 < /j, < 1 (see (2.23) and (2.24)), the distribution of the order parameter in the second order phase transition (c.f. discussion in Chapter 11), the dynamics of disordered systems, and many others. The Pascal distribution is the particular case : 7 = 1, A = k. The Bose-Einstein distribution corresponds t o : 7 = A = A: = 1.

5.3.2 Langevin and Fokker-Planck equations leading to the generalized Gamma function

Rate equations provide a standard way of describing the evolution of the population. The effect of coupling of different microscopic (internal) degrees of freedom in rate equations, as well as the linear couplings to other degrees of freedom like external reservoirs, create dissipative and fluctuation processes. They are frequently represented as a noise acting on a chosen degree of freedom. The resulting equations are of the Langevin form :

-Xi = YijXi + FijkXjXk + ... + F[0) + XjFJi} , i = l , 2 , . . . (5.25)


where T's are time-independent matrices and F's are the fluctuating forces characterized by :

(F}i\t)FP{t'))=G$'\t-t)

(F?\t))=0 3 = 1,2,...

Implicit summation over repeated indices applies in equation (5.25). In the usual case where the fluctuations are due to the coupling to a

reservoir close to the equilibrium, Gjk are stationary and depends only on the time difference t — t . Moreover, if it is sufficient for the dynamical description of collective variables {xi(t)} to consider a coarse-grained time scale which is large compared to the correlation time of fluctuations, then G can be reasonably approximated by the Dirac correlation :

(F}i>(t)FP(t')) = 2Qfp6(t-t)

where the positive scalar Q is a measure of fluctuations, independent of {xi}. In this limit, equation (5.25) describes a multidimensional process. If F^ = 0, (i = 1,2 . . .) and Ff1 + 0, then the process (5.25) is called an additive stochastic process and can be formally written in the more condensed form :

—Xi = Ci({xj}) + Fi

while for vanishing i*y ' and the Fj ' ^ 0, the process (5.25) is called a multiplicative stochastic process

—Xi = Ci{{xj}) + Gij{{xk})Fj

and the function Gij depends explicitly on all other modes. Whenever £i({xj}) is a linear function of the modes {XJ}, the Gaussian structure of the random forces (5.26) is conserved and the process remains Gaussian in accordance with the general results of Section 3.1.1. On the contrary, these equations can hardly ever be solved if Ci({xj}) is a nonlinear function.

5.3.2.1 One-dimensional Langevin equation with the multiplicative noise

After this general introduction, we shall discuss now an explicit class of such non-linear one-dimensional Langevin equation with the multiplicative


Gaussian noise . It includes many physical systems and is analytically soluble [A. Schenzle & H. Brand (1979)]. Moreover its general solution can be expressed in terms of the generalized Gamma distribution.

The Langevin equation, we are interested in, is :

dor

— = KX- x1+~< + xF(t) (5.26)

where (F(t)F(t )) = 2QS(t — t) and 2Q is the positive strength of the noise, usually proportional to the temperature of the system *. The parameters K and 7 are positive numbers. In some interesting cases, they can be identified with the parameters of the Hamiltonian of the physical system. In this way, various realizations of these parameters are precisely connected to known physical processes. This is the case for 7 = 2, associated with the laser model [V. DeGiorgio & M. Scully (1970)] describing fluctuations of the sub-harmonic electro-magnetic field generated in a non-linear crystal inside an optical cavity [W. H. Louisell (1960); N. Bloembergen (1965)]. The same class of equation appears for any non-linear wave mixing inside a laser [A. Schenzle k, H. Brand (1979)], such as Raman scattering, three-wave mixing, with possibly some other value for 7.

5.3.2.2 Explicit physical processes leading to the one-dimensional Langevin equation with the multiplicative noise

Chemical reaction kinetics provide simply a wide class of evolution equations similar to (5.26). The simplest one is certainly the autocatalytic birth process :

A+X^X+X.

Denoting by k+ and A;_ the respective chemical rate constants associated to the production and disappearance of one X atom, one has the non-linear evolution equation for the concentration nx in the species X :

—r— = k+nAnx - k-n2x • (5-27)

"To simplify notation, we choose the unit of time such that the coefficient of the nonlinear restoring force x 1 + 7 is equal to 1. If one wishes to work explicitly with such a coefficient, say 6, different from 1, then the Langevin equation writes : ^ = K I -bx1+~/ + xF(t), and the coefficients K and Q should be replaced everywhere by /t/b and Q/b, respectively.


We suppose now that by chemical means the concentration TIA is maintained constant, except for some fluctuations : TIA = n°A + SUA- Then equation (5.27) can be written as :

^ = k+nAnx - k-n2x + nxF (5.28)

at with the random force F given by : F = k+SnA- Equation (5.28) is identical to equation (5.26) with the value 7 = 1. One can readily imagine from this simple example, that more complicated chemical systems or interactive social groups [W. Weidlich (1973)] can yield the same kind of non-linear evolution equation with other values (essentially integer values) of the parameter 7 [J. de la Rubia & M. G. Velarde (1978)].

As a last generic example, one can cite the damped anharmonic oscillator which can be applied in many physical contexts [H. Haken (1975)]. This equation of motion reads :

.,d2x .dx n 3 Mw+xTt=-ax-Px •

Adding a random Langevin force and considering the overdamped oscillator for which one can put formally M = 0, one obtains again the equation (5.26) with the value 7 = 2.

5.3.2.3 Solution of the one-dimensional Langevin equation with the multiplicative noise

The complete evolution of a system governed by the Langevin equation (5.26) can also be described by the Fokker-Planck equation acting on the probability distribution of the variable x :

+ g £ ^ 2 p ( M ) ) • (5-29) The steady-state solution (dP(x, t)/dt — 0) of the Fokker-Planck equa

tion (5.29) [R. I. Stratonovich (1963)] :

P(z,0) o c z t - 1 e x p ( - - ^ r J (5.30)

is nothing else but the generalized Gamma distribution (5.22).


As discussed on page 129, the most probable value of x is x* = 0 if Q > K and x* > 0 if Q < K. This is a reminiscent of a phase transition driven by the strength of the noise. The quantity x* could be interpreted as the order parameter and the large-noise domain Q > K (respectively, the low-noise domain Q < K) should correspond to the disordered (respectively, ordered) phase. This leads to the phase diagram shown in figure 5.4.

Q

disordered phase

->

A

/ /

/

/ ordered

phase

K

Fig. 5.4 Sketch of the phase diagram for the model described by the non-linear Langevin equation (5.26). The positive parameter Q is the fluctuation strength and K is the coefficient of the linear restoring force.

5.3.2.4 The limit case with vanishing random force

The case Q = 0 has a simple interpretation. If there is no random force, the equilibrium state is determined by the minimum of the potential energy :

U ••

KX r2+7

+ 2 (2+7 )

One recognizes the standard Landau-Ginzburg model of the phase transition. Different shapes of the potential are shown in figure 5.5. From this picture, it appears that the case 7 = 2 should be special, since it corresponds to the case where the potential energy is analytical in the current variable x. So, this value of 7 is expected to be found in many physical situation. This


-

•

K<0

K>0 /

/

Fig. 5.5 The potential U = - K X 2 / 2 + x2+~*/(2 + 7) for 7 > 0 in arbitrary units. The equilibrium value for the variable x is : x* = 0 if K < 0 and x* = /c1/"1' if K > 0 (this figure shows the case 7 = 2 and K = ±4) . This serves as the mean-field model of the phase transition when no random force is applied.

is effectively the case, and this includes popular first scaling (c.f. the KNO scaling in Section 5.5) forms for pp and e+e~ collisions [S. Barshay (1982); K. Chou et al. (1983)]. Multiparticle production process can be considered as a dissipative process driven by the initial kinetic energy in the collision. It is an open question whether the equilibrium can be reached before the hadronization *. In the following chapters we shall discuss the relation between the perturbative QCD rate equations and the fragmentation-inactivation binary process which is a driven off-equilibrium process with dissipation effects included in the form of the inactivation function. In particular, the off-equilibrium critical behaviour in such a theory can be discussed rigorously. The similarity between the stochastic process (5.26) and the time-dependent Landau-Ginzburg theory was noticed [P. Carruthers & I. Sarcevic (1987)] and the possibility of an off-equilibrium phase transition in multiparticle production was suggested.

"One should however mention an unquestionable success of statistical thermal model in describing hadron multiplicities in high energy e + e - , pp and pp collisions [F. Becattini (1998)].

Other Examples of Integral Transforms 135

5.4 Other examples of integral transforms

As explained in the beginning of the section, Poisson transforms are expected to play a central role in Physics. But they are indeed not unique. One can introduce in a similar way binomial or Pascal transforms (k > 1) and they have about the same asymptotic properties as the Poisson transforms since they approach a S—function when (n) and n tend together to infinity and their ratio remains finite.

An example of this kind is the Bose-Einstein transform . We define it

P(n) = / Jo

«»>*>" - / ( * ) * , (1 + (n)z)"+ 1 •

with the usual normalizations (5.2) corresponding to the constraints given in (5.3). Its generating function is :

GBE-TM = r , . J{X\, i dx / •OO

+ (1 — u)(n)x

which implies the relations on the moments :

(n(n-l)---(n-q + l)) 1 d*GBE-T (n)i (n)9 dui

q\(x")

u = l

(5.31)

These relations are similar to the relations (5.11) obtained for the Poisson transforms. Moreover, for the asymptotic n —> oo, (n) —> oo and fixed, finite n/(n)y we obtain the first scaling form :

(n)P(n) ~ / exp f '-^- \ J-±-!-dx

even though the Bose-Einstein distribution itself does not tend to the Dirac function under these conditions.

5.5 KNO scaling limit

Possibility of compressing the information contained in multiplicity distributions P(n) for different average values (n) was investigated intensely in high energy collisions following the original hypothesis of Polyakov [A. M.


Polyakov (1970a); A. M. Polyakov (1970b)] and Koba, Nielsen and Olesen [Z. Koba et al. (1972)]. KNO prediction was based on assumption of validity of Feynman scaling for the many-body inclusive cross sections. Later, the relation of a KNO scaling to a phase transition in the Feynman-Wilson gas was emphasized [N. G. Antoniou et al. (1984)]. This hypothesis, which carries now the name of Koba-Nielsen-Olesen scaling (KNO scaling), states that data for different mean multiplicities (n), i.e. for different collision energies should fall on a unique curve when (n)P(n) is plotted as a function of reduced variable n/(n). This KNO scaling limit, which is a special case of the first scaling law discussed in Section 3.5.2 for critical systems, is defined by the asymptotic behavior of (n)P(n) = <&(n/{n)) when both n, (n) —»• oo but the scaling variable z = n/(n) is fixed. The average multiplicity (n) serves here as a scale parameter and the normalization conditions (3.24) imply that moments of the scaling function are independent of (n) (c.f. Section 3.5.2).

Another properly normalized scaling function obeyed by P(n) is :

^(z) = z$(z)

meaning that data for different (n) should fall on a universal curve when nP(n) is plotted as a function of scaled variable z = n/(n). This form of KNO scaling removes the influence of possible statistical errors in (n) on the accuracy of the scaling function.

5.5.1 Extension of the KNO scaling rule

The average multiplicity (n) depends on the system size N, whose meaning can be different according to the investigated system (e.g. the total mass, or the collision energy). The KNO scaling means that the iV-dependence of P(n) is due to the N—dependence of its first moment {n(N)) only :

P(n)EEPw(n) = P ( n )(n) .

One can then interpret any deviation from the scaling function for some N as a signal that the shape variation of P(n) cannot be reduced to the (n)—scaling exclusively. The problem becomes then how to transform out from P(n) the AT—dependence of its higher order normalized moments. In this context, Hegyi (1997) discussed more general scaling transformations which make reference to higher order moments (nq(N)) of P(n). The idea goes as follows.

KNO Scaling Limit 137

Let us introduce the homogeneity relations for the probability distributions :

p«)(„) = ^ _ p ( i - D ( n ) , j = l,2,... ( n ) j - i

(5.32)

P ( 0 )(n) = P(n)

From (5.32), the A?"—dependent normalization averages :

w, = Enp( j )(n) n

are defined recursively by :

W j - i

(n)0 = (n)

Consequently, the quantity (n)j involves only averages of the form (n2 ) for i < j *. The generalized scaling hypothesis corresponds then to the same homogeneity relation as in KNO scaling, which is extended to P ^ (n) :

(n)jPW(n) = $ ) ' f e ) • (5-33) Equation (5.32) can be iterated, leading to the sequence of distributions :

P(n)(iV) - • pW(n) -* P ( 2 )(n) -+ . . . -> P ( j ) (") -> • • •

until one removes the iV—dependence from the scaling function and achieves the collapse of data for different N after the rescaling (5.33) by (n)j.

It remains to be seen whether this equivalence hypothesis (5.33) is going to be satisfied at extremely high energies in the hadron - hadron collisions, replacing the original KNO scaling hypothesis. In this case, the ultimate scaling of multiplicity distributions could be understood as the fixed point property of nuclear matter.

'Closed form for the {n)j axe : {n)j = ] ^ _ 0 ( " 2 ) a S with the exponents : on =

(—1),+-' I . )• The first few averages are then : (n)i = (n 2 ) / (n ) , (n)2 =

(n 4 ) (n ) / (n 2 ) 2 , (n) 3 = <n8)(n2)3/<n4>3<«>, etc.

Chapter 6

Featuring the Correlations

Up to now, discussion of the statistics of random variables was mainly concentrated on general features of the probability distributions and their generating functions. For that reason, only limited number of moments of probability distributions were mentioned in this discussion. In phenomeno-logical applications, however, most of the useful information about random systems is collected in the form of low rank moments and their combinations. Indeed, special features of systems appear clearly in some complicated combination of moments or some particular correlation functions. This is the idea of this chapter to put in hands of the reader all tools, definitions as well as the explicatory discussion issued from the moments and the correlations of the probability distribution of the random variable. As usual throughout this book, the random variables are supposed to be positive, but generalization to real or even vectorial variables is straightforward and will be omitted.

6.1 Moments and their generating function

This section is devoted to remind notations and definitions for various types of moments and their combinations in the analysis of probability distributions. For convenience, this section is divided into small subsections corresponding to the proper definition of each particular moment.

139

140 Featuring the Correlations

6.1.1 Moments

Given a count probability distribution P(n), where n = 0 ,1 ,2 , . . . , one can define the ordinary moments of the distribution :

oo

/zq = (n*) = ^ V P ( n ) g = 0 , l , 2 , . . . 71=0

with the corresponding moment generating function :

oo

M{u) = ^enuV{n) = Y , ^ . (6.1) n=0 q "'

The running argument can be a complex number *. Because P(n) is a probability distribution, one has : A4(0) = 1, and one gets the moments by the derivatives of M. :

(„«> - m M { u )

dv.Q u=0

Another moment generating function is the Stieltjes transform of the

distribution P ( n ) . It is usually defined as :

5 W - E S = E ^ H 5 • (6-2) n=0 q

The Stieltjes transform is less used in physical applications, but has been extensively studied in Mathematics because of its relation with the Pade approximants [G. A. Baker (1975)].

6.1.2 Cumulant moments

Another useful generating function is the cumulant generating function as defined by the logarithm of the generating function M :

InM{u) = YJ'^u<' . q

*Note that we shall not discuss here the proper existence, continuity or derivability of the infinite sums written down in these definitions or properties, and we give preference to the usual pragmatic approach of the physicist on these questions.

Moments and Their Generating Function 141

The cumulant moments rCq 3JT6 the combinations :

Ko = 0 ,

« i = (n )

K2 = (n2) - (n)2 (6.3)

K3 = (n 3 ) -3 (n 2 ) (n ) + 2(n)3

K4 = (n4) - 4(n3)(n) - 3(n2)2 + 12(n2)(n)2 - 6(n>4

As a check, all these cumulant moments with the exception of K\, are equal to 0 for the Kronecker distribution P(n) = (5n,n0, for which one has (n<) = <n>*.

Often the normalized cumulant moments j q are used instead of the Kq. They are defined as :

* = & • ( 6 - 4 )

6.1.3 Factorial moments

For the discussion of fluctuations in small finite systems, e.g. in narrow phase space bins in the case of multihadron production in high energy strong interactions, it is useful to replace moments by their corresponding factorial moments. The factorial moment £q is :

e, = < n ( n - l ) - - - ( n - g + l)> . (6.5)

Factorial moments can be obtained by calculating the derivatives of the factorial moment generating function , introduced before in Section 5.1.3 :

oo

Q(u) = Y,unP(n) = MQnu) . (6.6) 71=0

Namely, one gets the identity :

WQ{u) (n(n -l)---(n-q + l)) =

dui u = l

or equivalently

g{i + u) = Y J ^ . (6.7) q


Note also that the probability distribution can be recovered easily as :

P(n) = i dnGiu)

n\ dun u = 0

where the constraint u = 0 removes all but the n-th term of the series. One deduces the inverse formula :

' 3=0 J'

expressing the probability distribution in terms of the factorial moments.

6.1.4 Cumulant factorial moments

A convenient measure of correlations in small finite systems is provided by the cumulant factorial moment fq, which is defined by the expansion of the logarithm of CJ(1 +u) :

lng(l + u)^^2^uq . (6.9)

One finds

« q]

/o=0

/ i = (n>

f2 = (n(n - 1)) - (n)2 (6.10)

/ 3 = (n(n - l)(n - 2)) - 3(n(n - l))(n) + 2(n)3

which are similar relations to those expressing the cumulant moments in terms of the moments (see page 141).

6.1.5 Normalized moments

For the purpose of studying the scaling properties of fluctuations it is often more convenient to work with the normalized moments. An example of the normalized cumulant moments has been given previously in Section 6.1.2. The other ones are essentially the normalized factorial moments :


and the normalized cumulant factorial moments :

JCq = fq/(n)* . (6.11)

Moments Fq and Kq will be shown in Chapter 7 to be averages over the related correlation functions. Factorial moments arise with the integration of inclusive cross-section over phase space while the factorial cumulant moments are obtained by a similar integration over suitably denned multi-particle correlation functions.

Writing (6.7) and (6.9) as :

one derives the expressions for the Fq moments which are usually easy to measure in terms of mostly lower order JCq moments :

Fi = ] C ( o ) FjKq-i j=0 \ J /

with Fo = 1, Fi = 1, /Co = 0, /Ci = 1. For lowest few moments Fq, one finds :

Fi = l

F2 = l + K2

F3 = l + 3fC2 + /C3 (6.12)

F4 = 1 + 6K2 + 2,K?2 + 4/C3 + £ 4

F5 = 1 + 10/C2 + ISKl + IO/C2/C3 + IO/C3 + 5£ 4 + /C5

These are the general relations relating any (normalized) moments to their corresponding (normalized) cumulant moments.

The ratios between normalized cumulant factorial moments and normalized factorial moments are sometimes used for applications. They are defined as :

Hq = ^-=ff . (6.13)


6.1.6 Bunching parameters

An alternative to the normalized factorial moments is provided by the bunching parameters [S. V. Chekanov & V. I. Kuvshinov (1994)], which are defined as :

n P ( n ) P ( n - 2 ) Vn = p / V ^ 3 ' • n > l 6.14

n — 1 P(n — 1)J

They allow to reveal spiky structure of events by investigating the probability distribution near the multiplicity n. It is easy to see that rjn = 1 for all n in the case of a Poisson distribution. Conversely, the general solution of the equation (6.14) corresponding to a constant j]n (rjn = n < 1) is the Gaussian distribution :

P(n) = ah-ljfl'i

with the positive parameters a and b to be determined by the normalization conditions (5.2). One recovers the Poisson distribution for the particular case n = 1.

6.1.7 Combinants

Another interesting set of functions are the combinants [S. K. Kauffmann k M. Gyulassi (1978)], defined when P(0) ^ 0 by the equation :

oo

,1 ln</(u) = lnP(0) + ^ R q U ' P=I

= lnP(0) + f; ff; j=l 3 \n=l

(6.15)

These are useful when the probability distribution is essentially positive only close to origin. The first few lowest combinants are given by :

P(l) Ri P(0)

2

2 P(0) 2 \V{0))

= P(3) p(2)P(i) i m i ) 3 P(0) P(0)2 + 3 VP(0)


P(4) P(3)P(1) 1 / P ( 2 ) \ 2 P(2)P(1)2 1 / P ( l ) \ 4

4 P(0) P(0)2 2 \P(0)J P(0)3 4 \P(0)J

The combinants can be conveniently expressed as an infinite series of the cumulant factorial moments fp :

1 " ( _ I ) P

y ' p =0 F"

This is the exact analogue of relation (6.8) for the combinants. The reverse relation can be found in the equivalent way, expressing In. G{u) as a series of u by use of (6.15) and (6.9). One finds :

CO |

/p = EdbrR* • (6-16) q=p V* f)

The calculation of combinants Rq require only a finite number of P(j) with j < q, what may be advantageous in certain applications. In addition, the combinants share some features with the cumulants. Firstly, the set of Rq provides a similar measure of fluctuations as the set of fq. Like cumulants, the combinants allow easily to see the deviation from the Poisson distribution for which only Ri remains nonzero. Secondly, combinants have the additivity property of cumulants. When random variable is composed of statistically independent random variables which give :

Six) = l[gU){x)

for the generating function, the corresponding combinants are additive in those of the independent components.

6.1.8 Existence of the generating functions

In the previous sections, we detailed moments and their generating functions. An existence of various moments is clear, but existence of the corresponding generating functions is not guaranteed. An example is the log-normal distribution .

In Section 2.3.1, we have seen that moments of the continuous log-normal distribution (2.37) are given by : (wg) = exp(g2)/2. If one considers


its discrete Poisson transform distribution, one knows from (5.11) that the factorial moments are :

(n(n - 1) • • • (n - q + 1)) = (n)« exp(g2)/2 .

Hence, the factorial moment generating function (6.7) becomes :

q=0 q-

This series has a radius of convergence equal to 0, so it is always diverging even if all moments are perfectly well defined. In this case, there is no unique relation between the distribution and the corresponding moments. Consequently, several different distributions could have these same moments.

6.2 Some tools specific to the moment generating functions

Studying generating functions of probability distribution is quite analogous to working in the reciprocal space with Fourier transforms instead of considering the real space. Some features are then more clearly seen with these functions [H. S. Wilf (1990)].

6.2.1 Singularities of the moment generating function

Algebraic singularities of the moment generating function near the origin is the sign of the algebraic tail of the probability distribution [E. M. Hendriks et al. (1983)]. To show this result, we consider the function M(u) (6.1) for small negative u and we suppose that the distribution P(n) decreases like a power law :

P(n) ~ l/na

when n tends to infinite. Because of the normalization of the probability distribution, a must be larger than 1. On the other hand, we do not constrain the distribution by demanding the existence of certain moments, hence all values of a > 1 are allowed.

Some Tools Specific to the Moment Generating Functions 147

Writing M. for a small u as an integral :

1 l~°° xa~^ M(u) = _ . . / dx v ; T(a) J0 e*-« - 1

we get the expansion [R. M. Ziff (1977); H. Gould & K. Holl (1981)] :

M{u) = T(l - a ) ^ - 1 + ] T ^a~^u>

for non-integer value of a. The £ is the usual Riemann—C, function or its. analytical continuation for negative values of its argument [M. Abramowitz & I. A. Stegun (1964)].

For integer values of a, the expansion is a little bit more complicated :

M{u) ^L_ (_in(_u) + c + (a))+ £ j \

3 # " - i

with C is the Euler constant and ip the digamma function ip(z) = T'(z)/T(z) [M. Abramowitz & I. A. Stegun (1964)].

In both cases, the generating function can be expanded as a sum of an algebraic singularity with a logarithmic term in the case of the integer a, and of an entire function. One has then the correspondence :

M{u) ~ (-•u)a <=> P(n) ~ n " a _ 1 / r ( - a ) [a non-integer]

M.{u) ~ — ua ln(—u) <=> P(n) ~ a\/na+1 [a integer]

M{u) ~ -(-u)a ln(-u) <==> P(n) ~ n""a _ 1 l n n / r ( - a ) [a non-integer]

between the small—u singularity of the moment generating function Ai, and the large—n tail of the probability distribution P(n).

6.2.2 The Stieltjes series

The Stieltjes series constitute a general sub-class of the moment generating functions <S, as defined in (6.2), when the probability distribution has finite moments fxq of positive integer order.

The fundamental result making these series important is that if >So is a Stieltjes series, then it is the same for the function Si defined by the


identity :

$,(«)= *ff> • (6-17) 1 - S'o(0)uSi(u)

This means that there exists a probability distribution for which <Si is the moment generating function (6.2).

Relation (6.17) allows clearly to expand the generating function So as a continued fraction with all coefficients positive [G. A. Baker (1975)] :

5b(«) = 5b(0) / ( l - S&(0)Si(0)u/(l _ 5i(0)5a(0)u/ • • •)) •

The converse is also true. If <S has a continued-fraction expansion with positive coefficients, then it is the moment generating function of some probability distribution. This representation is very rapidly convergent to the limit generating function and this feature can be used efficiently to get an information about the probability distribution from first few moments [C. Bender k S. Orszag (1978)].

The problem of uniqueness of the probability distribution defined by the moments \iq is crucial for the application of the above scheme. A general sufficient condition has been found by Carleman [G. A. Baker (1975)] :

If

OO

X>,-1/a9 = ~ g = l

then fig are the moments of a unique probability distribution. Carleman's condition is rather general in Physics, but note that its proof

requires in principle analysis of higher order moments.

6.3 One example: the Poisson distribution

Each set of moments and generating functions has its special merits and most naturally describe certain distributions. Factorial moments and cu-mulants are usually defined with respect to the unconditioned (Poisson) process. Conditioning a sample by selecting events of a given multiplicity allows to separate out e.g. the correlations due to the non-Poissonian overall multiplicity distributions, but it introduces new correlations which can be regarded as unphysical because they depend on the way these events have

One Example: The Poisson Distribution 149

been selected. On the other hand, variation of correlations with the overall multiplicity in high energy collisions is very sensitive to the underlying dynamics.

Let us exemplify here the collection of moments for the Poisson distribution :

p(n) = M2e-<»> . n!

The moment generating function is :

M{u) _ -<n>( l -e") (6.18)

from which one deduces the moments, which appear to be polynomials of

< " > * • :

<n>

(n) + (n)2

(n) + 3(n)2 + (n)3

\ 2 , c / „ \ 3 , / _ \ 4

Ml =

M2 =

P3 =

/ i 4 =

M5 =

n) + 7(n)2 + 6(n)3 + «

n) + 15(n)2 + 25(n)3 + 10(n)4 + (n)5

The cumulant generating function is the logarithm of the function M :

InM(u) = -(n)(l - eu)

from which the cumulant moments write :

K0 = 0

(n> for any 9 ^ 1

The factorial moment generating function is identical to the moment generating function M. written for the argument In x :

0(1 + u) = e<n>u

and the factorial moments themselves are then :

£q = (n)q for any q > 0 .

*These polynomials (n») s Q?((n}) verify the recurrence relation : Qq+i(z) = z(Q'q(z) + Qq(z)), with Q0(z) = 1.


The logarithm of the generating function Q gives :

In 5(1 + u ) = (n)u

from which one derives the cumulant factorial moments :

/o = 0

h = (n) / , = 0 for any q > 1 .

In summary, the normalized factorial moments for the Poisson distribution are quite simple since :

Fq = 1 for any q

and :

Ki = 1

Kq = 0 for any q ^ 1 .

This feature makes this distribution very particular from the point of view of the factorial moments. Alternatively, one has the bunching parameter equal to r]n = 1 for any value of n > 2.

Finally, the combinants for the Poisson distribution are Rg = 0 for all q > 2, with Ri = 1 and InP(O) = -{n).

The same kind of exercise can be done when a particular distribution is expected. For example, if one focuses on the Bose-Einstein distribution :

P(„) = <">"

the generating function :

W^T)=1 + {n)u

appears quite simple and the set of corresponding moments can serve as a test for any experimental distribution. Following this line of reasoning, one can define quite generally :

1 _ v ^ iZq , 0(1 -u)-2^ q\

u

Infinitely Divisible Distribution Functions 151

with the normalized moments wq :

<n)°

( n ) i = 1

W2 = 2 - ^ n ~ ^ 2 (n)2 (n)

ro3 _ 6 6 ( n ( n - 1 ) ) | ( " ( " - W " - 2 ) ) (n)3 (n)2 (n)3

Non-vanishing moments roq for <7 > 2 indicate and quantify deviations from the expected Bose-Einstein law, according to the order q of these moments.

6.4 Infinitely divisible distribution functions

Particularly important in physical applications is the class of infinitely divisible probability distributions . These are the distributions for which the generating function Q (see the definition (6.6)) has the property that for all positive and real a, [£7(u)]a is again the generating function of some probability distribution. The fundamental remark about these probability distributions is that the function Q is the product of k identical generating functions :

g(u) = g1/k(u)x...xg1/k(u)

with integer k. Since the product of probability distributions corresponds to the distribution of the sum of independent random variables (c.f. Section 2.2.1), this means that if the distribution of a random variable X is infinitely divisible, then X could be considered as the sum of an infinite number (k —• 00) of identically distributed independent random variables.

The set of all infinitely divisible distributions has been entirely characterized [W. Feller (1971)]. The main result is that the characteristic function , which coincides formally with Q(e~tu) is equal to :

4>x(k) = e-fe2/2ff2 x exp / I exp(—iku) + ——- ) w(u)du

l + u :

with, in addition, a possible shift of the whole distribution. The weight


function w should satisfy some weak conditions* which are easily satisfied in physical situations.

It is instructive to comment each term of this characteristic function. The first Gaussian term is the characteristic function of a Gaussian distribution. The following term is the product of Poisson distributions with a distribution of the mean values (n) (according to (6.18)). The last term corresponds to a certain variable. One can prove more rigorously this intuitive result : the random variable X whose distribution is infinitely divisible, corresponds to the sum of a certain variable, a Gaussian variable and of an infinite number of independent Poisson variables. The stable laws, as discussed in Chapter 2, are particular cases of infinitely divisible laws. To this class belong Poisson, Gaussian, Pascal and log-normal discrete distributions. In general, all distributions P(n) corresponding to the Stacy distribution (see Table 5.1) with the scaling exponent 0 < 7 < 1 are also infinitely divisible.

Other characterizations are known for this class of probability distributions. One alternative is the following. If G(u) denotes the moment generating function, then it can be written as (6.15) :

00 0 0

ln0(u) = lnP(O) + 5 3 R , u « , 0(1) = £ ) P ( n ) = 1 . g = l n = 0

For infinitely divisible distribution one has :

00

^ R , = - l n P ( 0 ) < o o , R „ > 0 . g = l

This means that the probability of not detecting any particle does not vanish. The generating function for these distributions takes then the simple form :

a ( « ) = e 3 c p f f ; R , ( u « - l ) J .

/.From this and relation (6.15), follows useful relations between count prob-

*w is positive, regular and tends to 0 when u —• ±00, whereas (u2) exists and is finite.

Composite Distributions 153

ability ratios P(n)/P(0) and combinants [I. Szapudi k, A. S. Szalay (1993)]

P W - i f ^ T ! P(n~g) p(°) nh q p ( ° )

R _ P ( ? ) i y „ Rp ( g - P )

The combinants of all ranks are all non-negative if and only if the probability distribution is infinitely divisible.

About this class of probability distributions, one can make an intriguing remark that most of the successful phenomenological distributions used in the analysis of various count probability distributions such as the distribution of galaxies, distribution of hadrons in high energy collisions or distribution of photo-counts, are infinitely divisible. Even if the deep reasons are missing, it is worth mentioning this result because it guides our intuition towards some specific distributions, such as the stable distributions or the Poisson transforms.

6.4.1 Truncating the multiplicity distribution

We have shown before that the combinants Rq for infinitely divisible distributions are positive definite. Because of (6.16) this is the same for the factorial cumulant moments fq and for the corresponding normalized moments K.q.

For a truncated distribution, one cannot expect the distribution to remain infinitely divisible, and this means that at least one Rq should be negative. Kq are then not all positive, while the normalized factorial moments Fq remain positive by definition (6.5). Therefore, their ratio (6.13) may change the sign [I. M. Dremin et al. (1994)]. This provides not only an easy check of the infinite-divisibility but also helps to understand consequences of truncation of the experimental distribution.

6.5 Composite distributions

Generally, the observable probability distribution is the result of a sequence of more or less independent events, corresponding often to different physical processes. An understanding of the physical mechanism which gives rise to


the observed probability distribution requires then a separate understanding of contribution of different elementary distributions. This brings us to the study of composite distributions and the conditional probabilities.

6.5.1 Conditional and joint probabilities

Let us first consider the distribution which is composed of two populations " 1 " and "2", with n = n\ + n^. From the probabilistic point of view, the random variable n is the sum of the two random variables n\ and n,2 and one knows that if n\ and n-i are uncorrelated then the probability distribution of n is the convolution product of the distributions of n\ and 7i2 as written in (2.10).

One defines then the conditional probability P(n |m) for finding n under condition that n\ is known, and the joint probability P(ni,n2) to get at the same time the values «i and n^. It exists a relation between these probabilities, which writes generally as :

P(ni,na) = P(n)P(ni|n)

where

P(n)= Y. p("i>»2) n l . " 2

nl"t"TV2==n

and

P ( « i K ) = P(ni,na) ( J ^ P f r i . n a ) J

The natural choice for P(nj|n) is the binomial distribution :

P(«i |n) = - ^ 7 P " 1 ( 1 - P ) T 1 2 (6-19) n\\ni\

where n = n\ + n^. This corresponds for example to the case where some physical process generates n particles, but only n\ can be observed because the experimental device is not perfect and cannot detect the particles of species "2". In this situation, n\ is the observable and one asks if one can get information about the physical n-distribution. The probability p in the above distribution is the probability for a created particle to belong to


species " 1 " . This binomial choice is a good approximation for a given n in the cell if particles are weakly correlated, i.e. if the correlation length between particles is short. The independence of the partitioning of the members as implied by (6.19) does not mean of course that ni, n2 are uncorrelated in P(ni , n2).

Let us discuss the case when P(n) belongs to the class of Poisson transforms and P(ni\n) is binomial. Many analytical results can be derived in this case. Suppose that we are interested in the probability of observing particles " 1 " when those of type "2" are not observed. In addition, let us suppose that that the n-distribution is the Poisson transform (5.1). One can write :

P (m)u n i = ^ P ( n ) u n P ( n 1 | n ) u - n 2 .

Summing over all possible values of n i , one gets the factorial moment generating function Q$u) of the "1"-distribution :

Qiiv) = Q(pu + l-p)

where Q(u) is the corresponding generating function of the n-distribution. This shows clearly that the probability distribution of n is quite similar to the "1"-distribution. For example, if n-distribution is Poissonian with the mean value (n), then ni-distribution is also Poissonian with the mean value (nj) = p{n). Another example is when n is given by the Pascal distribution with the cell parameter k and the average multiplicity (n). Then P(ni) is also the Pascal distribution with the same value of k and the average multiplicity (n$ =p(n) [P. Carruthers & C. C. Shih (1984)].

6.5.2 Clan structures

Let us suppose that observed particles arise from a two step sequential process in which the production of c clusters (clans) with probability P(c) in the first step is followed by decay of each cluster (j) independently into n,j particles. In this case :

c

P ( n i , n 2 , . . . , n c ) = P(c) J J P ^ K ) .


The probability of observing n particles in the whole system is :

P(n) = £P(c)£f[P(i)(ni) , 5>,-=n • C Uj j = l j

The composite generating function Q{u) is therefore :

gin)=^p(c)^n^Po)K)=£p(c) n ^ M C Uj j = l C j = l

where Q^ is the generating function for the cluster j . If one assumes the same probability distributions for all clusters, the index j in the generating function can be omitted and one writes the product appearing in the preceding equation as the c-th power of the common generating function

YlUi £o')(u) = £c(u)- T h i s l e a d s t o :

G{u) = g{c){Giu)) (6.20)

where Q^ denotes the generating function of the cluster production :

a(c)(«) = £ p ( c K • c

iFrom the identities (6.9) and equation (6.20), one can deduce the systematic relations between moments :

(n) = {nc){np)

(n2) - (n)2 = «n2) - (nc)2)(np)2 + «n 2 ) - (np)2)(nc) (6.21)

where n c denotes the number of clans and np is the number of particles per clan.

The idea of clans can be found already in the work of Anscombe (1950). Making reference to biological systems Anscombe remarked : "If colonies or groups of individuals are randomly distributed over an area in a Poisson distribution we obtain a negative binomial distribution for the total count if the numbers of individuals in the colonies are distributed independently in a logarithmic series distribution * ". Clan concept has been rediscovered in the high energy multiparticle production [L. Van Hove k, A. Giovannini

"i.e. the Pascal distribution (5.15)


(1987)] in order to explain the wide occurrence of the approximate Pascal distribution.

A clan is then a group of particles of common ancestor and each clan contains at least one particle. Clans are assumed to be produced independently. Moreover, one obtains the Pascal distribution whenever the distribution of particles emitted in an average clan is the particular Pascal distribution (5.15) for the parameter k ~> 0 and (np)/k —¥ est :

P(p)(np) = (np) n

and the Poisson distribution of independent clans :

P(nc) = MV<nc> . nc!

In this case, the moment generating function of the composition of the two distributions is :

gW=(i + (i-u)JL^J -K><nP>(l~/3)//3

which is the generating function of the Pascal distribution (5.10) with the parameter :

k = (ne)(np)(l-0)/0 .

In this interpretation, the Pascal distribution of observed particles is the compound of a Poisson distribution and a limit Pascal distribution. The parameter 1/fc can then be interpreted as a measure of aggregation of particles into clans, since it corresponds to the ratio of probabilities to have two particles in the same clan Pm(2) and two particles in two separate clans P ( l ' ) ( 1 ) ' P f i "^ 1 ) . respectively :

1 _ P(i)(2) k P ( 0 (1)P ( 1»)(1)

Moreover, the parameter 1/fc corresponds to the second normalized cumu-lant (6.4) : 1/fc = 72, i.e. to the two-particle correlations . Following this reasoning, one could think that the clan structure is a source of intermittent fluctuations in high energy collisions [L. Van Hove (1989)] (c.f. Chapter 7).


6.6 More about the Pascal distribution

Let us continue the discussion which started in Section 5.2 of properties specific to the Pascal distribution . The moment generating function for this distribution is (5.10), and the corresponding cumulant generating function is therefore :

InM(u) = -kin ( l + (1 - eu)^-

Expanding this function as a series of the current variable u and identifying coefficients of uq/q\ gives the normalized cumulants :

7i = 1 1 1

72 = 7 + 7 T k (n) 2 3 1

73 = To + 77-7 + k2 k(n) {n}2

7 4 _ fc3 + k2{n) + k(rf* + (rt>3 ( 6 - 2 2 )

24 60 50 15 1 75 = TT + 77777 + TTT-To + 77777 + fc4 fc3(n) k2(n)2 k(n)3 (n)4

One should mention that this result about normalized cumulant moments is a particular case of the general relation between the normalized cumulant moments of P(n) in the class of Poisson transform distributions. To see this point, one defines first the moments jq as the coefficients of the formal series :

/ • O O

\nM{z)(u) = ln / f(z)ezudz Jo

9

The first moments write :

7i :

72 :

73 :

= 1

= <*2> -

= (z3) -

-1

- 3 ( 2 2 ) + 2

More about the Pascal Distribution 159

7 W = (z4) - 4<z3) - 3(z2)2 + 12<z2) - 6

7 t o = (z5) - 5{z4) - 10<z3)<z2) + 20(z3)

+ 3 0 ( z 2 ) 2 - 6 0 ( z 2 ) + 2 4

where the z-moments are defined by (zq) = / zqf(z)dz. The normalized cumulant moments of P(n) can then be expressed in terms of these z-moments of their Poisson weight :

7i

72

73

74

75

These relations imply several useful inequalities for phenomenological applications.

6.6.1 The limiting forms

For a fixed parameter k, the first-scaling plot of the Pascal distribution is shown in figure 6.1. It converges to the limiting Gamma distribution in a characteristic way, namely, the discrete form of the Pascal distribution in the high—z tail region approaches the limiting form from above. This is a general behaviour for the Poisson transforms. The same is true for the small—z tail, but this could depend on the Poisson weight . Similar behaviours are seen in figure 5.1. The deviations between these two distributions are not small, telling that the fluctuations in P(n) are rather large.

As the k parameter increases for a fixed (n), the Pascal distribution is getting narrower and the maximum shifts towards smaller z. In the limit (n) —> oo, the Pascal distribution reduces formally to the Poisson

= 1 to , 1

_ (z) (z) 3 1

~73 +72 Wj + W ~ 7 4 + 7 3 ( n ) + 7 2 (n)2 + (n)3

- - t o + ,vto i 2 . + - t o _^_ , A') _HL - 7 5 + 7 4 { n ) + 7 3 ( n ) 2 + T 2 ( n ) 3

+ (n)3


<n>P„

10"

10 -3

10"

Fig. 6.1 The double-logarithmic plot of the Pascal distribution (5.12) for fc = 2 and different mean values (n). The histograms are for (n) — 4, 8, 16, 32, 64, 128, 256. The larger the value of (n), the closer is the histogram to the scaling form f(z) = iz exp(—2z), which is shown in the figure as the thick continuous line. Note that on both ends, i.e. for large—z and small—z tails, the scaled Pascal distribution approaches the scaling function from above.

distribution for which

7? (n)9- i

In this case, the normalized cumulant moments of Poisson and Pascal distributions differ for q > 2 by terms of order (2 9 _ 1 - l)(n)2_«/fe. The coefficient 2 9 _ 1 — 1 in front of higher order terms increases fast with q, so the convergence is slow.

On the level of factorial cumulants one finds : /Ci = 1 and Kq = 0, 5 ^ 1 , for the Poisson distribution, whereas for the Pascal distribution one finds :

K„ ( g - 1 ) ! <7>1

which goes to infinity when q increases.

More about the Pascal Distribution 161

6.6.2 High-energy phenomenology

The Pascal distribution in the context of high energy physics has been put forward first in cosmic ray physics by Mc Keown and Wolfendale (1966). Full phase space studies have shown that both (n) and k depend on the primary energy EQ :

(n(E0)) ~ El

k(Eo) ~ 0.4(1 - e-1-8 1 0"4 £°)

At E0 = 30 GeV, (n) ~ 4 and k ~ 465, whereas at E0 = 104 GeV one finds (n) ~ 18 and k ~ 3. Onset of correlations is controlled by the parameter 1/fc which is very small at lower energies, meaning that the Poisson multiplicity distribution is a good approximation. The general tendencies seen in the cosmic ray physics have been later confirmed at the accelerator energies [A. Giovannini et al. (1977)]. More detailed investigation by UA5 Collaboration in pp collisions at the center of mass (cm.) energy \/s =200 GeV, 546 GeV and 900 GeV has shown that multiplicity distributions in full space and in symmetric rapidity intervals are quite well fitted by the Pascal distribution. The problems begin first to show up at the highest energy for which a shoulder is visible in the experimental distribution.

Analysis of the available data in the framework of the clan concept have shown several regularities. First of all, the average number of clans (nc) increases linearly with the rapidity cut yc at a fixed cm. energy and as yc approaches full phase space, (nc) is converging to a constant value. Secondly, (nc) in a fixed rapidity interval is independent of energy in the region of linearity mentioned above. Finally, (nc) is larger in e+e~ than in hadron-hadron or lepton-hadron reactions, whereas (nc) is smaller in e+e~ and lepton-hadron reactions than in hadron-hadron reactions. The "shoulder problem" showed that analyzing the data, one has to examine more carefully different topologies of jets. The shoulder problem in e+e~ annihilation results from the superposition of jets of different topologies and the phenomenological regularity of Pascal distribution is approximately restored in the sample of events of the same type.

Chapter 7

Exclusive and Inclusive Densities

In this chapter we shall give the essential definitions and concepts for the multiparticle distributions and correlations. The correlation observables are very important in the present studies of high energy collisions and nuclear collisions at intermediate energies. The recently proposed methods of studying these correlations using the scaled factorial moments, made possible a considerable progress in this field. The methods to study these distributions at small scales, i.e. the scaled factorial moments and the correlation integrals, are presented and summarized in this chapter.

7.1 Generalities and variables

We shall be interested in the studies of the distribution of particles or fragments produced in collisions of elementary particles or nuclei. Most frequently the correlation studies are performed for one particle specie, should it be charged particles, pions, kaons etc. It is straightforward to generalize it for the case of several particle species, such as the opposite charge particle correlations or flavour correlations.

The total inelastic cross section 07 can be written as a sum of the cross sections an for the production of exactly n particles in an event :

0 0

n=0

Furthermore, one can define the probabilities P(n) = crn/o'i of observing n particles in an inelastic event. In order to study the distributions of particles

163

164 Exclusive and Inclusive Densities

in phase space, one has first to define probabilities of observing particles at given points in the phase space. Generally, the 'position' (momentum) variable can be one or many dimensional. Most often the variables are the rapidity, azimuthal angle and the transverse momentum or any combination of these. For the definition of the longitudinal and transverse variables it is often required to perform a rotation, so that the longitudinal variables are denned by the sphericity or thrust axis and not by the beam axis. This change removes from the fluctuations studied, those due to different orientation of the event axis with respect to the beam axis. The invariant Q2 distance between particles and the mass difference for the fragments distribution can also be used as the variable in which the multiparticle distributions are studied.

The one-particle distribution is the probability of observing a particle at a given point j/i , irrespective of the number and positions of other particles :

P1U/1) = — T~ •

An experimental estimator of the one-particle distribution is :

Pi (yi) = \Y^Kxk ~ Vi)j

where Xk are the positions of the observed particles and the brackets mean the average over the events. The experimental resolution induces in most cases some minimal binning in the measured variables. Then, the information accessible to the observer is the distribution integrated over some phase space cell u>i of the size of the experimental resolution :

- - f W< JUi

pi{y)dy

This is due to the fact that in a given event one only knows the number of observed particles in each of the phase space cells of minimal size. This leads naturally to the study of multiplicity distributions and their moments in different phase space cells.

The information about n-particle correlations is contained in the n-particle distribution function pn{yi,- • • ,Vn)- This quantity denotes the probability density of observing n-particles with momenta yi,...,yn, irrespective of the number and positions of other particles. These distribution

Generalities and Variables 165

densities are related to the n-particle inclusive cross sections :

1 dna (7/ dy1 • • • dyn

The estimator of the n-particle distribution is :

Pn{Vl, • • • ,Vn) = ( Yl 5(yi ~ Xh) • • • 5(Vn ~ Xjn)\ .

The exclusive distributions P (n ;y i , . . . ,yn) describe the distributions when the multiplicity is exactly n. These two are related by :

Pn{V\, • • • ,Vn) = P(n; j / i , . . . ,y„)

+ YI1 p ( n + J ; yi' • • • > 2/™. yi • • • > v',) I I dy'i

and

P(n;yi,...,yn) = pn{yx,... ,yn)

+ ^i-1)3 — / Pn+j(yi,•••,yn,y'i---,y'j)Y[dy\ j=i 3' Ju *=i

In particular, the void probability P(0) (the probability of the event with the zero multiplicity) is :

00 1 » n

P(0) = l -53- /p (n ;y [ > . . . , y ; ) J Jdy ; . n=l n - ^ i = l

The above results can be expressed also in terms of generating functional [N. N. Bogoliubov (1946); L. S. Brown (1972); K. J. Biebl & J. Wolf (1972)]:

00 1 f n 1 G(u(y)) = 1 + ^ "1 / Pn{n\yi,...,yn)u{y1)---u{yn)Wdy'i

n = i n ' 7 n i = i

= a(l + u(i/)) (7-1)

with u{yi) {i = l , . . . , n ) being the arbitrary function of yi in fi. pn(yi, • • • 1 Vn) and P(n; j / i , . . . , 2/n) can be recovered by a suitable functional differentiation of Q and Q{u) taken at u = 0, respectively.


The n-particle distribution can be written using the n-particle correlation function :

Pn{yi,- • •, yn) = pi(yi) • • • pi{yn) + Cn(yi, ...,yn)

and one can define the reduced correlation function :

cn(yi,---,yn)= , , -,—r . (7.2)

A large amount of data exists for the two-particle correlation function C2. This function was studied for the like-sign particles in order to extract information on the size of the emitting region from the Bose-Einstein correlations [W. A. Zajc (1991)].

7.2 Cumulant correlation functions

The n-particle correlation function consists mainly of statistical combination of lower order correlations. In order to study genuine n-particle correlations, one has to define the n-particle cumulant Kn(yi,..., yn), which enters into the expression for pn together with cumulants of the order lower than n. The first few densities are :

pi{yuyi) = Pi(yi)pi(y2) + K2(yi,y2)

(03(2/1,2/2,2/3) = Pi(yi)pi(y2)pi(y3)

+ ^2pi(yi)K2(yj,yk) + K3(yi,y2,y3)

M2/i>2/2,2/3,2/4) = p\{yi)p\{y2)pi{yz)pi{yi) + ^2lpi{yi)pi{yi)K2{yk,yi)

+ 'YjPi{yi)Kz{y3,yk,yi) + ^K2(yi,yj)K2{yk,yi)

+ K4(y1,y2,y3,y4) (7.3)

where the sums are taken over all permutations of the set {yi,..-,yn} without the transposition inside the factors of the sums. Notice that : £2(2/*, yj) = K2(yu yj). In general :

Pn(yu---,yn)= J2 Yl^1^"'P1^(h)\K^r)---K2{;-)}(i2) {U}nP

erm

•••[Kn(.,...,.)...Kn(.,...,-)]{ln) (7.4)

Cumulant Correlation Functions 167

where the subscripts (k) (i = 1 , . . . , n and U = 0,1,2, . . . ) mean that there is ^i-terms in the product of pi's, /2-terms in the product of .ftVs, etc., and YA=I ^* = n- The ^-arguments in Ki are filled by n possible moments {2/1, • • •, 2/n} taken in any order. The sum over permutations in (7.4) is a sum over distinct way of filling these arguments. There are precisely :

n! (l!) 'i(2!)^...(n!)'W1!Z2!.../n!

terms for any factor product. The relations (7.3), (7.4) can be also inverted :

#2(2/1,2/2) = £2(2/1,2/2) -P i (2 / i ) / ? i (2 /2 )

#3(2/1,2/2,2/3) = £3(2/1,2/2,2/3) - £ P i ( 2 / i ) P 2 ( 2 / 2 , 2 / 3 )

+ 2/>i(j/i)p(2/2)Pi(2/3)

#4(2/1,2/2,2/3,2/4) = P4(2/i, 2/2,2/3,2/4) (7 .5)

- X ^ 1 ^ 1 ) ^ 3 ^ 2 ' 2/3,2/4) - X]P2(2/1,2/2W2/3,2/4)

+ 2 ]Tpi(2/iVi(2/2)P2(2/3,2/4)

- 6pi(yi)pi(y2)pi(y3)pi(yi)

The sums in (7.5) are taken over all possible permutations of { j / i , . . . , yn} without the transposition inside the factors of the sums.

When the parametric function u(y) is replaced by the constants u, then the reduces to the factorial moment generating function (6.7) for the multiplicity distribution :

00

g{u) = Y, p(n)(l + UT = SO- + «) n=0

= i + £ ^ / M 2 / i , . . . , 2 / g ) r K = i + £ > 9 (™) g=l q' jQ i = l q=l q'

where £q are the factorial moments, as defined in (6.5). The n-particle cumulant measures the statistical dependence of the

whole n-particle set. The n-particle cumulant is zero, if anyone of the n particles is independent of the others. The second order cumulant is equivalent to the two-particle correlations, but at the three (and higher)


particle level the study of the cumulant distribution requires the subtraction from three-particle correlation function of the combinatorial part from the lower order correlations.

7.3 Scaled factorial moments

The studies of the multiplicity distribution cannot show the structure of the correlations between the momenta of the produced particles. In the independent particle production, the probability of producing a particle does not depend on the fact whether and how many other particles are produced. On the contrary, if some correlations are present then the production of q particles enhance the probability of the production of the particle (q + 1). As a result, the multiplicity distributions are broader than the Poisson distribution. This gives an information on the global number of produced particles, i.e. on the integrated correlation functions and not on the correlations between particles with definite momenta. The integration of the n-particle distribution over a domain Cl of the phase-space gives the factorial moments of the multiplicity distribution in that domain [A. H. Mueller (1971)] :

(n(n - 1) • • • (n - q + l))n

and correspondingly the scaled factorial moments :

P _ Ldyi---JndyiPg(yu--->yg)

{hdy pi(y))

If the one-particle inclusive distribution is approximately constant, i.e. in the 'plateau region', we can rewrite the above relation as :

F q = ~& d y i ' " d y i d i ^ • • • ' » « ) (7-8)

= / dyi pi(yi)

/ dyx--- / dyq pq{yx,... ,yq) Jn JQ

(7.7)

Scaled Factorial Moments 169

where the dq represents the g-particle reduced density :

4 , ( y i , - , y g ) = f ^ 1 ' - - ' ^ . . (7.9) Piivi) • • • pi(yq)

The use of the reduced density is common in phenomenological parametriza-tions of the g-particle distribution in the cases where the one-particle distributions are factorizing.

7.3.1 Intermittency with the scaled factorial moments

Bialas and Peschanski (1986) proposed to study the dependence of the scaled factorial moments on the resolution in the rapidity. The idea was to study the structure of the particle density in rapidity in the high energy event. They showed that the scaled factorial moments averaged over M bins of width 6y in the total rapidity window of length A y = MSy correspond to the moments of the probability density in these bins :

^^1...^^MP(x1)...,^)^fE^) = ( ( M D M ^

N(N - 1) • • • (AT - q + 1) { }

where rij is the number of particles in the bin j in the event of the total multiplicity N in the whole rapidity interval A y , Xj = Mpj, and pj is the probability to find a particle in the j-th bin. An average of the right-hand side of the above equation over a large number of events with fixed total multiplicity N should converge to the left-hand side.

The multiplicity distribution in different bins Pmui(ni,..., UM) is a convolution of the probability distribution P and the multinomial distribution :

N\ Pmu*(ni,. - - , « M ) m\---nM\MN

/•OO /-0O

x / dx\--- I (LXMV{XI, ... , X M ) I " ' Jo Jo

VM

Thus, the factorial moments of the multiplicity distribution in different bins give the moments of the probability distribution P. The intermittent


behaviour * is denned as a power law dependence of the scaled factorial moments on the number of bins :

Fg~(Mr* = ( — ) ~(Jy)-"«

where vq is called the intermittency exponent of rank q. The intermittency postulates scaling of the multiparticle spectra.

This analysis was applied to the JACEE event [T. H. Burnett et al. (1983)], suggesting that the rapidity density fluctuations are of a non-statistical origin [A. Bialas & R. Peschanski (1986)]. The scaled factorial moments can also be averaged for a sample of many events. This allows to perform similar studies also in low multiplicity events such as discussed in e + e _ or hadron-hadron collisions.

The scaled factorial moment is calculated for each event in a definite binning, i.e. for each event the sum over all the M bins is taken, and then the average over all events is taken. Generally, the events in the sample have different multiplicities so the normalization N(N — 1) • • • (N — q + 1) in equation (7.10), which accounts for the Bernoulli's character of the statistical fluctuations around the studied probability P, is replaced by {N)q

where (N) is the mean multiplicity in the sample of events. This gives the horizontally averaged scaled factorial moments [A. Bialas & R. Peschanski (1988a)] :

M 9 _ 1 M

Fq = 7M7" X ^ ' ( n J - ! ) ' •' ("i - 9 + 1)> j = i

where (• • •) means an average over events. The horizontally averaged scaled factorial moment measure also the

fluctuation of the total multiplicity. In this case, Fq(AY = 1) ^ 1 and gives the moments of the total multiplicity. These fluctuations arise from the integration of the correlations over the whole phase-space.

*We shall use the name intermittency signal or intermittency pattern to describe the increase of the scaled factorial moments with the resolution, but not necessarily a power law. We shall also call an intermittency signal a stronger one, if the corresponding local slopes of the dependence of the scaled factorial moment on the resolution are bigger.


7.3.2 Correcting for the shape of the one-particle distribution and the lack of the translational invariance

Generally, even if the particles are uncorrelated and all correlation functions are such that Kn = 0, the scaled factorial moment contains a spurious dependence on the scale due to the shape of the one-particle distribution p\{y). So the horizontal scaled factorial moments should be corrected for this dependence by a factor [K. Fialkowski et al. (1989)] :

Mi-1 M

j = i

The corrected scaled factorial moments Fq/Kg are less biased by the variations in the single-particle spectrum. Therefore, Bialas and Gazdzicki (1990) proposed to look at the fluctuations in the variable :

y("-fir*«tf> <7'11)

where 0 < Y(y) < 1 . This change of variables before an intermittency analysis is especially important for rapidly changing pi, such as for the transverse momentum distribution, where this procedure was shown to better follow the true intermittent correlation than the corrected horizontal analysis. In the following, if not said explicitly, we shall use the notation Fq

for the scaled factorial moments corrected for the shape of the one-particle distribution.

One can also define the vertically averaged scaled factorial moments [A. Bialas & R. Peschanski (1988a)] :

p = J_ y - ( W j ( n j - l ) - - - ( n j - q + 1))

M j ^ (nj)"

This corresponds to the average of the scaled factorial moments calculated in each of the M bins over all bins. This form of averaging is equivalent to the previous one for the case of a flat one-particle distribution. Usually, the data show little difference between the vertical scaled factorial moments and the corrected horizontal scaled factorial moments .

From the theoretical point of view, it is often easier to relate vertical


scaled factorial moments to integrals of the g-particle distributions :

„ ,<• x _ J _ v - Jymin+(j-i)Sy ayi Jymin+U-i)Sv Vq p^yu •••>yi>

M 3 = 1 Uymin + U-l)Svdy Pl(y»

For slowly varying pi(j/), the scaled factorial moments can be written using the reduced densities as follows :

i i M ry-min+jSy rVmin+jSy

F q ^ = T^MluYl / dyi • • • / dyi d«(yi> • • •' y«"> {dy)l M ^{Jymin+U-l)8y Jvmin+U-l)Sy

Generally, one assumes translational invariance of the reduced densities and, consequently, of the scaled factorial moments in different bins. The reduced densities depend then only on the relative variables yq — yj so that one has :

i r$y r^y Fq{Sy) = Jtyfi I dyi'"L dy^Vl'---'y^ • (7-12)

The comparison of the horizontal scaled factorial moments to the phe-nomenological parameterization of the reduced density is difficult because the contribution of different bins is weighted by the single-particle distribution pi(y) for that bin. The vertical scaled factorial moments have the contribution of each bin scaled so that all of them enter on equal footing to the sum over the bins. For sufficiently small bins, they are closer to integrals of the reduced density (7.12) than to the horizontal moments even if the one-particle distribution is not flat. In the next sections we shall extensively use the relation (7.12) between the scaled factorial moments and the reduced densities, assuming a simple and approximately translationally invariant parameterization for the reduced densities.

7.3.3 Unphysical correlations due to the mixing of events °f different multiplicities

It is expected, at least in the rapidity variable, that the correlations have two parts : a long range part describing the total multiplicity correlations and a short range part describing small scale behaviour of the production process [UA5 Coll. (1987)]. Thus, the two-particle reduced density (7.9) can be written as :

d2(Ap) = 1 + Cir(Ap) + Csr(Ap)


where the long range correlations C/r are a smooth function for Ap ->• 0, e.g. Cir(Ap) oc exp(—Ap/£), and the short range correlations Csr can have very strong dependence for small scales, e.g. the power law Csr(Ap) oc (Ap) - " . It is expected that the short range power law behaviour can be extracted by fitting the scaled factorial moment by the expression [K. Fialkowski (1991)] :

F2{u) = 1 + B + C u r ' 2 .

One should notice however that it is not a priori obvious that the short range correlations describe mainly correlations induced by the production process and are well separated from the long range correlations describing mixing of events of different multiplicities, for example due to the mixing of different collision impact parameters in the event sample. Generally, if events in different classes have very different one-particle distribution, then mixing of different types of events could induce strong short range correlations, even though each particular class of events does not show any increase of the fluctuations at small scales. An example could be the mixing of two types of events characterized by two exponential transverse momentum distributions with different slope parameters. Even though each class of events has no correlations and all events are similar to the particular average in one of the classes, the average of the mixed sample is very different from any typical event of a particular class. The whole sample would show fluctuations which do not describe the fluctuations occurring in any production process but which are due to the mixing of different types of events, each with no fluctuations. This problem shows that in order to understand the structure of correlations, it is important to measure the scaled factorial moments in separate bins without bin averaging, and also some bin-to-bin correlations such as the scaled factorial correlators.

7.3.4 Dimensional projection

It was observed by Ochs and Wosiek (1988) that the two-dimensional in-termittency analysis in rapidity and azimuthal angle gives a stronger in-termittency signal than the one-dimensional analysis in the rapidity alone. This was interpreted as an effect of the pencil-jet structure in the branching model. Later Ochs (1990) showed that the effect is quite general and it is to be expected that the one-dimensional projected distributions have no longer scale-invariant fluctuations, even though the system possess such a


structure in its full-dimensional version. Bialas and Seixas (1990) observed in a phenomenological model of singular multiparticle distributions in three-dimensional momentum space that the effect of the dimensional projection depends on the range of the transverse momenta over which one averages the fluctuations. Similar conclusions have been drawn also in the model of phase-transition for the nuclear collisions [P. Bozek & M. Ploszajczak (1990)].

Our discussion will concentrate on the second scaled factorial moment, for which it is particularly easy to relate its behaviour with the singular structures of the two-particle density £2(^1,^2) (see equations (7.8), (7.9)), where f i , r% are some n-dimensional variables of particles " 1 " and "2", respectively. For slowly varying pi(r), one gets for the second scaled factorial moment F2 :

1 M" r f

i = 1 JSli JSli nsn where the n-dimensional volume fi is divided in Mn cubes fi; of volume SCI. The intermittent fluctuations in the density distribution or a power law behaviour of the factorial moment is equivalent to the existence of singularity in the reduced density d2. In the following we shall assume, that this singularity is dominating and hence, in the calculation of F^1' we shall omit all non-singular terms in d2. This approximation yields the power law behaviour of the scaled factorial moment and not of the scaled factorial cumulant. Let us consider two translationally invariant densities with different parametrizations of the singularity :

• The factorized parameterization :

4fac\xuXux2,X2) = - • % -, „ n (7.13) | z i - X 2 h \Xx~X-i | ( » -1 )^

where fl = [xi,Xi] • The isotropic parameterization :

4 i a o ) ( n , r 2 ) = , , C* (7.14) ri - r 2

where Ci is some constant. Results do not change if one rescales one of the components of r*in (7.14), allowing in this way for non-isotropic shapes of d2. In this case, by a suitable coordinate transformation which changes


only the integration domain CI, one can bring the singularity back to the isotropic form (7.14). This latter modification is negligible for small bins, and hence the behaviour of the factorial moments is similar in these two cases. Thus, it does not change the qualitative behaviour of the scaled factorial moment for the projected distributions, i.e. the flattening of the scaled factorial moments for small bins in rapidity. However, the variation of the integration region fi changes the resulting effective slope in the one-dimensional intermittency analysis and can be analyzed by the finite-size scaling method [P. Bozek k. M. Ploszajczak (1992a)] (c.f. Sections 9.4.1 and 9.4.2).

Both factorized d2 and isotropic d2 types of singularities, as defined in (7.13) and (7.14), give a power law behaviour for F2 with the exponent v2. However, they have a different behaviour in the (n — 1)-dimensional subspace X. The (n — l)-dimensional scaled factorial moment can be obtained from the density d2 by integrating over Xi :

M " _ 1

f f - 1 ) ( & ; ) = - ! - V / dxl f dx2 f d*! / dX2d2(x1,Xl;x2,X2)

The factorized singularity d% (equation 7.13) gives a power-law dependence also in (n — 1) dimensions :

Fkn~X)(Su>)~j£(8u)v' .

On the contrary, the reduced isotropic density has no more a singularity in X variables and, consequently, it does not show a power law behaviour of the scaled factorial moment. In particular, in the two dimensions such as for the (y—^-distribution, the one-dimensional scaled factorial moment in the limit SX —*• 0 can be written as :

if> {5X) = j ^ j dXd2{X) . (7.15)

where d2(X)= fAxdxi fAxdx2 d2(xi,x2,X) and X = X\ — X2 . The intermittency exponent at the scale SX is given by :

dlnF^(SX) "2(SX)- dlnSX • ( 7 ' 1 6 )


After a substitution of (7.15) into equation (7.16) one obtains :

SX d2(6X)

f0XdXd2{X)

For SX -» 0, the intermittency exponent v2 (SX) tends to zero if the density d2(X) has no singularity at X = 0 . Using the similar arguments as above, one can argue that for finite bins 6X one should observe a flattening of the scaled factorial moment.

^From the above argument it is clear that an even stronger effect of the dimensional projection is observed when projecting a three-dimensional singular density. In this case, the flattening of the one-dimensional scaled factorial moment is stronger and the effective slopes are smaller than the ones resulting from a two-dimensional to one-dimensional projection. The experimental results confirm that the strongest intermittency exists in the three-dimensional analysis, and that the intermittent signal decreases with the number of the effective projections of the multiparticle distribution [CELLO Coll. (1991); KLM Coll. (1989a)]. This could be taken as an evidence for the non-factorized type of singularity in the reduced density.

7.4 Scaled factorial correlators and bin-split moments

The scaled factorial moment in a rapidity window Sy is equivalent to the ^-particle distribution function integrated from the scale 0 to Sy. This makes difficult to disentangle a true scaling behaviour of the multiparticle distributions in a certain range of rapidity separations from other effects, which could be present at the limiting scales. This concerns especially the behaviour of the g-particle distribution for particle separation going to 0. The presence of some limit on the scaling behaviour in this region, can change dramatically the dependence of the integrated n-particle distribution on the upper integration limit Sy [A. Bialas & K. Zalewski (1989); P. Bozek & M. Ploszajczak (1991b)]. On the other hand, the limit Sy -> 0 tests only a limited part of the q-particle phase-space (| yq — yj |< Sy -> 0). To cure this disadvantage of the scaled factorial moments, Bialas and Peschanski (1988b) proposed to study the scaled factorial correlators which are the observables relating the fluctuations in separated bins. The scaled factorial correlator Fij for two bins of width Sy separated by the distance

Scaled Factorial Correlators and Bin-Split Moments 177

L is defined as :

F (Sv L) = ( n i ( n i - 1) • • • (m - g + l)n2(n2 - 1) • • • (n2 - g' + 1)) 9'q'(y' ( n i ( n 1 - l ) . . . ( n 1 - g + l ) ) . . ( n 2 ( n 2 - l ) . . . ( n 2 - 9 ' + l)>

where ni (respectively : n2) is the number of particles in the first bin (respectively : in the second bin). Scaled factorial correlator defined in this way, is then averaged over all pairs of bins distant by L in the rapidity window A y , what corresponds to the vertical averaging. The scaled factorial correlators in the random cascade model (c./. Chapter 9 for the discussion of the a-model) are independent of the bin size Sy and exhibit a power law dependence on the bin distance L :

Fq,q, ~ 1/Z/W . (7.17)

uqtqi in the above expression is called the intermittency exponent of the scaled factorial correlator Fqtqi. The scaled factorial correlators can also be directly related to the integrals of the (g+g')-particle distribution function :

I ^ / rVmin+jSy l-Vmin+jSy F<i,i'(Sy^L) = T7 2Z[ dyi-- dyq

rymin+L+jdy ry™™+L+JsV rVmin-t-^-rjoy rymin-t-Li-tJoy / dyq+1 ••• dyq+q, pq+q, ( 2 / 1 , . . . , yq+q>)

Jymin+L+(j-l)Sy Jymin+L+{j-l)Sy y

fVmin+jSy / dyqpq(yi,...,yq)

Jymin + (j-l)5y

+L+j5y \

/ dyq. pq,(yi,...,yq.)\ +L+(j-l)5y I

rVmin+jSy rymin+jSy

dyy l)5y

rVmin + L+jSy rVmin+L+jSy dy\

,+L+(j-l)5y

where M = (AY — L)/5y is the number of the bin pairs in the rapidity window \ymin,ymin + Ay] . For slowly varying single-particle density, the scaled factorial correlators can be approximated by simple integrals of the (l + g')"Par*ic^e reduced distributions.

The scaled factorial correlators and scaled factorial moments are all related to the n-particle distribution function and so there exist relations between them. These relations are true for any type of translationally invariant distributions. In the case of self-similar distributions, one obtains relations also between the intermittency exponents of the scaled factorial


moments and scaled factorial correlators :

Vq,q> = Vq+qi — Vq — Vqi .

Moments similar to the scaled factorial correlators are the split-bin correlators [D. Seibert & S. Voloshin (1991)], i.e. the bin of width 6y is divided in two parts and the correlations in the "left" and "right" part of the bin are calculated :

F(SB) = J_ v ^ ( "p r ) M ^ ( n i ) ( n r > '

This definition corresponds to the scaled factorial correlator Fiti for the case Sy = L. Due to the approximate independence of the scaled factorial correlators on the bin width Sy both in the random cascade model [A. Bialas k R. Peschanski (1988a)] and in the experiment [NA22 Coll. (1991c)], scaled factorial correlators and the split-bin correlators are largely equivalent. The advantage of the split-bin moments is that they allow the analysis of the fluctuations in continuum observables such as the transverse energy in certain subdomains of the rapidity and/or azimuthal angle. The split-bin moments are sensitive to the scale on which they are calculated and are not contaminated at all scales by the zero scale behaviour [K. Haglin &: D. Seibert (1991)]. This could be important if the experimental data are contaminated by some double track counting, which introduces spurious correlations at the scale of the resolution of the detectors.

7.5 Scaled factorial cumulants

The (/-particle distribution is the sum of a trivial product of the single-particle distributions and of the g-particle correlation functions. Generally, one studies the behaviour of the scaled factorial moments which tests the whole g-particle distribution function. Self-similar behaviour in this variable is expected in the random cascade model, such as the a-model (see Chapter 9), but generally it is not clear whether the self-similar behaviour is present in the scaled factorial moments, in the scaled factorial cumulants or whether the power law correlations are only a part of the complicated mixture of correlations in the multiparticle distributions. The scaled factorial cumulants have the advantage of testing the genuine ^-particle correlations,

Scaled Factorial Cumulants 179

and so it is always interesting to test their contribution to the higher order correlation for each process.

The cumulant correlation functions can be studied using the scaled factorial cumulants :

r Jndyi---Jndy<iK<i(yi>--->y<i) A-g = — - 5 .

( / n d » P i ( y ) )

If the one-particle inclusive distribution is approximately constant, we can rewrite the above relation as :

}Cq= 9fl I dyi"j dyq ki(yi> •••>%)

where the kq represents the ^-particle reduced cumulant:

k , , Kq{yi,...,yq)

PliVl)• • • PliVq)

Obviously, one has in this case the relation between scaled factorial moments and scaled factorial cumulants (equation (6.12)), analogous to the relations for non-integrated quantities (7.3). In the case when the multipar-ticle densities are not translationally invariant, the above relations, except for the simplest one : F2 = 1 + fC2, are only approximate. Thus, it is required to study the scaled factorial cumulants directly from the experimental data.

Analogously one can define the cumulant correlators , obtained by the integration of cumulant distributions in separated bins :

M^ / rVmin+jSy j-Vmin+jSy

£q,q'(&y>L) = U7 z21 / dyi--- I dyq -+( j - l )*w

1 J \ I rvmin+joy pym,

, =1 \Jymin+(j-l)$y

rVmin+L+jSy rymin+L+jSy ^

/ dyq+l •• dyq+q> Kq+q, ( y i , . . . , yq+q>)

( rymin+jSy rymin+jSy

/ dyi--- I dyq Pi(yu ... ,yq) rVmin+l>+J$y i-Vmin+L+jSy \

/ dyi--- I dyq. pq>{yi,...,yq>) Jymin+L+(j~l)Sy Jymin+L+(j-l)Sy J

where M = (Ay — L)/5y is the number of the bin pairs in the rapidity window [ymin,Vmin + Ay] .


7.5.1 Correlation integral

The correlation integral allows also for the direct use of the Q2 distance * as a variable in the definition of the multiparticle correlations. The method of scaled factorial moments use definite binning in the phase-space which induces strong fluctuations when changing the scale. Due to the statistical errors, this limits the lowest scales accessible by this method. Another method to measure the correlations in phase-space cells is the correlation integral method [P. Lipa et al. (1992a)]. The scaled factorial moment are the integrals of the many-particle densities over some phase-space cell. The vertically or horizontally averaged scaled factorial moments are given as averages of such integrals over phase-space cells of size Si. If it happens that several particles lie in the same cell, then the scaled factorial moment has a large value. However, it could happen that particles are close to each other but in two or more neighboring cells. Thus scaled factorial moment have strong fluctuations in their values depending on whether it happens that a cluster of correlated particles is in one, arbitrarily chosen phase-space cell or not. The correlation integral gives a solution to this drawback by choosing the integration domain only by the condition (for the second moment) |pi — P2I < <5y and that both particles lie in the observed phase-space. Then, the correlation integral of the second order takes the form :

JasdPldp2p2(pi,P2)

In3dPidP2Pi{pi)Pi(P2)

where fls is the strip (or in higher dimension a tube). One can see that the integration region is smoother than for the scaled factorial moment. This makes the result more stable and the statistical errors are smaller. To define the tube, especially in higher dimension, one has to decide what is the distance in momentum between the two particles. This can be the usual distance : yj{yi — 1/2)2 + {4>i — <fo)2 + {PT 1 — PT 2)2, or the maximum of the distance in each variable. It can be also the four momentum square distance. According to the chosen distance one defines the tube Sls of size Sy as the region where any interparticle distance in the cluster is smaller than Sy. Obviously, for orders higher than two this requires also a definition of the topology of the cluster, i.e. the definitions of all the distances in the

*Q2 = —(p\ —P2)2, where pi and P2 are four-momenta of particles " 1 " and "2", respectively.


for the GHP topology,

Cq(5y)=( dPl • ••dPgPq(Pl,. . • ,Pg)Q(Sy ~ \pi -P2|)©(<fy ~ \P2 ~ f 31) " * "

9 ( J y - | P « - l - P , l ) ) /

( / dpx • • • dpqpi(p!) • • • p1(pq)e(6y - \pi - P2\)9(Sy - |P2-P3 | ) - - -

®(8y - \Pq-l - Pq\))

for the snake topology and

c (5 v_ / dpi • • • dpqpq(pi,... ,pq)@(6y-\pi - p2\) • • • @(Sy - [pi - pg\) V J dpi • • • dpqpxipx) • • • piipg)Q{Sy-\p1 - p2\) • • • &(6y - \px - pq\)

(7.18)

for the star cluster topology, respectively. The star correlation integral is most often used in the experimental

analysis because the computing time of the numerator in (7.18) is proportional to N2, where N is the multiplicity, in contrast to the GHP and snake integrals which require a computing time proportional to Nq and Nq/ql respectively. Inserting the estimators for the multiparticle densities in the equation for the star correlation integral, one obtains after a change of variables :

q[y) (EJE^e^-br-Pri)*-1) ' In the numerator, the sphere count around the particle i\ is raised to the factorial power [q — 1] = (q —1)1 and in the denominator to the usual power q — 1. In the denominator we take the particles from different events by the procedure of event mixing. A simpler procedure which is often used, consists of taking the first particle from one event and inserting it into another event and then calculating the sphere counts around the inserted particle. This simple procedure introduces some bias inversely proportional to the multiplicity but is computationally faster. The correlation integral can be defined also for the calculation of the cumulant integrals and for the calculation of the so-called differential factorial correlations, i.e. differences of the correlation integral at two different scales 5y\ and 6y2. This corresponds to the integration of the correlations at scales between 8y\ and

Scaled Factorial Cumulants 181

• • •

a)

• •

• • •

b)

• •

• • •

0 Fig. 7.1 Examples of the cluster topologies used in the definition of the correlation integral of order 4 : (a) the GHP topology, (b) the snake topology and (c) the star topology.

cluster that are taken into account. Several topologies were proposed [P. Lipa et al. (1992a); H. C. Eggers et al. (1993a)]

(1) the Grassberger-Hentschel-Procaccia topology (GHP topology), where the size of a g-particle cluster is given by the maximum of all the q(q — l ) /2 distances between any particle in the cluster (figure 7.1a),

(2) the snake topology, where one calculates the maximum of the (q— 1) distances between the first and the second, the second and the third particle and so on (figure 7.1b),

(3) the star cluster, where one takes as the size of the cluster the maximum of (q — 1) distances of all the particle of the cluster from one chosen particle in the cluster (figure 7.1c).

According to this, the g-particle correlation integral can take the following form :

Cq(Sy)={ / dPl • • • dPqPn{Pl, • • • ,Pq)Q(Sy ~ \Pl ~ P2 |) • • • ©(<fy ~ \pi ~ Pq\)

©(<ty - \P2 ~ P3\) • • • ®{Sy ~ \P2 ~ Pq\) • • • @(<ty ~ \Pn-l ~ Pq\) J /

( / dpi • • • dpqp^pi) • • • pi(pq)Q(Sy - | p i -p 2 | ) - - -©(<5j / - |pi ~Pq\)

e(Sy - |p2 - Pa|) • • • &(Sy - |p2 - Pq|) • • • ©(<fy - |p , - i - P,|) J

Linked Structure of the Correlations 183

Sy2 similarly as in the factorial correlators. The method of the correlation integrals is now widely used in the experimental analysis because it gives results on the integrated multiparticle densities with lower errors than the scaled factorial method.

7.6 Linked structure of the correlations

As discussed above, the density correlations often contain trivial background contributions. In order to extract physical correlations, one uses cumulant moments constructed by systematic removal of lower order density correlation. The factorial cumulant moments fq are just integrals of the corresponding cumulants.

7.6.1 Linked pair approximation

Much attention has been devoted in the high energy phenomenology [P. Carruthers &c I. Sarcevic (1989)] and in the astrophysics [P. J. E. Peebles (1980)] to the study of correlations in the framework of the linked pair approximation . These investigations have shown that the two-particle correlation functions have a large combinatoric contribution to the higher order cumulants. The cumulants provide a convenient measure of the statistical independence of the arguments of the n-particle density pn. In the case of the full statistical independence of the arguments, pn factorizes into the product of n single-particle densities pi, and the n-particle correlation function Kn vanishes.

The linked pair approximation builds the reduced correlation functions (7.2) of order q out of reduced two-particle correlation functions :

M l . - , ? ) = - ^ [ * j ( 1 . 2 ) M 2 . 3 ) - M « - l , 9 )

+ permutations] (7-19)

as a linked chain of (q — 1) products of k2(i, j). There are qq~2 terms in the bracket [•••], such that one can expect Aq to be representative of average kq. Moreover, the correlation functions are symmetric in their arguments.

The linking coefficient Aq is a free parameter to be determined and Ai = A2 = 1 by definition, except for the Poisson distribution where A2 is


not defined *. The linked pair approximation structures do not have a solid theoretical justification, although Peebles (1980) showed that the random fractal cascade process could lead to the equation (7.19) for the three-point correlation function.

Integrating kq (equation 7.19) over the phase space under the consideration, one obtains the normalized factorial cumulant moments Kq = fq/{n)q. Assuming the translational invariance and using the strip approximation to evaluate cumulant moments, it was shown that moments K.q in the linked pair approximation satisfy the recurrence relations :

Kq = AqK%~1 . (7.20)

This allows to write the direct relation between the coefficients Aq and the probability distribution as :

f ; ( l + u r P ( n ) = e x p ^ ^ § ( u ( n ) / C 2 ) 9 ) • (7-21) n=0 \ 2 q /

The examples of linking coefficients for certain useful models are shown in Table 7.1. Note in particular, that the coefficients Aq do not depend on the value of the parameter A; for the Pascal distribution.

The linked pair approximation implies also the scaling features of the combinants (6.15). If normalized factorial cumulant moments Kq satisfy the linked pair approximation (7.20) then the ratios R n / (n) at a fixed multiplicity n should be the scaling functions of (n)K.2 :

X ' J = 0 J'

= rq((n)IC2) .

7.6.2 Linked approximation in the conformal theory

Another linking structure of correlations is found in conformal theories [I. M. Dremin (1993)]. The conformal invariance of the theory means its invariance at a homogeneous deformation : x —> Ax, and an inversion : x —> x/X2 (A = est). This invariance imposes strong restrictions on

*For the Poisson distribution, the two-particle correlation functions &2 vanish, so Aq cannot be denned by (7.19). But the relation (7.21) allows to define them as Ai = 1, and all the other Aq = 0 for q > 2.


Table 7.1 Linking coefficients for some distributions. The hierarchical models of Schaeffer [R. Schaeffer (1984); R. Schaeffer (1985)] and Balian-Schaeffer [R. Balian & R. Schaeffer (1989a); R. Balian & R. Schaeffer (1989b)] refer to the general choice Aq = q<i-2-"2^q~1^1' with integer values of v. Only relevant values of v are cited in the table.

Model

Poisson

Pascal

Fry (1984)

Hamilton (1988)

Balian-Schaeffer (v -

Schaeffer (y -

Balian-Schaeffer (u •

= 0) (1989)

= 1) (1984)

= 2) (1989)

Aq

A! = 1 , Aq = 0 if

( 9 - 1 ) '

^ ( ^ / 3 ) ' - 2

^ 3 / 3 ) * - 2

qq~2

4(2g)*-3

4 3 (4g) ' - 4

q>2

the Green functions and vertices of theory [E. S. Fradkin (1965); A. M.

Polyakov (1970b)] and has been shown to have striking consequences at

a critical point, particularly in two dimensions [A. M. Polyakov (1970b);

A. A. Belavin et al. (1984)]. In such theories, the reduced correlation

function of order q is :

N kq(l,...,q) = Y[[k2(i,j)\

1^-Vf(l31,...,t3p)

where

N g(g ~ !) - «(« ~ 3 ) 2 P 2

/ in the above expression is an arbi trary function of p variables (3i (I =

1 , . . . , p ) , which are given by different products of xfj = (xi — Xj)2 variables,

where i, j are different particle indices. Taking into account the q(q — l ) / 2


permutations of all particle indices, one finds :

/C, ~ Aq{K.2y/2 . (7.22)

At the second-order phase transition, absence of the typical scale implies the power like behavior of cumulant correlation functions. For the scaled factorial moments, the conformal symmetry implies a linear increase of intermittency indices with the rank. The experimental data of UAl Collaboration on pp collisions are well reproduced by the recurrence relations (7.22) [I. M. Dremin (1993)].

7.6.3 Linked approximation for the A-scaling

In the non-critical system at equilibrium or in the shattering phase of the binary fragmentation process (see Chapter 10), the cluster probability distribution satisfies the second scaling law (3.26) [R. Botet et al. (1997)]. This is a special case of the A-scaling (c.f. Section 3.5.4) :

<n)*P(n) = ft ( = ^ ) = * ( * A )

with A = 1/2. This scaling relation allows to compute the factorial cumulant moment generating function as :

/

oo exp(uzA)$(zA)dzA . (7.23)

-oo

This means that / i = (n) and :

—V = est ,

constant meaning here independent of (n). One can even be more precise. From relation (7.23) and the definition of the cumulant factorial moments (6.9), one gets the formal identity :

from which the first few cumulant factorial moments follow :

-A. - (z2 \ p& ~ \ZA)


(4)

(4>-3(4) 2 (7-24)

<4>-io(4><4>

All (z^) refer to the moments of the function 3>(ZA)-

It is easy to see that in the second scaling limit, relations (7.24) imply the equation (7.22). However, contrary to the case of a conformal symmetry, the factorial cumulant moments in these non-critical systems decrease with the system size as :

/C ,~ l /<n)? / 2 .

First scaling (3.23), as occurring in the second-order phase transition, is worth a special attention. In this case A = 1, and the linked pair approximation is recovered. The coefficients Aq are independent of the system size and the first few ones are given by :

Ml

<212)3

7.6.4 Counts and their fluctuations

Simplest observable in any measurement is the count number of objects appearing in a given fixed cell, e.g. volume for galaxies, time interval in photo-count experiments, or rapidity interval for produced particles. Such observations aim at finding the frequency of occurrence of particular number of objects and the form of the probability distribution P(n) gives necessary information about the fluctuation in the count. Hubble (1934) was first to analyze count statistics for galaxies as seen in photographic plates.

A_ f!A

A _ ftA

A _ A 5 A ~

A3 =

A4 =

A5 =


He found a log-normal distribution of counts :

P(n) ~ exp -[(ln(n/(n))2/2<T2]

with a ~ 0.45 instead of Poisson distribution expected for randomly distributed objects. This was the first important result pointing to the non-randomness in the luminous matter distribution. Later studies based on the Zwicky catalogue [F. Zwicky et al. (1961)] of clusters of galaxies pointed out to a good fit obtained with the Gamma distribution (5.20) with k ~ 6 [P. Carruthers & D.-V. Minh (1983)].

On the other hand, some values of the hierarchical constants Aq have been measured for visible Universe. The value of A3 has been found to be equal to A3 = 0.43 ± 0.07 [E. Groth & P.J.E. Peebles (1977)], or A3 = 0.27 ± 0.02 [P. J. E. Peebles (1980)]. One should not be surprize that the values for A4 [J. Fry & P.J.E. Peebles (1978)] and A5 [N. A. Sharp et al. (1984)] are quite uncertain.

A particularly useful quantity in astronomical studies is the void probability (c.f. Section 5.1.1) :

/ OO y. \

P ( 0 ) S P ( 0 ; f t ) = e x p f £ ^ ( - < n > H

where fi stands for the investigated volume and P(0), (n) and Kq refer to this volume. One important feature of this quantity, which is valid for the class of Poisson transform distributions, was given in (5.7) :

(-(n)r9"PW(0) K)~ n! 0<n>» •

Partial derivatives are used here, since they are calculated at a fixed volume CI. This expresses that within this class, the whole probability distribution P(n) can be reconstructed as soon as the function P(n) (0) is known with good enough accuracy. Note also that P(„)(0) is just the Laplace transform of the Poisson weight f(x) (see equation (5.1)), giving access to the distribution / by the inverse Laplace transform of P(0).

Probability P(0) is symmetrically dependent on the entire hierarchy of correlation functions [S. D. M. White (1979)]. Notice that for the Poisson distribution with mean (n) :

lnP(0) = - ( n ) .

Erraticity Concept 189

Constructing then the quantity :

lnP(O) X~ (n)

where (n) is the average multiplicity of particles in the domain fl, one can show that if the linking coefficients Aq are independent of both (n) and the domain size ft, as it does for the hierarchical correlations, x c a n D e written as the following scaling law :

9 = 1 H'

provided this quantity is positive. For the Poisson distribution, x = 1 f° r

all values of (n). The effects of clustering appear as x < 1 [J- Fry (1986)]. For example for the Pascal distribution, it appears to be equal to :

= ln(l + (n)/C2) ^ 1

{n)K.2

independently of the value of the parameter k of this distribution. The shape of the x((n)) curve is then characteristic of a whole class of distributions.

The function x c a n therefore be interpreted as a measure of the void probability in the domain fl which normalizes out the contribution from uncorrelated particle (fragment) emission. Thus, the overall shape of the scaling function for x < 1 is affected only by the clustering properties of produced particles(fragments) involving correlations of all order. The presently available observational data for the void probability in the galaxy clustering exhibits the hierarchical scaling [J. Fry et al. (1989)].

7.7 Erraticity concept

Observed particle multiplicity is an integrated information over a long history of the system. For that reason, fluctuations due to, for example, the critical behaviour could be smoothed over. To find them experimentally, one should search for spatial fluctuation patterns which are localized 'in time'. Hwa's paradox [R. C. Hwa (1996)] of apparent similarity of climatic conditions in Oregon and Florida, provides a nice illustration of this problem. Looking to the annual precipitation of certain regions in Oregon and


Florida, one could hastily conclude that the climatic conditions in these two regions are similar. This would be of course incorrect because the rainfall patterns are completely different : drizzly in Oregon and stormy in Florida. The apparent similarity of the weather conditions in these two states comes from the integration of daily rainfall over the entire year, which are then averaged over many years. To learn about the nature of precipitation and, hence, the climatic conditions in Oregon and Florida, it is necessary to study spatial and temporal fluctuations in the precipitation pattern. Similar conclusion applies to the studies of hadronization properties in e.g. ultrarelativistic collisions and in the multifragmentation process in intermediate energy heavy-ion collisions.

Study of the scaling features in the multiparticle spectra and determination of the fractal parameters is essentially different from the analysis of dynamical systems in which one studies the time sequences and the divergence rate for different trajectories. In multiparticle systems we do not have at our disposal any time sequence to estimate variability of the considered system. Since the information about the system and its dynamics is contained in the ensemble of events obtained in similar dynamical conditions, variability of the multiparticle system could be associated with the variability in the events pattern in the studied ensemble. Hwa (1996) has suggested that the measure of the variability in the sample of events could be the event-to-event fluctuation of the scaled factorial moment Fq of order q. The superscript which denotes a given event V , runs over all events in the studied sample of N events. The variability of the studied process could be quantified by looking to the features of the probability distribution P(Fq) of Fq (e = 1 , . . . , N), such as the scaling features for systems of different sizes, the form of the tail of P(Fq) for large Fq or the features of the moments :

Ep,q = (F>)/(Fq)*

of P(Fq) for different bin-sizes. The scaling behaviour :

Ep,g oc M+*V>

where M is the number of bins, has been called erraticity [R. C. Hwa (1996)]. The erraticity index is defined by :


The erraticity analysis is not suitable when event multiplicity n is low, i.e. the average bin multiplicity n = n/M is very small, because in this case most of the bins do not participate in the characterization of the event pattern. In this case, scaled factorial measures of the multiplicity should be replaced by the measure of the void distribution (gap distribution) in, for example, the rapidity [R. C. Hwa (2001)].

Let us consider an event with n particles. For this event there are (n +1 ) gaps {5yi, 5y2,..., 8yn+i}, which specify the pattern of the considered event. To quantify this pattern of voids, let us define the cumulative variable X such that dn/dX = est (0 < X < 1) (c./. equation (7.11)). The particles are located at X\, X2, • • •, Xn and their mutual separation is : Xi = Xj+i — Xi with J27=o Xi = •*•• ^ o r e a c n e v e i u ; o n e c a n define two kinds of moments :

i=0

(7.25)

and corresponding erraticity measures :

Sq = (SqlnSq)

(7.26)

oq = (S q lnS q ) or their normalized variants :

S, = Sq/SW (7.27)

The denominators in (7.27) correspond to the respective statistical contributions which can be calculated by randomly distributing produced particles in the X-space. The erraticity, which is certainly a very important conceptual step, remains at present not sufficiently explored and significance of different erraticity behaviours has not been yet fully explored.


7.7.1 Wavelet representation

Wavelet technique is increasingly popular in analyzing the event patterns and in locating the clustering in the analyzed data. The standard correlation densities characterize the correlations between bins which are resolved at a certain scale. As such, they do not provide a natural description in terms of large scale structures (clusters or voids). To disclose this information from the information contained in bins of given size, one has to perform a basis transformation provided by the wavelet transform [S. Mallat (1989); I. Daubechies (1988)]. The wavelet basis is constructed from dilatations and translations of one single initial wavelet, called the mother wavelet, and provide a self-similar and orthogonal basis. The basic idea is to analyze the data at different scales or resolutions. As such, it is obviously a useful tool in exhibiting possible scale invariant properties of the multipar-ticle spectra. The multiscale localization property of the wavelet transformation allows to select contributions from different scales in the random signal and localize small and large scale structures separately. Good review of this subject can be found in [C. K. Chui (1992); M. Farge (1992); Y. Meyer (1990)].

Any one-dimensional sample of a point distribution e(y), such as the rapidity distribution of the pions in the interval [0,1] with resolution Ay, can be represented as a histogram of M = 2R bins :

e(y)=eW(y)= J > f W*) (7.28) j=o

where R = mod(|lnA2/|/ln2) + 1 is called the resolution scale. Function $Rj(y), called scaling function or mother function, is defined by the equation :

*R-ij(y) = * (2 f l - 1 j / - j ) = £ bi$(2Ry - 2j - i) = i

= J2b&R.v+i(y) (7-29) i

The coefficients ej- are the amplitudes of the orthogonal expansion of energy density e^ in the basis {$RJ} at the scale R. Within a given scale, the functions {$R,J} are orthogonal with respect to the shift index j . The histogram function at the next finer resolution is the average of


adjacent bins at the finest scale. Obviously, some details are lost compared to (7.28). The difference between e^ and e^R~1^ can be expressed in terms of the difference functions : ^Rîj(y). They are defined by the following equation :

* * - i j ( y ) = *(2 f i"12/ - j) = '£i{-i)ib1-i*(2Jy - 2j - i) i

= Y,(~mi-i$j,2j+i(y) (7.30) i

and the equation (7.29). In general, the difference function îj(y) is called wavelet or father function. Coefficients 6; in (7.29) and (7.30), which have to satisfy certain conditions of admissibility [I. Daubechies (1988)], define a filter in the wavelet analysis. Once an admissible set of bi is chosen, the solutions $ and ^ can be found by numerical iteration of equations (7.29) and (7.30).

Equations (7.29) and (7.30), which are called the dilatation equations, are fundamental equations of the wavelet analysis. Like {$R-i,j{y)}, the functions {& R-ij(y)} are orthogonal with respect to the shift index j within the given resolution scale R — 1. They are also orthogonal to the functions {&R.-i,j(y)} at the same scale but are not orthogonal, e.g. to the functions {$R,j(y)}-

The multiresolution analysis goes as follows. We begin by the distribution e(y) at the resolution scale R. In the next step, &RJ (y) are expressed in terms of functions <&R-.ij(y) and ^Rîj(y) at the lower scale R — 1, according to equations (7.29) and (7.30). Then, $R-ij(y) are rewritten in terms of functions <&#_2,j(y) and ^R—2,i(y) at the next lower scale R — 2, etc. Going from any given scale R' to the next lower scale R' — 1, only the difference between these two resolutions is memorized in terms of the difference functions ^w-i,j{y)- Consequently, the amplitudes ej ' of the binned representation at the finest scale R are transformed during the multiscaling decomposition into the amplitudes i\j ' of difference functions at rougher

scales : 0 < i < R— 1, and the amplitude £Q ' of the scaling function at the roughest scale i = 0 :

e(R)(v) = ( £ E 4 ? * ^ ) ) + 40)*o,oG/) . (7.31)


Equation (7.31) defines the linear wavelet transformation :

£ = W(bk)e , e = [s0R), e[R),..., e g ^ J

depending on the coefficients bo,. plitudes :

ft**!

.., bn of the filter

1 £(y)VjAy)dy

The transformed am-

are called the wavelet amplitudes. The wavelet correlations, i.e. correlations between the wavelet am

plitudes ijtk, provide statistical information about the fine subclustering structures (clump correlations) inside larger structures (clusters, filaments or voids) and help in disclosing the hierarchically organized processes [P. Flandrin (1992)]. The wavelet correlation densities are constructed by applying the wavelet transformations either to the correlation densities or directly to the evolution equations for the corresponding generating functions. The discussion of the wavelet transform densities and wavelet correlation densities for specific random cascade models can be found in [M. Greiner et al. (1995)].

Orthogonal wavelets provide a multiresolution representation of a random signal, which dissect the random signal into contributions from different scales. If applied to hierarchical random processes, such as discussed in Chapter 9 and 10, the multiresolution analysis is expected to provide very simple correlation structure, once represented in an appropriate wavelet basis. Indeed, it was shown [M. Greiner et al. (1995)] that the Haar wavelet basis represent exactly normal coordinates for the p-model (see (9.22)). In the same model, the D4 wavelet does not provide exact normal coordinates, nevertheless, the covariance matrix is nearly diagonal.

The idea of multiresolution analysis is to find representations of the density distribution e(y) at various scales. Unlike the Fourier transformation which requires the information in the whole physical space, the wavelet transformation requires only the local information in space. It is then possible to reconstruct only a portion of a signal or only its local contributions to a given range of scales. This allows a journey down or up a hierarchy of scales and to view the studied "image" (distribution) with different magnifying or reducing "glasses". This property of the wavelet transformation allows to attempt the unbiased separation of the interesting physical correlations from the various backgrounds which are difficult to separate


otherwise. In the usual correlation method analysis, different scale correlations, also those coming from the background, are mixed together in the same phase space. On the other hand, the discrete wavelet transformation has an extra dimension : the scale which can be used efficiently to separate these correlations. These are main reasons of great success of wavelet analysis in practical applications. In the next section, we shall present few typical examples of wavelets.

7.7.1.1 Simple examples of wavelets

The simplest example is provided by the Haar wavelet (see figure 7.2). The Haar wavelet scaling functions :

*jg(l/) = *(H)(2*y-j) = if j2

elsewhere

~R <y<(k + l)2~R

(7.32)

are constructed from the unit box function $Q 0 ' (y) by a discrete dilation with a factor 2R and a translation governed by an integer parameter j . They are orthogonal with respect to the shift parameter j . The Haar wavelet difference functions : ^R_I Ay) = ^^H\2R~ly — j) are defined with :

*£?(*)

' 1

- 1

.0

if 0 < y < 1/2

if 1/2 < y < 1

elsewhere

(7.33)

The functions ^n-i,j(y) a r e a^ so orthogonal with respect to the shift parameter j within a given resolution scale. Moreover, they are orthogonal to the scaling functions ^R21 Ay) at the same resolution scale.

One can express the scaling functions 3># j(y) at a given resolution R

by functions ^p,2iAy) and ^R2\ Ay) at a next inferior resolution. For

example, for R = 1 one finds :

<Ho\y)

s W ,

1 2 L <Oy)

l r

2 L {H),

<Ho}(y)

,(»),

(7.34)

n,o'(y) = ~ ny{y)-ny(y)


For the choice (7.32) and (7.33), only &o = b\ — 1 are different from zero in equations (7.29) and (7.30). The drawback of extremely simple wavelet is its discontinuity which translates into a poor localization in Fourier space. Different choices of 6» correspond to different filters in the wavelet analysis and, hence, correspond to different wavelets $(y). Somewhat better localization properties in Fourier space are obtained with the choice [I. Daubechies (1988)] :

60 = i ( l + V3) , &1 = 1(3 + ^3)

b2 = i ( 3 - V 3 ) , 63 = i ( l - \ / 3 )

which yields the D4 wavelet. These two cases are examples of discrete wavelet transform. In most

applications, however, one uses continuous wavelet transform [M. Farge (1992)]. The most commonly used complex-valued wavelet is the Morlet wavelet:

*(Z/) = exp(7 z l fc* • y) exp(- | y |2 /2)

which is a plane wave of wavevector fcq,, modulated by a Gaussian envelope of unit width. Higher-dimensional generalization of the Morlet wavelet has an interesting property of angular selectivity which gets better as | k-% | increases. Unfortunately, an increased angular selectivity is accompanied by a decreased spatial selectivity.

Another complex wavelet, used mainly in quantum mechanics applications, is the Paul wavelet (see figure 7.2) :

*m(y)=T(m+l)/ VV ' 1 + m • (7.35)

Complex wavelets with zero Fourier coefficients for negative wavenumbers are particularly suitable to analyze time-evolution of the distribution since in this case the wavelet preserves the direction of the time.

Commonly used real-valued wavelets are m-th derivatives of the Gaussian :

^(V) = ( - 1 ) " ^ (e-M '2) (7.36)


V(y) 1.5

0.5

-0.5

-1.5

a)

i

-

vGO 0.5

0

b)

r 7 ,

V 1

I . I ,

4 -2 0 v 2 4

Fig. 7.2 Three examples of wavelets : (a) discrete Haar wavelet, (b) continuous real-valued Marr wavelet ("Mexican hat") (equation (7.36) for m = 2) and (c) continuous complex-valued Paul wavelet (equation (7.35)).

Among those wavelets, most often used is the Marr wavelet, the so-called "Mexican hat", which corresponds to m = 2 in equation (7.36) (see figure 7.2). One may notice that the generalization of Marr wavelet to higher dimensions is isotropic and cannot distinguish between different directions of the propagation.

Chapter 8

Bose-Einstein Correlations in Nuclear and Particle Physics

The Bose-Einstein interference, or equivalently, the Hanbury-Brown and Twiss effect (HBT effect), is the principal issue of correlation studies in relativistic collisions of leptons, hadrons and nuclei. The effect of correlations between identical bosons which obey Bose-Einstein (Fermi-Dirac for fermions) statistics instead of the Maxwell-Boltzmann statistics is also one of the classical problems of statistical physics. Bose-Einstein correlations have been first studied for gases but nowadays it is known that they are important in the theory of superconductivity, theory of superfluidity, quantum optics or astronomy. The photon interferometry was first proposed as a way to measure the spatial extension of extragalactic sources [R. Hanbury-Brown & R. Q. Twiss (1957)]. Hanbury-Brown and Twiss (1957) showed that the angular sizes of distant objects can be measured in independent detectors from correlations of signal intensities. The interference of indistinguishable particles was later proposed as a measure of the spatio-temporal extension of the interaction region in elementary particles or nuclear collisions [G. Goldhaber et al. (1960); G. Coccini (1974); G. I. Kopylov & M. I. Podgoretsky (1972); D. H. Boal et al. (1990); NA22 Coll. (1996b)]. This method, which is usually referred to as the HBT measurements, became nowadays a standard technique.

This traditional name is misleading because of important differences between interference studies in astronomy and in nuclear physics. In astronomy, the source separation is much larger than the detector separation what implies that the interference pattern should be studied in the coordinate space by looking to the difference of arrival times. In nuclear or particle physics, on the contrary, the source separation is much smaller

199

200 Bose-Einstein Correlations in Nuclear and Particle Physics

than the separation of detectors and, therefore, the interference pattern is studied in the momentum space by looking to the momenta difference.

Even though the intensity interferometry in nuclear and particle physics was introduced more than 30 years ago, several basic questions concerning the form of Bose-Einstein correlation function remain unanswered. On the other hand, the level of sophistication both in the theoretical descriptions and in the experimental studies has increased very much, in particular in the field of heavy-ion physics [W. A. Zajc (1991)].

8.1 Basic features of Bose-Einstein quantum statistical correlations

Intensity correlations appear due to the symmetrization of the two-particle final state. Suppose that a pair of particles is observed with respective momenta qi and q2. The amplitude has to be symmetrized over unobservable variables and, in particular, over the emission points x\ and x2. If final state interactions can be neglected, the amplitude of such a final state is proportional to :

A(quq2) oc -L[e*»»*i+*»*» + e*9i"+*n'i] . v 2

If the particles are emitted incoherently, the observed two-particle spectrum is proportional to :

P2(qi,q2) oc / dxipifa) / dx2p1(x2) | A(qi,q2) |2

and the two-particle intensity correlation function is defined as :

fl.(gl,ft) = ? ( ?'?\=l+ |di(gi - t t ) | a •

This function carries information about the Fourier-transformed space-time distribution of the particle emission :

dx{q) = d1(qi-q2)= / dxe^d^x)

as a function of the relative momentum q. As compared to the unsym-metrized case, Bose-Einstein correlations modify the momentum distribu-

Basic Features of Bose-Einstein Quantum Statistical Correlations 201

tion of the pair of particles in the final state by a weight factor :

(1 + cos[(gx - q2) • (xi - x2)]) .

A large amount of data exists for the two-particle correlation function. These correlations are much stronger than those between the unlike-charge particles [DELPHI Coll. (1990a)]. This means that the two and many particle correlations are mainly due to the Bose-Einstein interference mechanism.

The correlation function was studied for like-sign particles in order to extract information on the size of the emitting region from the Bose-Einstein correlations [W. A. Zajc (1991)]. For a Gaussian source density profile, the correlation function is related to the radius of the radiating region R :

D2(q) = l + Xexp(-R2q2/2) . (8.1)

In deriving this formula, we have assumed that the sources emit coherently and are randomly distributed over the sphere of radius R. In many phe-nomenological parametrizations one includes the chaoticity parameter A in front of the exponential term in (8.1). The parameter A equals 1 for a completely chaotic source and the correlation function rises up to 2 as q —> 0. The chaoticity parameter is generally found to be different from 1.

Obviously, reduction of A may be caused by experimental difficulties. On the other hand, it may also reflect physical effects which could lead to the coherence of the source, such as for example in the laser problem, or in the presence of the boson condensate (the pion condensate). Let us now recall some of them. HBT measurements of pions produced from a disordered chiral condensate would give reduced A [J. D. Bjorken et al. (1993); J. D. Bjorken (1997); K. Rajagopal & F. Wilczek (1993); S. Gavin (1995)], though the rescattering of particles in the collision volume tends to destroy phase correlations and to restore chaoticity. In the MIT atom laser [M.-O. Mewes et al. (1997)], magnetically trapped and evaporatively cooled sodium atoms are extracted in coherent states from the Bose-Einstein condensed system. Since the extracted atoms do not exhibit an HBT effect, the chaoticity parameter A would be 0 in this case. Another physical effect reducing A is the production of pions from long-lived resonances. For example, three pions coming from the decay of r), which has a lifetime ~1.2 A/c, would appear to be produced at an enormous distance of order A from the collision volume (sic\). Similarly, pions from the decay of LJ would appear at a


distance of 24 fm, whereas decay of 7/ would yield pions at some 800 fm, etc. Hence, the physical picture we have is those of the collision volume surrounded by an enormous halo of pions from long-lived resonances. This halo is expected to contribute to an enhancement of HBT correlations at very small q. One should mention that pions in the halo constitute a small fraction of all produced pions. The majority of pions come from short lived resonances, e.g. p —> TTTT or A —> NTT, and are produced well within the collisions volume and, hence, are irrelevant for our discussion.

8.2 Parametrization of the H B T data

The two-particle interferometry correlation function is defines as [S. Pratt et al. (1990)] :

D2{qi'q2) ~ l + [ / S ( * i , « i ) ^ i ] - [ / 5 ( « 9 l f t ) d * f l ! 9 ] ( 8 - 2 )

where S is the source function (the Wigner transform of the single particle density matrix), q = q2~qi and P = (1/2)(E\ +E?, <fi +9*2)- -D2 is positive definite even though S(x, p) may take both positive and negative values.

Obviously, it is very difficult to measure the full six-dimensional two-particle correlation function. In the case of the spherically symmetric static source, the two-particle correlation function depends only on the value of the momentum difference between the particles. If one is interested also in measuring the lifetime of the source, then it is useful to measure also the correlation functions in different domains of the energy difference of the two bosons. However, generally one can have more complicated dependence on the momentum, an example of this is the Bjorken-scaling source, which leads to different type of correlations in longitudinal and transverse variables [K. Kolehmainen & M. Gyulassy (1986)]. Other scenarios lead to Q2 — — (Pi — P2)2 as a one dimensional variable for which the two-particle correlations should be studied, e.g. in the string decay [M. G. Bowler (1987)]. If one is looking for a new behaviour in the Bose-Einstein correlations, one is interested in some general covariant features universal to different type of collisions. Hence, the natural variable for the exploratory studies of the small scale correlations should be Q2. The Q2 variable can be used also as a distance in defining the many particle correlations [H. C. Eggers et al. (1993b)]. Then one should simply measure the n-particle cu-

Parametrization of the HBT Data 203

mulant in the variable being the sum of the Q2 of all the pairs of particles. The simplest way to analyze HBT data is to parameterize the correlation

function D2 as a Gaussian in q. Expanding the correlation function (11.23) for small q to second order and ignoring the P-dependence of S(x, P) we have :

D2(q) = 2 - A"((SM^> - MM) + • • •

where x = x\ — xi- This form suggests a parametrization [F. B. Yano h S. E. Koonin (1978); M. I. Podgoretskii (1983)] * :

D2(g) = l + A e x p ( - A " ( ( ^ ^ ) - ( ^ > ( ^ » ) •

Various versions of this parametrization result from different physical assumptions about the production region. Assuming :

one finds :

D2(q) = l + \eM-Q2R2) .

The sign of time-time and space-space components should be the same. Therefore, a better single-size parameterization would be :

\X^Xvj \X^ij \Xi,j = O^i/ix,

which yields :

D2(q) = l + \exp-(q2R2 + q%T2) .

In this expression, r (T 2 = (t2) — (t)2) is essentially the duration of the collision and R (R2 = (r) — (f)2) the radius of the collision volume.

At the next level of sophistication, one may take the evolving geometry into account. One should mention however that the interpretation of HBT data is not unique and depends on those geometrical assumptions about the interaction region. Here we shall introduce the parametrization which nowadays is commonly used in interpreting present HBT measurements

•Recently, a Buda-Lund parameterization [T. Csorgo & B. Lorstad (1996)] was advocated as an improvement with respect to the covariant Yano-Koonin-Podgoretskii parameterization for non-expanding sources [F. B. Yano & S. E. Koonin (1978); M. I. Podgoretskii (1983)].


in high-energy physics. Let us consider a pair of particles of total three-momentum P and relative three-momentum q. Since q • P = 0, therefore q° = q • v, where v is the cm. velocity of the pair of particles. Then q^Xfj. = q- (f—vt) and :

< A " « ^ } - (^><av» = ((q • (f - vt))2) - {q- (f- vt))2 .

Let us now define a three-dimensional coordinate system in which the longitudinal direction is along the beam axis, the outwards axis (x-direction) is along the transverse component of P = {q\ + q2)/2, and the side axis is the y-direction. In this system vy vanishes. Moreover, since the ensemble of events is symmetric under y —¥ —y, therefore all cross terms involving y vanish as well. In this case, the parameterization of the correlation function involves four radii [S. Chapman et al. (1995)] :

D2(q) = l + A e x p -{q2outR

2out + qHdeRtide + <l?ongR?ong + IqoutqiongRll)

(8.3)

where the radii parameters are :

Rlut = ((* - Vxtf) -(X- Vxt)2

Ride = (y)2 - (y)2

Rlng = ((z-vzt)2)-(z-vzt)

2

R2ol = (ix ~ vxt)(z - vzt)) - (x - vxt){z - vzt)

It should be noted that R2t need not to be positive.

This parameterization is now a common tool in interpreting HBT measurements. Many informations on the spatio-temporal development of the collision volume (the flow) can be extracted from such analysis [S. A. Voloshin k. W. E. Cleland (1996)]. The experimental dependence of the two-particle correlations on the momenta of particles indicates that the systems produced in collisions are expanding [U. Heinz et al. (1996)].

8.2.1 The space-time structure of the multiparticle system

The information one can obtain from the data on quantum interference about the space-time structure of the multiparticle system created in the collision has certainly a limited scope as it provides the information about freeze-out stage of the collision. Nevertheless, the possibility of visualization of the emitting source, even though not free from serious assumptions,


Fig. 8.1 Reconstructed space-time region of the particle emission in S + P b nuclear collisions as measured by the NA44 Collaboration (after [A. Ster et al. (1999)]).

is a challenging experience both in particle and nuclear collisions. The mj-dependence of the data on Pb+Pb nucleus-nucleus collisions [NA44 Coll. (1999)] shows that the radius of the system decreases with increasing transverse mass of the particles as lfy/msf. The my-dependence of the radii parameters is expected for the longitudinal direction [A. N. Makhlin Sz Yu. M. Sinyukov (1988)]. In the transverse direction, this m^-dependence of radii is usually interpreted as the evidence for a transverse hydrodynamic flow. However, a similar ex l/y/mr dependence for the Bose-Einstein correlation function was found in e+e~ annihilation reactions [NA44 Coll. (1999)] where the notion of hydrodynamic flow is not common.

The NA49 experiment on Pb+Pb collisions [NA49 Coll. (1998); NA49 Coll. (1999)] shows that the longitudinal flow of particles agrees well with the Bjorken in-out picture [J. D. Bjorken et al. (1993)]. Moreover, the particle emission starts rather late (the lifetime of the system is of the order of 8 fm) and the duration of pion emission is only of the order of 3 fm. This short emission time contradicts widely accepted scenario with long-lived mixed phase and suggests an explosive scenario in three-dimensional


Fig. 8.2 Reconstructed space-time region of the particle emission in {ir+ /K+)p collisions as measured by the NA22 Collaboration (after [R. Hakobyan (1999)]).

expansion phase. The NA44 data on S+Pb nucleus-nucleus collisions at 200 A-GeV [NA44 Coll. (1995)] shows a similar Bjorken shape of the source (see Fig. 8.1), but the particle emission in central region starts earlier (~4 fm) and is finished only ~1.5 fm earlier.

In (n+/K+)p collisions at 250 GeV/c [NA22 Coll. (1998a); R. Hakobyan (1999)], similar in-out picture is found. However, the particle emission in hadron-hadron reaction starts almost immediately after collision and lasts about 1.5 fm, as shows figure 8.2. Thus we seem to have a continuous and consistent variation of source parameters going from Pb+Pb and S+Pb nucleus-nucleus collisions to hadron-hadron collisions.

8.2.2 HBT measurements in condensed matter and atomic physics

Study of HBT correlations are not restricted to astrophysics and nuclear or particle physics. Recently, considerable interest in these studies has been seen in condensed matter and atomic physics. The HBT correlations have


been measured for the first time in atomic system [M. Yasuda & F. Shimizu (1996)] by observing the time correlations in laser-cooled ultracold beams of bosonic 20Ne atoms. The correlations in the 20Ne beam are expected from a kinetic energy spread of atoms in the beam source and the correlation time is the inverse of the beam temperature. It was found that indeed the HBT correlation function begins to rise at time separations less than ~ 0.5 x 1 0 - 6 s and the correlation function increases in the region of small separation times by a factor of 2 as compared to larger times.

Such an atomic bunching phenomenon should not be seen in a measurement of HBT correlations in the MIT laser due the coherence of the atomic the beam in this experiment [M.-O. Mewes et al. (1997)]. The corresponding correlation function would be flat. In general, loss of HBT correlations probes the onset of Bose-Einstein condensation not only in atomic systems, but also in condensed matter systems such as the Bose-Einstein condensate of paraexcitons, i.e. the particle (electron) - hole , (S = 0)-like state, in cuprous oxide Cu 2 0 [Jia Ling Lin & J. P. Wolfe (1993); G. M. Kavoulakis et al. (1996)]

Another interesting application of HBT effect is in light scattering from atoms trapped in optical lattices [C. Jurczak et al. (1996)]. Optical lattices are periodic structures of micron-sized potential wells induced by the light shift of the atomic sublevels in a standing wave exhibiting a gradient of polarization. Atomic transport in optical lattices proceeds through escape and recapture of atoms from and by neighbor optical potential sites, generally with opposite circular polarization. It is then appealing to try to trace atomic jumps between potential wells through intensity correlations between cross-circular polarizations of the light scattered by the atoms. In this experiment, optical lattice has been created with an arrangement of four lasers in which atomic rubidium at a density ~ 2 x 109cm-3, filling about 10~4 of the lattice sites. The lasers also scatter from the rubidium, and the time correlations in the scattered light of two different polarizations effectively measure the atom-atom correlation functions in the lattice. From these experiments, the spatial diffusion of the loosely packed atoms in the optical lattice could be measured. Other applications of optical lattices and HBT intensity correlation spectroscopy can be found in the research of the anomalous diffusion and escape lines [A. Hemmerich (1994); S. Marksteiner et al. (1996)].

Finally, HBT effect has been proposed recently as a probe of the time


and space structure of bubbles in sonoluminescence [S. Trentalange & S.U. Pandey (1996); Y. Hama et al. (1996)].

8.3 Bose-Einstein interference in models

An important problem of practical importance is how to include effects of Bose-Einstein quantum interference in the calculation which disregards identity of particles. Let us suppose that we have an amplitude for production of n particles : An (q), where q = qi,... ,qn (qi's are here the three-vectors), which is calculated ignoring the identity of particles. In the correct quantum mechanical calculation, one should replace Ah (q) by the amplitude :

An = Y,^\Q{P}) (8-4) {P}

where the sum runs over all permutations {P} of momenta qi,...,qn. This generalization (8.4) is an oversimplification of the physical situation where, in general, one has to average over unmeasured parameters. This can be conveniently taken into account in the language of the density matrix. In the momentum representation we have :

p£Hq,q) = E-^ 0 ) (? . {«})^0)*(?.{«» (8-5)

where the summation runs over all those quantum numbers {a} which are not measured in a given experiment. The momentum distribution of particles is given by the diagonal elements of the density matrix :

and is normalized :

JnW{q)dq = l .

Using (8.4), one can derive the density matrix pn(q,q), corrected for the identity of particles and, finally, the multiparticle density :

Q»® = h E Pr?\q{p},q{p>}) = T,ne{pn)&q{p})} • M ' {p},{p'} {p}

Idealized Picture of Independent Particle Production 209

The sum in (8.6) runs over all permutations {P} and {P } of the momenta qi,... ,qn, and the factor 1/n! corrects for the fact that the phase space for identical particles is n\ times smaller than the phase space for N non-identical particles. Notice that the symmetrized momentum distribution iln(q) is not anymore normalized to 1. This implies also that the multiplicity distribution will change as result of the symmetrization.

8.4 Idealized picture of independent particle production

We assume now that if the particles are distinguishable, then their production is independent in the sense that the multiplicity distribution of produced particles is Poissonian :

p(°)(n)= (0) , (8.7) n!

and for a given multiplicity n, the density matrix factorizes into a product of single-particle matrices :

P<£)(q,q')=Pi0Hqi,q'i)p{0)(q2,q'2)---P{0)(qn,qn') • (8.8)

An independent particle production is an idealization which, even though is not expected to describe all details of the real experiment, reveals hopefully some generic features of the problem. The problem of quantum interference can be solved analytically in the case of Gaussian density matrix [T. Csorgo & J. Zimanyi (1998); Q. H. Zhang (1999)]. Main results of such an analysis remain true also for an arbitrary density matrix [A. Bialas & K. Zalewski (1999)].

Explicit expressions in terms of the eigenfunctions and eigenvalues of the single particle density p(°) have been found for the generating function of the multiplicity distribution [A. Bialas & K. Zalewski (1999)] :

^fLW=exPfe^(^-l)J,j

where the positive coefficients Jj are given by :

Jj = J Pi0)(qi,q2)p(0)(q2,qz) • ••P{0)(qj-i,qj)p{0)(qj,qi)d3qi. ..d*qj


for j > 1 and J\ = 1. The coefficients Jj depend explicitly on the single particle density p^°\q,q ). Nevertheless, using the properties of the generating function of the multiplicity distribution (c./. Chapters 5 and 6), one can calculate the average multiplicity :

oo

and all cumulants :

^ = S{p$<»>?o>4 •

Notice, that since all Jj are positive, therefore all cumulants Kq are positive as well. This implies in particular, that the symmetrized distribution is always broader that the Poisson distribution.

Analogously, elegant expressions for the generating functional for the multiparticle correlation functions in momentum space could be derived in the independent particle production scenario [A. Bialas & K. Zalewski (1999)] :

Q(u(q)) = exp f £ MW.(Jj(u(q)) - J,-(l))

where

JjHq))=J<Ql)p{0)(quq2)u(q2)P{0)(q2,q3) • • • u(qj)p^(qj,q1)d

3q1 . ..d3qj

All correlation functions can be derived explicitly from this generating functional. In particular, all correlation functions Kp(qi,... ,qp) and the single particle distribution fl(qi) can be expressed in terms of a single Hermitian function L(qi,qj) [A. Bialas & K. Zalewski (1999)] :

n(q) = L(q,q)

#2(91,92) = L(qi,q2)L(q2,qi)

K3(qi,q2,q3) = L{ql,q2)L(q2)qz)L{qz,q1) + L(qu q3)L(q3, q2)L(q2,qi)

Kp(qi, ...,qp) = L(qu q2)L{q2,q3) • • • L(qp, qx)

+ permutations of (q2,..., qp) (8.9)


where oo

L(q,q) = ^2(n){0)jE^(q,q) j = l

and

ZU)(q,q) = J P{0)(q,q2)P{0)(q2,q3) • • • p<0)G&,?V«2 • ••#* •

These expressions represent very strong constraint on the observed particle distribution, saying that all effects of the HBT symmetrization can be expressed in terms of a single function of two momenta, i.e. all higher order correlations can be derived from the two-particle correlation function . One should remember however, that these expressions are valid only in any model which assumes independent production in a sense given to this notion by equations (8.7) and (8.8).

Functions L(qi, qj) can be constructed explicitly assuming a factorizable Gaussian form for the distribution parameters of the single particle states for the non-expanding static source and the Poisson multiplicity distribution with the mean value (ra)(0) when the Bose-Einstein correlations are switched off [T. Csorgo & J. Zimanyi (1999)]. In this case, the mean and second factorial moment of the multiplicity distribution can be expressed in terms of the combinants (c.f. Section 6.1.7) :

oo

9=1

OO

(n(n - l ) ) = (n)2 + ^ q ( q - l ) R q

9=2

where

and

7± = ~(l+x±\/l + 2x) , x = R2aj, .

In the above equation : aT = 2mf, f = T + a2/{2m), R2 = R2 + (mT)/(a2a^). a denotes the width of the wavepacket. R and T are radius


and temperature of non-relativistic thermal Gaussian distribution [S. Pratt (1993)].

The multiplicity distribution for the large values of n depends on the ratio (n) ( 0)/7±

/ . If :

(n)(0) < ncr < 7+ /2

then the mean multiplicity (n) is finite. Otherwise, it becomes infinite. Hence, at the critical value 7+ ' of the parameter («)(o), the multiplicity distribution approaches a singular point which represents the Bose-Einstein condensation . In this case, almost all particles occupy the eigenstate corresponding to the largest eigenvalue of the uncorrected density matrix.

The width of the momentum spectrum decreases due to the symmetriza-tion and the width of the two-particle correlation function increases. This effect becomes stronger when the system approaches criticality, i.e. close to the critical point the determination of the size of the emission region from the width of the correlation function becomes impossible. The associated multiplicity distribution is very broad, i.e., the probability of an event with multiple produced particles is not negligible. This argument was put forward [S. Pratt (1993)] as a possible explanation of the 'Centauro' events [C. M. G. Lattes et al. (1980)]. However, since the observed events show a rather large relative momentum of the order of 1 GeV between particles belonging to the same group, therefore the phenomenon of Bose-Einstein condensation seems to be an unlikely explanation of these events. Finally, let us note that the broad distribution of identical particles (pions) implies also very large fluctuations in the charged/neutral particle ratio.

8.4.1 Monte-Carlo simulations

Most of the studies of the multiparticle production in ultrarelativistic collisions is done using various Monte-Carlo codes. The inclusion of the Bose-Einstein symmetrization in these codes is an important practical problem. Several prescriptions have been proposed, but the simplest one is the Cracow algorithm which consists of modifying only the off-diagonal elements of the multiparticle density matrix (8.5) and keeping the diagonal matrix elements of the unsymmetrized matrix unchanged [A. Bialas & A. Krzy-wicki (1995)]. In this algorithm, each event generated by the Monte-Carlo event generator is given a weight which is calculated as the ratio of sym-


metrized distribution (equation (8.6)) and the unsymmetrized one. This simple prescription modifies minimally the original spectra. In the practical applications, the unsymmetrized density matrix has the form :

N

PNH^Q) =pN{q) Y[s(Qi-l'i) i=l

where Pjv(<z) is the probability of a given configuration as obtained from some Monte-Carlo event generator for classical particles and g is a Gaussian distribution. This prescription does not modify the diagonal elements of the unsymmetrized density matrix and does not introduce any additional correlations between emission points of the produced particles. Its application is successful in reproducing existing data on two-particle correlations and the experimental multiplicity distribution can be recovered by rescaling Bose-Einstein weights with a factor cVn [S. Jadach & K. Zalewski (1997)], where n is the global multiplicity of particles (pions), and c and V are the fit parameters. Such a rescaling restores original multiplicity distribution and both the single longitudinal and transverse momentum spectra [K. Fialkowski & R. Wit (2000)].

Another approach has been adapted in the Lund model [B. Andersson & M. Ringner (1998)]. In this procedure, the uncorrected matrix element representing the decay of one string in the Lund string fragmentation model [B. Andersson et al. (1983)] is :

•A!0)(<Z) 1 exp iK-b-)m)

where K ~ 1 GeV/fm is the string tension and b ~ 0.3 GeV/fm is the decay constant whose value is tuned to the experimental data. E is the area spanned by the string during its space-time evolution. Going to the energy-momentum space means the replacement :

£ -> 2«2S , b ->• b/2n2 .

Then the matrix element is symmetrized with respect to the exchange of n identical bosons [B. Andersson & W. Hoffman (1986)]. This prescription not only reproduces the two-particle correlations in e+e~ collisions but also predicts that the longitudinal and transverse correlations could be different because they are controlled by different physical mechanisms [B. Andersson & M. Ringner (1998)]. Moreover, the three-particle correlations are not


vanishing. One should mention that, at present, no symmetrization between particles coming from different strings is performed. This approximation, which is also made in the Cracow algorithm [K. Fialkowski & R. Wit (2000); K. Fialkowski & R. Wit (1997)], corresponds to the assumption that strings are created far away, one from another. It is plausible that neglecting this strong interconnection effect may turn out to be a severe approximation for more complicated processes such as the nucleus-nucleus or hadron-hadron collisions.

8.5 Bose-Einstein correlations in high-energy collisions

The experimental data on the multiparticle correlations does not provide a clear verification of the correlation scheme (8.9) which is expected for the Bose-Einstein interference in any model assuming an independent particle production. In S+Pb nuclear collisions, two-particle correlations are well seen but almost no genuine three-particle correlations have been found [NA44 Coll. (1999)]. This finding cannot be easily reconciled with the correlation pattern (8.9). The importance of the absence of three-particle correlations in heavy-ion collisions was emphasized long time ago [H.-Th. Elze & I. Sarcevic (1992)]. The UAl data on pp collisions also disagree with the correlation scheme (8.9), although in this case the three-particle correlations are too large [H. C. Eggers et al. (1997)] to agree with (8.9). On the other hand, NA22 data on 7r+p and K+p collisions agree well with the correlation scheme (8.9) [N. Arbex et al. (1998)]. Understanding of these different findings in the nuclear collisions and in the elementary processes remains a challenge. One should also say that although this simple picture of independent production of particles (8.7), (8.8) disagrees with the correlation data in both heavy-ion [NA44 Coll. (1999)] and pp collisions [H. C. Eggers et al. (1997)] (c.f. Section 8.5), nevertheless the degree of the deviation with respect to the Poissonian limit remains always a reference point of all analyzes of the correlations.

8.5.1 Higher order cumulants in pp collisions

Predictions for the structure of higher order Bose-Einstein correlations depend strongly on the chosen theoretical framework. Particularly interesting is the approach employed by Andreev et al. (1993) (called Andreev-

Base-Einstein Correlations in High-Energy Collisions 215

K2

1.6

1.2

0.8

0.4

q (Gev)

Fig. 8.3 Second order reduced cumulant for like-sign particles for pp collisions at 630 GeV [UA1 Coll. (1993)], together with the predictions of APW formalism [I. V. Andreev et al. (1993)] (see equation 8.10) (from [H. C. Eggers et al. (1997)]).

Pliimmer-Weiner formalism (APW formalism )), which is based on the classical current formalism. This general quantum statistical framework, which was used earlier in quantum optics assumes : (i) the Gaussian density functional for classical random currents (ii) isotropy in the isospin space. These two assumptions are sufficient to determine all higher order correlations in terms of normalized correlators dij, i. e. the on-shell Fourier transforms of the space-time classical current correlators, independently of the structure of the sources. Other assumptions in the APW framework are irrelevant for the structure of higher order correlations and specify the space-time evolution of the sources. Assuming constant value of the chaoticity parameter A and the real-valued currents one obtains [I. V. Andreev et al. (1993)] :

k2(qi2)=2\(l-X)d12 + X2di2

k3(qi2,q23,q3l) — 2A(1 - \2)[d12d23 + ^23^31 + d31du] + 2\Z dx2d2zd?,\

(8.10)


2

1

Max(?12,023,^3,) [Gev]

Fig. 8.4 Third order reduced cumulant for like-sign particles from the pp collisions at 630 GeV [UA1 Coll. (1993)] together with predictions of APW formalism [I. V. Andreev et al. (1993)] (see equation 8.10) (from [H. C. Bggers et al. (1997)]).

Predictions of the APW formalism for higher order correlations have been tested by Eggers et al. (1997) in the correlation integral formalism (c.f. Section 7.5.1) using the UA1 data for pp reaction at 630 GeV/c. Since the pion interferometry measures correlations in terms of pair variables (three- or four-momentum differences), the original variables of the p-particle density have been converted to p(p — l ) /2 pair variables (qi2,qi3,---qP-i P) defined as : gy = | qi - qj \= y/-(qi -qi)2- The following specific parametrizations for the two-particle correlator in terms of the four-momentum difference qij have been used in this analysis :

dij = exp(—r2q?j) (Gaussian)

dij = exp(—rqij) (exponential) (8-H)

d^ = l/qfj (power law)

Figure 8.3 shows the second order normalized cumulant of like-sign particles where the correlation integrals are calculated over bins spaced logarithmically between q = 1 GeV and 30 MeV. The two-particle Bose-Einstein correlations are very strong at small scale and follow a power law. The

^

1

1

(+++) and ( )

K * •

Bose-Einstein Correlations in High-Energy Collisions 217

Gaussian correlator is strongly missing the data *. The phenomenologi-cal implications of the above fact are twofold [A. Bialas (1992a)]. Firstly, because even at the highest presently achieved resolution one observes a rise in the two-particle correlations , the size of the source must be quite large. Since the momentum scales presently studied are about 30 MeV, we expect a source of the extension of at least 6 - 7 fm. Secondly, the power law dependence of the two-particle correlation on the momentum difference at small momentum requires a power law behaviour of the source density at large scales. It can be either an extended power law tail in the density profile of the emitting source or an extended fractal source.

The third order like-sign cumulant moment is shown in Fig. 8.4. The prediction of the APW framework for both exponential and power law correlator are shown with open squares and triangles respectively. The q-dependence of this framework does not depend on the functional choice of the correlator and fails at the same time to describe the data. One should remind however that the analysis of experimental data [H. C. Eggers et al. (1997)] used the assumptions of momentum independent chaoticity and real valued source currents. These common assumptions should however be verified against a more general version of the APW formalism.

8.5.2 Small-scale Bose-Einstein correlations

The presence of the power law Bose-Einstein correlations at small scales requires very specific shape fluctuations in the long-range region [A. Bialas (1992b); A. Bialas & B. Ziaja (1993)]. The n-particle interferometry reduced densities can be written as :

L'ny^ii • • • > "Vi)

_ / dxi • • • fdxn(Y^per exp(i(xikai +... + xnkan))dn{xi, ...,xn)

~~ n\(f dxdi(x))n

(8.12)

where dn is the n-point density of emitting sources and the sum runs over all the permutation of the indices en. The above formula supposes incoherent emission and neglects the final state interaction. In the case of uncorrelated

*In fitting experimental data with different two-particle correlators (8.11), an overall additive constant has been included in the expression (8.10) for £2(912)-


emission in space-time :

dn(xi,. ..,xn) = di(xi)---dn(xn) .

In this case, the n-particle cumulant can be written using the Fourier transform of the source density di(x) in the following way [A. Bialas & K. Za-lewski (1989)] :

dn(k1,...,kn) = Y,I[di{ki-kj) . (8.13) i jjti

If the source density d\ (x) has a power law tail :

di(x) ~ \jxD-i

its Fourier transform shows also a power law in some range of small mo

menta [A. Bialas & K. Zalewski (1989)] :

Thus, we obtain a relation between the power law tail in the source density

and the power law in the two-pion Bose-Einstein correlations which are

given in terms of the Fourier transform of the source density :

C2{q)~l/\q\^ .

The formula (8.13) implies that the higher order cumulants are expected also to have a power law dependence on the rescaling of momenta with an index

vP=K1 • (8.14)

Thus, the intermittency indices grow linearly with the increasing order of the correlations. There is not much experimental data concerning the higher order cumulants. Only the NA22 group measured the third order cumulant, with the preliminary result : i/3 = 2v<i. This does not follow the relation (8.14) and would mean that there are some additional correlations between like-charge pions. This could be due to the correlated emission of pions from the source. The correlations should be present on large scale in space, so that after Fourier transforming they would give a power law at small momenta. The presence of such strong power law correlations at large scales is difficult to understand, unless the time scales involved in their creation during the reaction are also large. Indeed, simply the

Bose-Einstein Correlations in High-Energy Collisions 219

K2(Q= 0.1) K2(Q= 0.7)

0.5 1.0

l/(dn/dri) 0.5 1.0 1.5

l/(dn/dTi)

Fig. 8.5 The second order order cumulant from the pp collisions at 630 GeV measured by UA1 Collaboration is plotted vs. the inverse of the rapidity density (from [B. Buschbeck et al. (1999)]).

causality constraint requires a time scale at least of the order of the space scale involved and probably in order to build up such strong correlations a longer time is needed.

8.5.3 Density dependence of the correlations

The UAl Collaboration [B. Buschbeck et al. (1999)] studied the dependence of the correlations between like- and unlike- pairs as a function of the particle density n (see figure 8.5). One can see the linear dependence of the the normalized cumulants on (dn/dy)"1. Like- and unlike- cumu-lants behave approximately in the same way as a function of the rapidity density dn/dy. For large Q (Q = y/(qi — 92)2> where q\ and qi are the four-momenta of particle " 1 " and "2", respectively) where the HBT effect is expected to be negligible, the cumulant vanishes in the limit of large density. On the other hand, for low Q where Bose-Einstein correlations are expected to be dominant, the cumulant tends to a finite value. If one considers the particle emission from a number Ns of independent sources then the particle density is :

dn dv dy sdy


where dv/dy is the particle density from a single source. Then, the normalized two-particle correlation function is [A. Bialas (1999)] :

where K2(y) depends on the particle distributions from one source. Hence, the emission from independent sources implies that the normalized cumu-lant is inversely proportional to the particle density, as seen also in the data (see figure 8.5) at large Q. The data for small Q, possibly dominated by the Bose-Einstein correlations, yields the normalized correlation function which approaches a finite value, in disagreement with the expression (8.15). This could imply correlations between sources. However this dependence, which is seen both for like- and unlike- particles, remains unexplained.

Chapter 9

Random Multiplicative Cascades

The simplest model where the self-similar density fluctuations occur is the self-similar cascade model. Those models were first introduced in the description of the energy dissipation in the fully developed turbulence by Mandelbrot (1974), Frisch et al. (1978), Zeldovich (1985) and Parisi (1985), and as a model of multiparticle production in high energy collisions by Bialas and Peschanski (1988a). Turbulent fluid must necessarily obey the Navier-Stokes equation, whereas the quantum chromodynamics (QCD) in principle describes all high-energy interactions and final states. Nevertheless, in many situations these theories cannot be resolved with the sufficient precision and, hence, the experimental data cannot be discussed. One looks therefore for models which reproduce certain essential features of data providing in this way the guidance in solving the exact theories. Hierarchical structures (cascades) are used because they can be easily made scale invariant and hence they can mimic scale invariance seen, e.g. in the turbulent energy transfer between different scales [A. N. Kolmogorov (1941)] as well as some deviations from it. In high-energy e+e~ and hadron-hadron collisions, particle production can be described to certain extend as the cascade of partons and hadrons. In the low energy nuclear physics, cascade models have been used frequently to study limits of cohesion of nuclear matter. In most applications, however, these models were lacking the self-similarity. The correspondence between the self-similar cascading models and the experimental data is established by assuming that measured densities result directly from such a cascade and then to compare correlations between measured densities in different intervals to those given by the model. In this way, one wants not only to answer whether such models reproduce the

221

222 Random Multiplicative Cascades

correlation data but also understand the statistical content of such models. The main ideas behind these self-similar models, is the independent evo

lution of different branches of the cascade with increasing resolution and the random multiplicative nature of the density fluctuations. As explained in Section 2.3, the multiplicative variables do not obey standard Central Limit Theorem, exhibiting large fluctuations even in the limit of large numbers. In the next section, we shall recall the definitions and the basic results of random multiplicative cascading. In the following section, we shall discuss features of the correlations in different multiplicative cascade models using the formalism of cumulant branching generating functions [M. Greiner et al. (1998)]. Results obtained in this formalism will be compared with those obtained in the multifractal formalism. Finally, important non-ideal effects of the density cut-off on the multifractal features of the random multiplicative cascades will be discussed.

9.1 Multiplicative cascade models

The random cascade is self-similar and thus gives scale-invariant density fluctuations in certain variable such as for example the rapidity. At each step of the cascade all rapidity subintervals are subdivided into A parts. At each subdivision the density in the resulting rapidity interval is multiplied by a random variable X. So that after n steps the density in a given interval of length ln = AY/Xn is given by a product of n independent random variables -X"(&),..., X(£n) (X(Q) > 0 and i = 1 , . . . , n) taken by X at all segments of the path leading to the given bin :

Wn = W0X(h) • --X^) = W(l...Cn (9.1)

where Wo is the initial density. The set Ci • • • Cn, where £j can take any integer value from 0 to A — 1, provides a precise characterization of the path leading to the considered bin. One should pay attention to a different meaning of subscripts in Wn and W^...^ : whereas in the first case the subscript counts number of steps n in the cascade, in the second case the subscript gives both a precise position of a bin after n-steps of the cascade and describes a unique path leading to this bin. Due to the self-similarity of the cascade, all variables X(d) have the same probability distribution P(X) and, therefore, the distribution of Wn is simply a product of distributions

Multiplicative Cascade Models 223

F(X). The random variables are defined so that :

(X) = fp(X)XdX = 1 , fp(X)dX = 1

To complete the definition of the model, we assume that at a given density the distribution of particles in the considered bin is given by the Poisson distribution with the average lnWn. This means that all correlations between particles are included in the density fluctuations.

n-6

X=2 — © —

k=2

AY

Fig. 9.1 The details of the cascade with A = 2, n = 6 and fc = 2 (for the explanation of the notation see the discussion after equation (9.2)). The cascade starts in the upper black dot. The factor Af(n - k) = A/"(4) in equation (9.2) is a sum over sixteen paths from the lower black dot to the bottom of the figure.

9.1.1 Weak intermittency regime

The normalized moments {W£)/{Wk}q of the particle density at a given resolution h = AF/A*, where h > ln, can be determined from the normal-


ized factorial moments in a fixed bin, averaged over different events. The density in a given bin of size lk in the random cascade consisting of n steps (see figure 9.1), is given by the product :

Wk = W0W(l...CkM(n-k) (9.2)

where

AA(n-fc) = ^X(C f c + i ) - - -X(C„) /A"- f c . (9.3)

The summation in (9.3) runs over all paths starting at the considered bin. The moments of Wk are then :

{WD = {pl){Wl.<k){M"{n-k))

and <W£...a) = fP(X)W^<kdX, what implies :

where

ln(X«) In A

(9.4)

and (Xq) = / P(X)XqdX. Assuming that {Afq(n-k)) tends to a finite limit when n —> oo, one obtains the power law behaviour versus the resolution for the scaled factorial moments :

Fg(lk) = Fq(AY)(AY/lky<

with the exponent vq given by (9.4). This is called the weak intermittency limit. In this regime, the self-similar cascade can be infinite and the total multiplicity has a limit distribution [A. Bialas k, R. Peschanski (1988b)]. The condition that for a given q :

(Mq(n-k)) <oo

for any k and n (k < ri) is equivalent to the condition :

vq<q-l (9.5)

with uq given by the equation (9.4). It can be shown that if (9.5) holds for a certain q = q0, then it holds for any q < qo- In this sense, one can define the weak intermittency phase of random cascading for q < qo.

Multiplicative Cascade Models 225

X=2

W(=X(C,) 1

iw2=X(y w, t

w, x.c,-. XW,

w 3 = w „ „ • • • w,=w01„ w3=w„„ • • • w3=w„„ w3=w,„

< A 7 »

Fig. 9.2 The a-model for A = 2. The first four steps are shown. For the explanation of the notation, see the discussion in Sections 9.1 and 9.3.

The scaled factorial correlators Fqtqi (equation (7.17)) in the random cascade model are independent on the width of the considered bins and have a power law dependence on the distance L between the two bins (7.17) with the exponents which are related to the exponents of the scaled factorial moments, as given in (7.54).

9.1.2 Strong intermittency regime

The strong intermittency limit of random cascading corresponds to :

vq>q-l . (9.6)

In this case (Nq(n - k)) diverges at n —>• oo and the power law behaviour of the scaled factorial moments is no more controlled by vq given in equation (9.4), but it takes the form [A. Bialas k K. Zalewski (1989)] :

Fq(lk) = F^AYXAY/lk)"-1 .


It can be shown that if (9.6) holds for certain q0 than it holds for any q > qo- Hence, in general one may have in random cascading two separate and distinct phases : the weak intermittency phase for q < qo (2 < qo < oo) and the strong intermittency phase for q > qo- The transition value 50 between these two cascading regimes, if both of them are present in a given random cascade model, depends on the specific features of the model such as e.g. the probability distribution P(X).

In the parameterization of intermittency effect using the singular distribution function, the scaled factorial moments are related to the integrals of the reduced distribution functions (equation 7.12). The singular scale-invariant distribution function in the variable y, e.g. rapidity :

dq(\yi, ...,Xyq)= X-^dgiy^ ...,yq) (9.7)

gives rise to the intermittent behaviour of the scaled factorial moment Fq ~ (5y)~^. However, this is correct only if the integral in (7.12) exists. This constraint gives bounds on the values of the intermittency exponents vq. Let us consider the case when all g-particle distributions have the form :

dg(yi,...,yq)~l [J U f c - ^ r " ) (9.8) ((npair3) ) sym

where the number n p o j r s of pairs \ yk — yi | is at least (q — 1) and at most q(q — l ) /2 , and the whole expression is symmetrized over the permutations of {?/i, . . . , yp}. If the integral in equation (7.12) exists in this case, then the scaled factorial moment Fq has an intermittent behaviour with the intermittency exponent vq = npairsv. For the distribution functions (9.8), the integral (7.12) exists if the intermittency exponents satisfy (9.5). In this case, one obtains intermittent behaviour of the scaled factorial moments for all scales, i.e. there is no need to introduce any regularization of the singularity in the distribution function.

9.1.3 Regularization of the scaled factorial moments in the strong intermittency limit

If the n-particle distribution (9.7) shows a self-similar behaviour with the exponent vq > q — 1, then to ensure an existence of the integral in (7.12) one should introduce some regularization for \ yk~yi \^> 0- Let us suppose that if the distance between any pair of points yk, yi in (9.8) is smaller

Multifractals and Intermittency 227

than a certain cut-off value I, then \ yu — yi | will be replaced by I. For 5y ^> I , the scaled factorial moments have then the following behaviour :

Fq(Sy) = a li-^iSy)1-* + b (5yyl"> + 0(5y2) .

Clearly, the above expression is divergent for I —¥ 0 if q — 1 < uq. For Sy ^> I, the first term is dominant giving the behaviour :

Fq(Sy) ~ (Sy)1-* .

The above regularization procedure means that a finite scale has to be assumed in the distribution function, below which the scale-invariance breaks down. This is equivalent to saying that the self-similar cascade has a finite length.

9.2 Multifractals and intermittency

The intermittency is profoundly related to the multiscaling of the probability density. Hence, before continuing with the discussion of the correlation in random cascading, let us discuss the relation between intermittency and multifractal formalism. The standard result for the multifractal set without any cut-off, is that the moments of the fractal probability density can be written using the multifractal singularity spectrum f(a) [G. Paladin & A. Vulpiani (1987)]. Let the probability density in a bin scale as la with its length I = AL/M = 1/M and let the number of bins having the scaling index a in the interval [a,a + da] be l~f(a)p(a)da, where /5(a) is a smooth function. Then, the moment rnq of the probability distribution can be written as :

r&max

mq(l) = / laq-f(-^p(a)da . •>Otmin

(9.9)

Using the saddle-point approximation to the above integral one obtains for

U O :

mq(l) ~ rT<«>

where

T ( ? ) = f(asP) ~ q<Xsp

and alp is the value where the function qa — f(a) has the minimum.


Another way of characterizing scaling properties in non-linear physics phenomena, chaotic dynamical systems and various fractal growth processes, is provided by an infinite set of generalized fractal dimensions , or Renyi dimensions [A. Renyi (1970)] :

iog(E£iP?(0) Dq = lim *——- (- (9.10)

H i->o q - 1 log I v ' where pj is the probability to find a particle in an interval j of length I = Sy/AY, and AY is assumed to have a unit length. The generalized dimensions (9.10) are defined for all real q. It can be shown [H. G. E. Hentschel & I. Procaccia (1986)] that for multifractals (inhomogeneous fractals) :

Dq < Dq> for all q > q

whereas for monofractals (homogeneous fractals ) :

Dq — D i for all q, q .

The multifractal singularity spectrum f(a) is connected by Legendre transformation to the sequence of mass exponents r(q) which in turn are related to the set of generalized dimensions [H. G. E. Hentschel & I. Procaccia (1986)] :

D - I M U " - l - q •

Intermittency in particle physics is defined as the scale-invariance of the factorial moments (c.f. Section 7.3.1) with respect to the change of the size of the phase-space cell Sy :

Fq(6y) oc (fy)-"«

when Sy —> 0. For the multifractal distribution we have :

where Pj(5y) is the probability for the particle to be in the bin j of size Sy and r(q) is given by :

Dn A = T<«> q-1

Correlations in Random Cascading 229

Since :

Fq ~ (^)-(?-1)D(pf1) where D is the support dimension, therefore :

uq = (q - 1)(D - Dq A) , q>2 . (9.11)

The quantity Dq A in equation (9.11) is the apparent Renyi dimension. Knowing the intermittency indices and the value of A, one can extract the true multifractal spectrum. It may be noticed that the condition of weak intermittency (9.5) is equivalent to the requirement Dq > 0.

From what is said above, one can see that the intermittency exponent vq is related to the fractal dimension Dq only in the weak intermittency regime (9.5). In the strong intermittency regime , the generalized fractal dimensions Dq vanish independently of the value of vq.

9.3 Correlations in random cascading

The problem of comparing the experimental correlation data with those obtained in the random cascade models is to know exact expressions for correlations in these models. With a clever change of variables from energy densities Wn to In Wn, the multivariate generating function of binary multiplicative cascade models can be derived analytically. Let us start with an initial interval of unit length on which some scalar quantity Wo (=1) is measured. In the first step, let us divide the initial interval into two pieces (A = 2) of equal length 1/2 and divide Wo randomly into two pieces WoX(Ci) (the value of C is 0 or 1 for "left" and "right" interval, respectively, and the subscript of £ denotes the step number in the cascade) :

Wi « i = 0) = W0X(Ci =0)=x0 = Wo (9.12)

in the interval "0" and :

W1(C1 = l) = W0X(Ci = l) = x1=W1 (9.13)

in the interval " 1 " . The multiplicative variables x0 and x\ (X(0) and X(l), respectively) are found randomly from a distribution, called the fragmentation function !F{XQ, X\). Notice that the subscript of W on the left side of equation (9.12) (or (9.13)) numerates the step of the cascade (step " 1 " in the case of (9.12) and (9.13)), whereas the subscript of W on the right side


of this equation is the index of the bin in the cascade of one step, i.e. "0" and " 1 " for "left" and "right" bins, respectively. This notation, which may cause a confusion in the cascade of only one step, becomes clear for longer cascades. In the second step of the cascade, each one of the intervals splits again. For example, interval "0" splits into two equal intervals "00" and "01" and the values of the fragmenting variable in these two subintervals are :

W 2 « i = 0, C2 = 0) = X(C2 = 0)Wi(& = 0) = zooxo = Woo (9.14)

and

W2«! = 0, <2 = 1) = X(C2 = 1)^1 (& = 0) = X01X0 = Woi

in the "left" and "right" intervals, respectively. Again, one should notice a different meaning of subscripts in W on left and right hand sides of (9.14). The subscript of x is the index of the bin and its digital length accounts for the step number of the cascade in which one finds this bin. The new random variables 2:00, #01 (xio, i n for the second interval) are determined by the same fragmentation function J-(xoo,xoi) = T(xio,Xn)). And this procedure can be continued : after n steps of the cascade, one ends up with 2n equal size intervals with random number in each interval characterized by the n-digit binary address C,\ • • • C,n with each & being either 0 or 1. The density belonging to a given bin Ci • • • Cn is given by the product of multipliers over all previous generations :

Wd-Cn =xZx-U---HibxZl • (9-15)

Obviously, this relation becomes additive for the logarithm of the density. This observation is crucial for the following discussion.

The structure of the random multiplicative cascade model is completely determined by the fragmentation function. This function determines also the multivariate statistics of a given cascade. The joint probability density P(ln Wo...00, • • •, In Wi...n) to find In W0...0o in the bin labelled (0 • • • 00) (the leftmost bin) and In W1...11 in the bin (1 • • • 11) (the rightmost bin) can be expressed fully in terms of the fragmentation functions at each branching. Assuming the same fragmentation function at each branching of the cascade one finds :

P(lnW0...oo,...,hiW1...11)


" / n n ^ci-o-io^ci-o-ii^o^i)

Cl-Cn=0 \ j = l

(9.16)

It is straightforward to generalize the whole formulation for the case of different fragmentation function at each branching [M. Greiner et al. (1998)]. This joint probability density can be changed into the multivariate generating function for cumulants in the variable In W. With the definition :

In .M(uo-oo, •••,«!..-n)

= In / exp f Yl uCl...c„ In WCl..<n j \

[I f[ d(\nWCl...Cn)]p(lnW0.. \C l -Cn=0 /

00 . . . , lnWV..„)

x e x p J2 "Ci-CnlnW^-C (9.17)

\C.-C»=o / .

where the tree parameters u^...^ are defined as :

l u Ci-0 = J2 wCi-CiO+i-Cn (9-18)

C i + i -C„=o

one finds after inserting (9.16) in (9.17) and rearranging the terms in the exponent of equation (9.17), that the integral (9.17) factorizes and :

n 1

In M(u0...oo, •••,u1...n) = ^2 ^2 Q(.uCi-Cj-iO,uCl..<j_1i) . (9.19) j = i C i - 0 - i = o

This form makes explicit the statistical independence of different branching in the cascade. The basic object in the sum (9.19) :

Q(UQ,UI) = In / dxodxi!F(xo,xi)exp(UQITIXO + uilnxi)


= ln(exp(«o ln:zo + «i lnari)) (9.20)

is the bivariate local branching generating function which describes the behaviour of the entire cascade and is related to the underlying fragmentation function via a two-dimensional Laplace transformation. It should be mentioned that one can invert Q(uo,ui) to find the fragmentation function J-(xo,Xi). Equations (9.19) and (9.20) provide the analytic expression for multiplicative cascades because the generating functions Q for each branching can be solved separately and analytically. For example, for the cascade consisting of two steps (n = 2) :

InM(u 0 0 , uoi, uio, «ii) = ln(exp(uoo In Woo + ln(exp(uoi In Woi

+ ln(exp(wio In Ww + ln(exp(un In Wn)}

one obtains :

lnAl(uoo,uoi,Mio,wii)

= Q(uoo + uoi,Uio + un)+Q("oo ,uoi ) +Q{uio,un)

= <3(wo, MI) + Q(u0o,uoi) + Q(uw,un)

where UQ = uoo + uoi and u\ = u\o + U\\. The basic solution (9.17) is applicable to any fragmentation function

or branching generating function. Thanks to the additive form for the branching generating function, the fragmentation function can differ from one generation to another, and even, from one branching to another. It can be generalized to higher-variate fragmentation functions as well. The only constraint on the utilization of the equation (9.17) is that the fragmenting variables at every branching must be independent.

The tree structure of (9.18) is shown in Fig. 9.3. Analogous structure for the cumulant branching generating functions Q is shown in Fig. 9.4.

Since Q at each vertex on the tree has only two parameters (the binary cascading), there are two basic kinds of branching cumulants : the same-lineage cumulants :

du0 du\

involving only derivatives with respect to one of two u parameters, and the


-©—

U 10 u 11

u 100 u 101 u 110 M 111

Fig. 9.3 The schematic representation of the tree structure. Each u is the sum of all u's in the tree below it.

splitting cumulants :

Qi+kQ

involving derivatives with respect to both "left" and "right" parameters uo and u\, respectively. This implies that any multivariate cumulant obeys the principle of common ancestry :

• A cumulant between any number intervals is the sum of all same-lineage cumulants at branching points which are ancestors of all relevant intervals.

• At the first splitting in daughters, the splitting cumulant of this branching point is added and the process stops, i.e. once at least one bin has separated into a different lineage, the last common branching point contributes and nothing after that.


Q(u0ux)

2Ko"oi) e

> Q(umumi) Q(unouiu)

InM-

Fig. 9.4 The schematic representation of the cumulant branching functions Q. The generating function In M. is the sum of all Q's on the tree.

I n g e n e r a l c a se , t h e m u l t i v a r i a t e c u m u l a n t of o r d e r n is g i v e n b y :

dn\nM{u0...oo, • • • , u i . . . n ) K, Cl-Cn du? • ••du?

Cl Cn 1=0

where we use the notation £ = (< j , . . . , £n) for the subscripts. For symmetric fragmentation functions and the same branching generating function at all branchings, the two-point cumulant is :

K a' = ( £ J C i C " J C i C ; d2Q{uQ,Ul)

k j = l du2

0 u=0

i = 0

(In Wc- In W ? ) - (In Wf) (In W?) (9.21)


9.3.1 Some examples of the branching generating functions

Before further continuing the discussion of the properties of the random multiplicative cascade, one should make a following remark. One may consider random cascade models which either conserve W at each step of the cascade or allow for the global fluctuations in W. In the latter class, one finds e.g. the a-model, which was discussed in Section 9.1. To the W-conserving (or "energy-conserving") class of models belongs the simplified version of the a-model (A = 2), the so-called p-model, which was proposed for the description of the classical turbulence and was compared to the high energy intermittency phenomena by Lipa and Buschbeck (1989).

Let us consider the multiplicative binomial process as given by the p-model. This one-parameter model is a simplified version of the two-parameter random cascading model, the so-called a-model (a/ . Section 9.1 and equation (9.25)). The fragmentation function for the p-model is :

^ o , x i ) = ^(6(x0 - (l+(3))+6(x0 - (1 -/?)))<f(zo +X! - 2) (9.22)

where /3 is an adjustable fluctuation strength parameter. Hence, xo has an equal chance to be 1 + (3 or 1 — /?, with xi taking the complementary value so that the ^-conservation is ensured at each step in this model. Inserting (9.22) into (9.20) yields :

'1 + /T Q{u0, ui)=-(u0 + ui) ln(l - /32)+ In < cosh

5 ( U ° ui) ln

1 - / 3 , )]} (9.23)

The derivatives of this branching generating function are needed to calculate the cumulant densities. For the first derivative one obtains :

dQ(u0,ui)

dun u = 0

dQ(uQ,ui)

du\ u = 0 | l n ( l /32)

For an arbitrary derivative one finds :

d^+n*Q{u0,ux)

du^du1? (-1) Baa"i+"»q(u0,m)

a< 1 +" 2 i=0

i.e. except for an alternating sign, the same-lineage cumulants and the splitting cumulants are identical in the p-model and it is sufficient to know :

dnQ(u0,Ul)

d< M = 0


infill) (£^^ 2 U-zSA^o" 1 !

1 , / 1 + / 3 -unln 2 V 1 - / 3

(9.24) i=0

to be able to calculate all correlations in this model. Since tanh is an odd-function, therefore all odd-derivatives of the branching generating function are identically equal zero.

The a-model with A = 2 is similar to the p-model except that it does not conserve the density W. The fragmentation function in the a-model

F{x0,Xl) = -[6(x0 - (l + /3))+S(x0 - (1 - P))]

x[S(x1-(l+(3))+S(x1~(l-/3))} (9.25)

Inserting (9.25) into (9.20), one derives the cumulant branching generating function in the a-model :

1 Q(u0, ui) = - (uo + ui) M l - PJ)

{ c o s h [ i W 0 l n ( B £ ) ] } + In < cosh

+ In < cosh

(9.26)

In jc 1 (l+P

For no = 0 or uj = 0, the branching generating function for both a-model and p-model are the same. Consequently, the same-lineage cumulants in both models are identical. The difference between these two models lies in the splitting cumulants which are identically zero for the a-model. The vanishing of these moments is also the reason for the energy non-conservation in the a-model.

9.3.2 Link to the multifractcd formalism

It is instructive to relate the cumulant branching generating function formalism and the multifractal formalism [M. Greiner et al. (1998)]. / ( a ) spectrum of singularity strength is related by a Legendre transformation to a sequence of mass exponents r{q). These are defined by the scaling


behaviour of the moments of the "bin energies" :

( £ (WCl..<nln)q) = dnV™ (9.27) \Cl-Cn=0 /

with ln = 2~n. Since the singularity strengths are related to the bin energies by :

W^» = (i»)a<" , C = (Ci C») (9-28)

therefore the moments in equation (9.27) should be understood as the scale-dependent measure for the one-point statistics of the singularity strength. Instead of densities {W?} and their correlations (W? W? • • •), in the context of multifractals one uses the singularity strengths a?. The definition of these strengths (9.28) means that the singularity strengths a ? are related to the density variables by :

ar=l--$—\aW* . c nln2 c

This means that the logarithm of the density variable Wp is related to the singularity strength (p a nd the singularity strength correlations (a? a ? • • •) are related to to the correlations between logarithms of the densities : (\nWp Inf . - •••).

Let us consider a symmetric fragmentation function of density-conserving multiplicative cascade process. An example of such a function is provided by p-model discussed above. Using the definition of the random variable W^...^ (see (9.15)) and the statistical independence of random variables x, one obtains :

<(^x-c,)'> = <4><4ca> • • • K-<J = W •

Inserting this expression into (9.27), one finds :

T(.)-^(^)-(«-i) + ^ - P-»> In 2

The extension of this relation for the case of asymmetric fragmentation function is straightforward.


9.3.3 Relation between branching generating function and multifractal mass exponents

The relation between the branching generating function Q and multifractal exponents can be seen by inspecting (9.20). One can see that :

Q(q, 0) = ln(exp(glna;o)) = ln(a;^) .

Inserting this expression into (9.29) one finds :

riq) = -(q-l) + Q{U°^2

Ul=0) • (9-30)

It is clear from equation (9.30), that one can consider multifractal exponents r(q) as the limit of the branching generating functions Q(UQ, UI). However, the branching generating functions provide more detailed informations than multifractal exponents because Q is denned in the full (uo — ui)-plane, whereas r(g) is restricted to the axes : UQ = 0 or u\ = 0. This implies that one can imagine different "energy-conserving" fragmentation functions which lead to identical multifractal exponents, while their branching generating functions are quite different. In other words, the multifractal formalism in contrast to the branching generating formalism cannot detect correlations at the branching. Below, we shall illustrate this point on the example of a- and p-models.

The multifractal formalism cannot also distinguish between "energy-conserving" and "energy-non-conserving" cascades. In the p-model, which is "energy-conserving", we can use (9.29) to extract the multifractal exponents r(q) from the branching generating function Q(u0,ui) (see equation (9.23)) :

T ( 9 ) = hT2ln 1 + /A 9 , / ! - / ? 91

The derivatives of r{q) :

dlr(q)

8ql = -Si,i + 1 a'Qtuo.ui)

,=o In 2 dul0 u=0

are related to the same-lineage derivatives of the branching generating function of the p-model.

Non-Ideal Random Cascades: The Cut-Off Effect 239

As we have discussed above, the same-lineage derivatives in both p- and a- models are identical what means that :

T\'l)p—model = TyQIa—model •

On the other hand :

Q{u0,ui)p —model

^ Q(u0,Ui)a—model •

Another advantage of the branching generating function formalism is that it can be adopted to the situation where the fragmentation function changes from one step (generation) of the cascade to another [M. Greiner et al. (1998)]. On the contrary, the multifractals require scaling of the probability density in each subinterval.

The branching generating function formalism has also some limitations which may turn out to be constraining in certain applications. First of all, this formalism requires the statistical independence of splitting functions at each branching and spatial ordering of cascades. A more severe problem is a problem of non-homogeneity of the calculated theoretical correlation functions, which are not invariant with respect to the spatial translations. The restoration of homogeneity, which seems to be satisfied in e.g. turbulence data, may destroy the possibility of inferring the branching generating function.

9.4 Non-ideal random cascades: the cut-off effect

In Section 9.1.2 we discussed the influence of the cut-off in the correlation function on their measure by the scaled factorial moment. We have seen that in order that the scaled factorial moments behave in the same way as the correlation functions, the singularity in the correlation functions must be integrable. A presence of some physical cut-off in the multiparticle cascade which would change the correlations and also the scaled factorial moments at some scale [P. Bozek & M. Ploszajczak (1992b)], is another aspect of the presence of the cut-off in the particle production process. It is obvious that physically one expects the presence of some limiting scale beyond which the scaling breaks down.

Let us consider the case when the number of particles in a given cell of the phase-space is generated according to the Poisson distribution with the mean proportional to the probability density in this cell. Increasing the


resolution, the total density is partitioned between different cells so that the mean density per cell is 1/M. Finally, one obtains very low energy densities in small cells. In the standard approach, these cells contribute to the multiparticle production by the generation of particles with a small mean multiplicity. For sufficiently small probability density this mean multiplicity can be much less than 1. One faces then the question, whether it is physically reasonable to generate particle from such low available density (energy). In fact, if one has in mind the picture that the density in one cell is proportional to the energy available in this cell, then one does not expect particle creation if the available energy is smaller than the pion mass.

Let us define the hadronization scale as a minimal energy necessary say to produce one particle in a given cell. We shall define the cut-off parameter e as a ratio of the so defined hadronization scale to the total available energy in the system. If one considers the random cascading as a model of the string fragmentation, then the cut-off could correspond to the ratio of the transverse mass of the particle to the energy of the string. In the fully developed turbulence, the effect of the cut-off is known as the intermediate dissipation range which causes the viscous cut-off to switch off the scaling in the energy density moments [U. Frisch & M. Vergassola (1991)]. Fluctuations in the distribution of the energy in the different phase-space cells make the effect of the cut-off visible already at resolutions larger than eO, i.e. before the mean value of the energy per cell reaches the hadronization scale.

9.4.1 Multiscaling dependence on the cut-off parameters

The cut-off e in the multifractal set means that we do not include in the calculation of the moments of the probability distribution those bins for which the density is smaller than the cut-off value. Recalling that the scaling of density with the bin is governed by an index a, one obtains a limiting value a for this index :

l* = (Wl) = e .

This means that for each length I, there exist a limiting value of the index a so that bins with a larger value of the index a (a > a) do not contribute.


Effectively, the limits of integration in (9.9) are modified :

mq(l) = f la,1-fWp(a)da .

This modifies the saddle-point estimate if asp > a. As a result, one obtains the following modification of the scaling law for the moments mq [M. Jensen et al. (1991)] :

r r T W if asp<a(l,e) mg(l) ~ I (9.31)

[ z«,-/(«(i,£)) if asp>a(l,e)

The above equation shows that the moments have a scaling behaviour down to some value of lum and for smaller length they follow a pseudoscaling law given by the second case of the above equation. This limiting value of the bin length is given by :

The scale lum where scaling breaks down depends on e and, through <^sp, also on the rank of the moment q. However, in the modified variables Yq, 0 :

(9.32)

all the dependencies of the type (9.31) for different values of the cut-off parameter are universal :

C - r ( g ) 0 if O < 1/a.p

r,(©) ~ { { q - 0 / ( 1 / 9 ) if 0 > l/asp

Now we shall use equations (9.31) and :

Fq{M)^M^Fq{l)-^L (9.33)

f,=

0 =

Inrriq Ine

InZ Ine


to describe the behaviour of the scaled factorial moments of the modified a-model. One gets :

'Fg(l)M9-1+T(") if aty < a(l/M, e)

Fg(M)~iFg(l)M9-1+-W+^(VM,,)-q /(a(i/M,£)) i f aiqJ<a(l/M,e)<a<$

Fq^M^-W-fW™^ if aty > a(l/M, e)

(9.34)

Prom the above equation one can see that the pseudoscaling behaviour of the scaled factorial moments is mainly due to behaviour of the first moment mi which enters in equation (9.33) for the scaled factorial moments. So that unlike for the multifractal moments mq, where the moments of different rank have different lum, in the scaled factorial moments the pseudoscaling sets on at the scale given by lum for mi :

Mlim = e-V"% .

One can define similar variables as for the two-scale Cantor set studied by Jensen, Paladin and Vulpiani (1991). The only difference is that before performing the multiscaling transformation (9.32) one has to rescale the moments and/or to rescale the length variable so that in the scaling regime {I > him) one has the scaling identity :

F,(Z) = r " « .

This can be achieved by studying the multiscaling in the variables :

Fq(M)/Fq(l) , M = l/l

where the second variable is used by convention in the intermittency studies in order to have positive exponents. The multiscaling variables now take the form :

a,- ta3*>/¥1/«, (9.35)

ln(l/e)


The scaled factorial moments for the random cascading model have a universal behaviour in these variables :

(q-l+T(q))e i f e < l / a #

(q - 1 + r(<z))9 + q - (76/(1/0) if l / a $ < 6 < l / a $

(q - 1)0(1 - /(1/G)) if 0 > l/a{qJ

(9.36)

9.4.2 a-model with the cut-off at small scales

Let us define the multiplicative random cascade model with the cut-off. In a given realization of the random cascade, we stop the generation of the cascade in a given branch if the density associated with this branch is smaller than e. This means that the corresponding bin is assumed to have zero density and does not contribute to the moments of the density. Obviously, by dropping these small density bins in each event from the averages over all events, we change the normalization of the density, i.e. mi(M) j^ 1. One has to take this change into account in order to have correctly normalized scaled factorial moments, i. e. in order to conserve the total density (energy). The rule of normalization implied by the equation (9.33) corresponds to rescaling all the bins with density higher than e by an amount corresponding to the density lost by dropping the bins with density smaller than the cut-off.

Results of numerical studies of scaled factorial moments in the a-model (A = 2) with the cut-off are shown in figure 9.5. The higher the value of the cut-off the sooner the initial scaling behaviour (Fq(M) = Fq(l)M

v*) breaks down. The dependence on the cut-off is universal and can be seen in the multiscaling variables Gq and © (see figure 9.6) [P. Bozek et al. (1995)]. The fluctuations caused by the cut-off are dominant for small scales and the effective slopes become much higher than in the original model without the cut-off. For the chosen parameters of the a-model, the (weak) intermittency index should be v^ ~ 0.05 whereas the apparent intermittency index in the pseudoscaling region is v£seu ' ~ 3.4. The exact numerical results show that the breaking of the scale-invariant regime (see figure 9.5) in the scaled factorial moments occurs already for larger scales than the limiting scale liim. This is due to the deviations from the saddle-point approximation. Nevertheless, the multiscaling dependence (9.36) is universal for different

Gq(Q) ~


In F2(M)

8 | . , • • 1

6 -f

4 f

i

2 I lr •

0 5 10 InM

Fig. 9.5 The second scaled factorial moment as a function of the number of bins for the a-model (A = 2) with cut-off in the weak intermittency limit. The open squares, the solid squares, the open triangles and the solid triangles correspond to the cut-off values : e = 10~5 , 3 • 10~5 , 6 • 10~5 and 2 • 1 0 - 4 , respectively. The solid line denotes the results of the standard a-model (6 = 0).

cut-off parameters (see figure 9.6) and the multiscaling analysis gives correct estimates of the cut-off scale, unlike the limiting scale seen in figure 9.5.

In a high energy collision, the value of the cut-off parameter should be given by the ratio of the mass of the particle to the total available energy. This leads to two predictions of the multiscaling for the high energy phenomenology :

• the intermittency should be stronger for heavier hadrons • the intermittency should decrease with increasing cm. energy of

the collision.

The first prediction could be difficult to test because it requires the particle identification in the intermittency analysis and a sufficiently high multiplicities of heavy particles. Moreover, a spurious effect could be present due to the flavour-antiflavour correlations in phase-space. On the contrary, the second prediction can be easily tested. Actually, the results of the EMU01 Collaboration (1991) on the one-dimensional intermittency analysis in the

QCD Cascade 245

G2(Q)

0 Fig. 9.6 The rescaled scaled factorial moment G% as a function of © = In M/ ln(l /«) , for the same parameters as in figure 9.5. The results for different values of the cut-off e are denoted by the same symbols as in figure 9.5.

nuclear collisions show that the intermittency indices decrease with the increasing incident energy (14.6, 60 and 200 GeV).

The above results are quite general. In the perturbative QCD cascade, we have an infrared cut-off which modifies behaviour of the correlations at small scales. We will see in the next section that, indeed, the dependence on this cut-off is universal in the rescaled multiscaling variables (c./. Section 9.4.1). In the fragmentation-inactivation binary cascading model (c.f. Chapter 10), the cut-off appears naturally as due to the inactivation probability and the size of the monomers beyond which the fragmentation process must stop anyway. Also in this model the dependence on the cut-off is universal in the rescaled variables.

9.5 QCD cascade

Quantum chromodynamics (QCD) resembles strongly quantum electrodynamics (QED), and quarks and gluons in QCD play the role analogous to


electrons and photons in QED. Quarks, like electrons, have the rest mass

different from zero. They carry a specific charge, called color, but differently

from electric charge, there are three kinds of color charges : " red", " blue"

and "green" (Nc = 3). Gluons, like photons have no mass, but contrary

to photons have color charge which is the combination of colors of quarks.

There are eight such combinations. Existence of color forces results from

gluon exchange (analogue of photon exchange between interacting charges

in QED) *. Large value of the coupling constant in QCD enables using

perturbat ive techniques in describing "soft" processes, e.g. interaction of

quarks and gluons in the hadron. The special feature of the Q C D is the

so-called asymptotic freedom which means tha t the coupling constant a,

decreases with the four-momentum transferred Q2 in the interaction :

" (llNc-2Nf)hi(Q2/A2) v " " " '

where Nf is the number of types (flavours) of quarks and A is the Q C D

scale. This means that for Q2 » A, the coupling constant becomes very

small and the theory becomes perturbative.

The studies of scale-invariant correlations in the per turbat ive QCD

star ted some years ago in the dipole model [G. Gustafsson & A. Nilsson

(1991)], in which fractal dimensions of the gluon distributions in the ;r-space

were calculated. The mult iparton correlations were studied also by several

groups in the perturbative gluon cascade [W. Ochs h J . Wosiek (1992);

Yu. L. Dokshitzer & I. M. Dremin (1993)].

In the double logarithm approximation and in the high energy limit,

where the cut-off is not important , the perturbat ive QCD cascade shows

intermittent behaviour for the case of fixed coupling constant, and a scaling

with scale dependent intermittency index for the case of running coupling

constant. The intermittency indices of the QCD cascade in the angular

variables are [W. Ochs & J. Wosiek (1992); Yu. L. Dokshitzer & I. M.

Dremin (1993)] :

where 70 :

7o = 6 < a ;s / 7 r

* Gluons which are charged particles can interact among themselves.

QCD Cascade 247

is calculated either with the running coupling constant (9.37) or for a fixed coupling constant : as = a>i'.

Ochs and Wosiek (1992) calculated recursively the many parton correlations in the double logarithm approximation. The many-parton distributions can be calculated from the generating functional Qp(u) :

Pn(P) = du(p1)...du{pn)

where P denotes the momentum of the parent parton and p = (p i , . . . ,pn)-In the double logarithm approximation, the generating functional obeys the following master equation :

*M-'(CW''^W-I«¥:fs) ' T p ( P ) is the phase space of the integration which in the double logarithm approximation takes the form :

Tp(p') = {p' -.P'<P, ep,P < G, p'eplp > Qo} .

Qo is a cut-off parameter. This leads to the recursive relation for the n-particle densities :

Pn (P)(h,---,kn) =dn (p)(fci,...,fc„)+ / d3P pn (P ')(fci,...,fcn) .

The term dn (p)(fci,..., kn) describes the contribution from the lower order densities. In the leading order the partons are angularly ordered, i.e. a parton with momentum P can emit further partons only inside the cone around P limited by P. There is also an infrared cut-off regularizing the singularity of the emission probability for collinear partons.

The correlation functions take the simplest form in the angular variables &P'P, 4> with the momentum dependence integrated over. The integrals can be calculated in the pole dominance approximation, which uses the fact that the main contribution comes from the region where the momenta of two partons are almost parallel.

By angular integrations, one can obtain the scaled factorial cumulant moments (6.11) in a cone with position 9 and opening angle 5 :

1 / f l 2\ '»- 1 -7o(Pe)(n- l ) /2n

K ~ 10{P6) V<*2


In the case of the constant coupling, for angles much larger than the cutoff 9 ~3> Qo/P , one obtains the power law in reduced two-particle correlations (7.2) :

/• a\ 7o /2

<*(*) * ( - ) .

We can clearly identify a scale-invariant regime in the two-particle correlations. This is true for high energy and angles much larger than the cut-off.

The running coupling constant changes this result since then the inter-mittency index becomes scale-dependent. Also as we approach the cut-off the correlations are changed. Generally, the correlation functions can be written as a function of one scaling variable :

ln(9/fl) 6 ~ ln(P9/A) '

This situation is similar the discussion of the self-similar cascade with an infrared cut-off (c./. Section 9.4.1 and 9.4.2), where the presence on the cut-off required the use of universal multiscaling variables of the form :

ln(AY/dy)

]n(E/e)

where E is the total energy and e is the cut-off. In QCD, the total available energy in the jet is PQ and the QCD scale is the effective cut-off in the cascade. The running coupling constant can also be expressed by the universal scaling variable e :

7o - y/a.

Thus in the variables i / ln(l /e) ln fCn and In 6, one should observe a straight line, i.e. a scaling behaviour. If one wants to take into account the cut-off modifications of the correlations, one should use the multiscaling variables (equation (9.35)). Then, the universal multiscaling variables for QCD cascade taking into account both the running coupling constant and the cut-off read :

ln(£n)

(9.38)

tf

QCD Cascade 249

ln(0/<5) ln(l/e)

In these variables one observes a scaling regime and universal deviations from the scaling for any value of the reduced cut-off e.

Chapter 10

Random Cascades with Short-Scale Dissipation

In this chapter we will detail the generic fragmentation model with build-in inactivation mechanism at short scales, which contains essential ingredients of most of the off-equilibrium deexcitation processes in dissipative environment. This model may be considered as a paradigm of all previous discussions, as it deals with an off-equilibrium collective system, which is simple enough to be analytically tractable but non-trivial since it takes into account some collective correlations, e.g. the overall-size conservation. Many previous results can then be recovered here exactly. Because of these features, this model has been used in various phenomenological applications, each of them characterized by its proper competition between fragmentation and inactivation mechanisms.

Fragmentation can be defined as the passage between one body characterized by an extensive quantity, say N, its size, and a set of bodies of the same nature but with smaller corresponding quantities : Si,S2,---,sn. When JV is conserved, then : si + S2 + •. • + sn = N. Such a conservative process is the most widely studied case. Otherwise, Si + S2 + • • • + sn < N and such a process is called a non-conservative process. Moreover, the fragmentation can be either sequential or instantaneous. The sequential fragmentation occurs when the fragments themselves are allowed to break up, what gives rise to a succession of small jumps in the space of configurations. The instantaneous fragmentation means that the typical relaxation time is much larger than the overall fragmentation time. This can be represented by one big jump in the space of configurations. One such model, the Mekjian model, will be detailed in in Section 11.2. The distinction between instantaneous and sequential cases can be made by investigation of microscopic

251

252 Random Cascades with Short-Scale Dissipation

physics which is responsible for fragmentation in each particular example. In the following, we will be interested in the sequential conservative

fragmentation since it is quite general and, contrary to the instantaneous break-up, it is believed to yield universal features which are independent of the precise mechanism governing the fragmentation. The idea of universality in fragmentation first appeared as a consequence of puzzling the power law behaviour of fragment-size (mass) distribution :

P(«) ~ l / s T

appearing in various experiments. In fact, many experimental data can be fitted by this simple law, independently of the sort of relevant interactions or orders of scale. Moreover, the exponents r are remarkably similar for any studied material and a particular value r = 2 looks like a limit value. Let us mention as specific examples: • gabbro fragmentation with lead or steel projectiles [M. A. Lange et al. (1984)] : r = 1.48 or r = 1.57 • mid-size meteorites [R. E. McCrosky (1968)] : r = 1.62 • crushed quartz, gneiss or granite [D. E. Grady &; M. E. Kipp (1985); W. K. Hartmann (1969)] : T = 1.63,1.71 or 1.74 • granite fragmentation in underground chemical or nuclear explosions [V. E. Schoutens (1979)] : T = 1.81 or r = 1.83 • broken coal [J. G. Bennett (1936)] : r = 1.83 • basalt fragmentation with polycarbonate projectiles [A. Fujiwara et al. (1977)] : r = 1.85 • sandy clays [D. E. Grady & M. E. Kipp (1985); W. K. Hartmann (1969)] : r = 1.87 • gravels [D. E. Grady & M. E. Kipp (1985); W. K. Hartmann (1969)] : T = 1.94 • stony meteorites [G. S. Hawkins (1964)] : r = 2.0 • asteroids [J. S. Dohnanyi et al. (1970); J. R. Donnison & R. A. Sugden (1984)] : r = 2.02 • nuclear fragments in the heavy-ion collision with 0.5 GeV < Eiab/A < 10 GeV [W. Trautmann et al. (1993)] : T ~ 2 ± 0.2. These multiple coincidences recall the history of universality in phase transitions and aggregation processes. By analogy with the scaling theory of phase transitions, the self-similar behaviour of fragment-size distribution has been often interpreted as the signature of critical behaviour, though

Random Cascades with Short-Scale Dissipation 253

a more precise analysis indicates that this sign is equivocal and other pertinent observables have to be considered [R. Botet & M. Ploszajczak (1994)]. For the aggregation phenomena, the self-similarity of P(s) has lead to the development of the concept of self-organized criticality [P. Bak et al. (1987)].

As said before, most of fragmenting systems are characterized by strongly off-equilibrium processes which cease due to dissipation. To take these features into account, we shall discuss the kinetic fragmentation model where a dissipative process stops randomly the sequential fragmentation [R. Botet & M. Ploszajczak (1992)]. In the following, we shall consider only a simpler version of this model, the fragmentation-inactivation binary model , where the kinetic rate-equation describe a purely binary process, i.e. any fragmenting cluster gives birth to exactly two fragments. This assumption can be removed and one can easily extend the model to take account of ternary or even higher order fragmentations.

The pertinent physical quantities in the fragmentation process are essentially the fragment size, which can be any extensive variable, e.g. the mass, the momentum, the energy, etc., depending on the example of the physical system, and the fragment multiplicity, i.e. the total number m of particles and/or fragments. Both are random variables characterizable by their probability distributions: P(s) and P(m). It is our experience that most of gross measures do not discriminate among models unless supplemented with more fine grained information concerning correlations of various kinds. The generality of the fragmentation-inactivation binary process, which will be discussed below, ensures the relevance of those results in different domains of physics including : multihadron production in high energy collisions, nuclear multifragmentation, fragmentation of atomic clusters, polymer fragmentation, photoelectron count distribution in optics, meteorite or asteroid fragmentation and, last but not least, the galaxy distribution. After all, the common feature of galaxy distributions, the QCD evolution and the turbulent cascades, is an underlying scaling and branching mechanism, even if obscured in observable limits. All these systems are clearly dissipative and hence apt for a description by a fragmentation-inactivation binary process.


10.1 Basic features of the fragmentation-inactivation binary model

In the fragmentation-inactivation binary model , the anscestor cluster of integer size N is relaxing via an ordered and irreversible sequence of steps. The first step is either a binary fragmentation, say (N) —> (j) + (N — j),

> • • •

H@ ©- -Q ($±

Fig. 10.1 Schematic representation of a topological sequence of the binary sequential fragmentation of the cluster labelled N in the fragmentation-inactivation binary model .

occurring with the probability /o = PF(N), or an inactivation event, say (JV) ->• (AT)* with probability i0 = pi(N) = 1 - pF(N). The asterisks in the superscript denotes an inactive fragment. Once inactive, the cluster cannot be reactivated anymore. The fragmentation leads to two fragments, of respective integer sizes : i and N — i, with the size partition probability oc !FitN-i- This fragmentation/inactivation process is continued for the fragments (see figure 10.1). Note the conservation of the total size at each fragmentation event. In the following steps, the relaxation process continues independently for each descendant cluster until either the low mass cutoff

Basic Features of the Fragmentation-Inactivation Binary Model 255

for monomers, i.e. fragments of size 1, is reached or all clusters are inactive. Instead of defining the model by the fragmentation probability p^(fc)

and the fragmentation rate function (fragmentation kernel) J-jtk-j, one can equivalently define it by the fragmentation and inactivation kernels. Fragmentation and inactivation occur with the respective probabilities per unit of time oc J-j^-j and oc Tk- In this representation, the fragmentation probability PF(&) is given by :

Therefore in the framework of the fragmentation-inactivation binary model , the knowledge of an initial state and these two rate-functions T and I, defines completely the fragmenting system and its evolution.

In the applications of fragmentation-inactivation binary model, the asymptotic regime, when t -> oo is of particular importance. This corresponds to the final system after complete dissipation, and this is sometimes the only observable state in the experiment.

10.1.1 Shattering transition

The composed-particle first moment:

Nc = l - (ni)/N

where (m) is the average number of monomers, plays the role of an order parameter when the total size N becomes infinite. If the probability pi(k) that no event occurs tends to 1, then Nc —> 1- This is called the infinite-cluster phase since there remains a large cluster of size of order N. On the contrary, when pi(k) —> 0, then limjv-><x> Nc < 1, i-e. a finite ratio of the total size is converted into finite-size clusters. This is the shattered phase. Therefore, as N ->• oo, one has in the fragmentation-inactivation binary model a distinct second order phase transition associated with the shattering, i. e. with the total destruction of an infinite cluster.

In other words, when the instability of the larger clusters is more important than the instability of the smaller ones, pp is an increasing function of the size and the system is in the shattered phase. Conversely, when the instability of the smaller clusters is more important, pp is a decreasing function of the fragment size and the system is in the infinite-cluster phase.


The transition line is characterized by the rigorous independence of the probabilities pp and pi on the size of the considered object at any stage in the fragmentation avalanche. It should be underlined that the asymptotic (t —> oo) fragment size distribution is rather independent of the functional form of rate-functions T and I on the transition line, whereas it depends strongly on their form in both shattered and infinite-cluster phases.

10.1.2 Scale-independent dissipation effects : the phase diagramme

Let us assume the homogeneous multiplicative fragmentation kernel: .Fy oc (ij)a, and the inactivation kernel: Ik = I\kP. This choice of homogeneous inactivation kernel corresponds to scale-independent dissipative effects , or, alternatively, to self-similar kernels according to :

J\i,\j — - -'i,j

(10.1)

The transition line (pF(k) = est) in this case corresponds to [R. Botet &; M. Ploszajczak (1992)] :

(3 = la + 1 if a > - 2

and

f3 = a-l if a < - 2 .

• At the transition line, the size-distribution is a power law :

ns(N) ~ NT-l/sT , l<^k<N (10.2)

and the exponent r is always smaller than 2. For example, if a = 0 then r = 2pp at the transition line.

• In the whole shattering phase, even though the system is not critical, its asymptotic size-distribution behaves also like a power law :

ns(N) ~ N/s2a+3^ , 1 « A « J V

but with the exponent always larger than 2, independently of the strength I\ of the inactivation rate. The value of the exponent r , which is the easiest to determine phenomenologically, does not fix unambiguously the

Various Approaches to the Fragmentation-Inactivation Binary Model 257

parameters of fragmentation and inactivation kernels and hence does not correspond to a unique fragment multiplicity distribution.

It is thus inevitable to carefully investigate the properties of the fragment multiplicity distribution in addition to the standard techniques of the fragment-size distribution in order to avoid a possible confusion, particularly dangerous in small and intermediate-size fragmenting systems such as atomic nuclei, fullerenes or metallic clusters.

10.2 Various approaches to the fragmentation-inactivation binary model

Similarly to what we have seen in Chapter 4 for the diffusion problem and in particular for the Brownian motion (Section 4.1), the can be investigated from different points of view. Each of these approaches can be interesting in itself, and all of them yield indeed the same final results. But according to the actual investigated feature one of those attempts can be more convenient than the others. We shall focus now on features of fragmentation-inactivation binary model on the critical line of the shattering transition (c.f. Section 10.1.2). One knows that this boundary separates the shattering phase from the infinite-cluster phase. Both phases are interesting in themselves. The infinite-cluster phase corresponds to the ordered phase : the size distribution is bimodal and the second scaling (c.f. Section 3.5.3) holds. On the other hand, the shattering phase is the disordered phase : the size distribution is a power law with the exponent T > 2 and exponential cut-off for large sizes. Moreover, the Gaussian first scaling (c.f. Section 3.5.2) is recovered. So, from the point of view of the scalings, this irreversible off-equilibrium process resembles the ordinary second-order phase transition at a thermodynamic equilibrium. Nevertheless, some features such as the power law behaviour in the whole disordered phase and the value of exponent r at the transition line are benchmarks of this model.

10.2.1 Fragmentation-inactivation binary model as a random multiplicative cascade

At each fragmentation step, one cluster, say of size k, splits up into two fragments of respective sizes j and k — j . In the terminology of the multiplicative cascades, this corresponds to a binary (A = 2) splitting of some


interval representing the cluster of size k into two intervals : j and k — j (c.f. Section 9.1). Moreover, the sizes of the fragments can be written as the size of the parent cluster multiplied by some random weight : k x X (with X — j/k) for one fragment and k x (1 - X) for the other fragment. The probability distribution for X corresponding to the self-similar fragmentation kernel Tij oc {ij)a is :

T{X0,X,) = ^ l ^ X r S i X o + X1-1) .

In fact, this is the same fragmentation function as in (9.22). The branching generating function (9.20) can then be derived in a standard way :

n< \ i ( r ( 2 a + 2)r(«o + q + l ) r ( m + a + l) , , \ , , n , . Odio, « 0 = In ^ r a ( a + 1 ) r{uQ + Ui+2a + 2) +1-PF) (10-3)

where we use pp as the fragmentation probability. In other words, pp is the probability that A = 2, while pj = 1 — pp is the probability that A = 1. Successive derivation of the equation (10.3) with respect to u0 and «i yields all cumulants in this model.

10.2.2 Fragmentation-inactivation binary model as a mean-field branching process

The fragmentation-inactivation binary process is a branching process , which can be mapped into the mean-field directed percolation on the Cayley tree *. Each node of a Cayley tree is occupied with a probability 1 and at each occupied point in time interval dt one chooses between three possibilities for the forward two bonds issued from the node : "fragmentation", "inactivation", and "no event" with respective probabilities : po = p2, pi = (1 — p)2 and p2 = 2p(l — p). These quantities are related to the constant fragmentation probability pp by :

Po P2

pF = (Po+Pi) (l-2p + 2p2)

*In general, one branching process may trigger another one. To take these correlations into account would in principle require the multidimensional generalization of the fragmentation-inactivation binary model. Assuming the direction of triggering to be random, the critical dimension is d = 4 [N. Jain & S. Orey (1968); S. P. Obukhov (1988); P. Alstrom (1988)], above which the scaling exponents in the generalized fragmentation-inactivation binary model can be derived from a single fragmentation-inactivation binary branching process [R. Botet & M. Ploszajczak (1996a)].

Various Approaches to the Fragmentation-inactivation Binary Model 259

At each fragmentation, a given ancestor cluster is replaced by two descendants and the fragmentation multiplicity increases by one unit. Prom one generation to the next, the number of fragments increases then by a factor 2po + P2 i n the average. At the percolation threshold where the branching tree barely survives : po = p~\ = (1 — £>2)/2, and therefore : PF = 1/2 — pi. Let us note in passing that the fragmentation-inactivation binary process at the transition line is also analogous to the process of directed self-avoiding random walk, because the previously activated sites of this tree-like process repel any subsequent reactivation [S. P. Obukhov (1988)].

The mean-field limit in a broad class of dissipative coupled systems exhibiting the self-organized criticality, can be described by such a critical branching process [S. P. Obukhov (1988); P. Alstrom (1988)]. Adding a simple perturbation mechanism to the relaxation rules of the fragmentation-inactivation binary model, one obtains the fragmentation-inactivation binary automaton which, for any intial condition in the so-called high-viscosity phase (0 < PF < 1/2), drives into the self-organized critical state without characteristic scales in space and in time [R. Botet & M. Ploszajczak (1996a)].

10.2.3 Cascade equation for the multiplicity evolution

Let us define the multiplicity of an event at a time t as the total number of fragments. Let us call P^/(m, t) the probability to get a cluster multiplicity m at a time t, starting from an initial cluster of size N at t = 0. The time evolution equation for the multiplicity is given by the following non-linear rate equations [R. Botet & M. Ploszajczak (1997b)] :

d^^+L/f^N\PN{m,t) (10.4)

N-l m-\

= Yl ^,N-j ^2 Pj(m', t)Pjv-j(m - m', t) + lNS(m - 1) . j — \ m' = l

The independence of the two branches starting respectively from the fragment of size j and N — j is clearly manifested in these equations by the multiplication of the corresponding probabilities.


In terms of the factorial moment generating function :

N

g J V ( u , t ) = ^ u m P J V ( m , i )

one obtains :

dG N~l

+ lN[u-QN(u,t)} (10.5)

with the initial condition :

<7i(u,*) = u

and the normalization condition :

a jv (0 , t )=0 •

Note that the partial derivative is taken at a fixed size N. The sum on the right hand side of equation (10.5) represents binary fragmentation of the primary cluster N into the daughter clusters of size j and N—j, respectively. The second term on the right hand side is responsible for the inactivation and, in fact, it is the dissipative term.

The exact equation (10.5) with the discrete variable j can be transformed into a continuous integro-differential equation with the continuous variable z = j/N, which varies from 0 to 1. This transformation is valid if N is large enough, but the small cut-off ~ 1/N must be kept.

10.2.4 Master equation

The basic master equations of the fragmentation-inactivation binary process, which are relevant for the description of fragmentation of initial cluster (mass, energy, etc.), can be found in [R. Botet & M. Ploszajczak (1994)]. They are easy in principle, but rather complicated in structure and will not be written in details here. Let us recall just the way to obtain them.

At a given time, the system is composed of two sets of fragments : the "active" ones which are allowed to break-up, and there are mo of such fragments, and the "inactive" ones whose number is no. The total multiplicity is m = TOO + no. One has then to define the transition probabilities from the system of multiplicity m to any system of larger multiplicity, i.e. the

Moment Analysis of the Fragmentation-Inactivation Binary Equations 261

disappearance of the system of multiplicity TO, and the transition probabilities between systems of multiplicities m! < TO to the system of multiplicity TO, i.e. the appearance of the system of multiplicity m. The first case corresponds to the fragmentation of any active fragments from the set of mo clusters which exist in the system. The latter case corresponds to the break-up of exactly m — m' active fragments from the set of m'0 existing clusters. A complication comes from the fact that the characteristic scale depends explicitly on the whole size distribution, so that one should not write this master equation in the physical time but instead in terms of another evolution variable, e.g. the number of events.

For that reason, master equations are usually written down for the probability to get exactly the configuration {ni,7i2, • • • , n/v} of nj monomers, n2 fragments of size 2 , . . . ,n^ fragment of size N, but direct equation for the multiplicity can be written as well.

In contrast to the cascade equation which connects various generating functions for different initial sizes at a given time, the master equation refers to generating functions for a fixed initial size but taken at different times.

10.3 Moment analysis of the fragmentation-inactivation binary equations

10.3.1 General equations for the factorial moments and cumulant moments

One can perform once the derivative of the fragmentation-inactivation binary cascade equation (10.5) with respect to u, and taking u = 1 one obtains exact linear recurrence equations for the multiplicity average :

^ ^ = J2 ^N-Mmh - (m)N) +IN(1 - (m)N)

(10.6)

(m)i = 1

which are easy to solve exactly for finite values of the size N, or even generally for special choice of the fragmentation and inactivation kernels.

Normalized factorial moments Fq and cumulant moments Kq of the


PJV(TTI) distribution, can be also easily calculated by the general relations :

1 doQN{u) 1 t™\4

(10.7)

i di]ngN(u) Kq {m)q

N du" u=l

For finite N and t —> oo, Fq moments can be found by solving exact nonlinear recurrent equations [R. Botet & M. Ploszajczak (1996b)] :

( N-l \ ( N-l

3=1 1=0

F0 = Fi = l

E ^ - i E "'Z^mF^iN - j) (io.8)

which are obtained by taking successive derivatives of equation (10.5). In view of possible applications of these equations for the description of jet cascading, it is important that they can be solved easily on a small computer, for sizes N ~ 104 — 106, to give a natural framework for the Monte-Carlo simulations for even bigger sizes, and then to find leading behaviour for asymptotically large N [R. Botet &z M. Ploszajczak (1996c)]. One should also mention that using known relations between factorial moments and other families of moments (see Section 6.1), one may obtain from (10.8) the exact evolution equations for each family of moments.

10.3.2 Moments of the multiplicity distribution at the transition line

Let us take !Fij = (ij)a, and let us focus on the asymptotic regime t —> oo. As it will be shown below, various classes of the multiplicity distributions can be defined in different domains of the parameters pp and a [R. Botet k M. Ploszajczak (1996c)].

On the transition line, an asymptotic cluster size distribution (10.2) is a power law with the exponent T depending on both the fragmentation probability PF and the exponent a of the fragmentation kernel [R. Botet &


M. Ploszajczak (1994)] :

T(T + a) _ 1 T(a + 2) r ( r + 2a + 1) pF T(2a + 3)

(10.9)

On the other hand, from equation (10.8) one obtains the recurrence relation for the multiplicity average :

2 N~1

(m)N =PI + —Y, ia(N - i ) ° H

with (m)i = 1 and $jv = %N + Y^i=\ ^j,N-j- F° r the power law trial function in the case a > — 1 , one finds

(m)N ~

with the two solutions :

/ \ PF {m)N ~1_2pp

and

(m)N ~ aNT~l

Finally, for pp = 1/2 one finds :

(m)N ~ -

aNb

for p F < 1/2

for pp > 1/2 .

lnAT . (10.10)

One can see clearly two particular domains : PF < 1/2 and PF > 1/2, which are called the Cayley regime and the Brand-Schenzle regime, respectively, as well as the mean-field percolation threshold for pp = 1/2. They have been evidenced for the first time in the Section 10.2.2. These domains are defined in the case a > — 1. The domain a < — 1 is called the evaporation regime and needs to be discussed separately.

10.3.2.1 Brand - Schenzle fragmentation domain (pF>l/2,a>-l)

From the general equation (10.5) for the moment generating function, one can extract the leading behaviour of the moments of any order when N is


large. One can calculate for example the asymptotic scaled moments of the multiplicity distribution [R. Botet & M. Ploszajczak (1996c)] :

(mi)N y ^ f = l ) fq2_TpFT{a + T)T{a + iy. ( 1 ( m )

{m)qN p F r ( a + l ) V e 2 - ^ ( r - l ) « + ^

Since this ratio is independent of the average multiplicity (m)jv, therefore the generating function GN(U) is a function of the variable u{m)^ and the first scaling holds, in accordance with the discussion in Section 3.5.2.

The relation (10.11) can be approximated around the maximum of the multiplicity probability distribution. Comparing to the moments of the generalized Gamma distribution (5.24), one finds that the asymptotic scaling function for the multiplicity distribution is a particular case of this function (5.22) with the parameters 7 = 1/(2 — T) and A = 1/(4 — 2r) :

$(z) = Bz^^'1exp -(bz)^ (10.12)

where B is the normalization constant and :

6 = (2 - r ) 2 - T ( r - l ) r ~ 1 .

The most probable value z* of the scaling function (10.12) depends on r and is given by :

{ 0 for r < 3/2

(10.13) (r - l ) - 0 - i ) [2(2 - T)]-(2" r)(2r - 3 ) 2 " r for r > 3/2

Analogously as in (5.23), the behaviour of z* at r = 3/2 is reminiscent of an equilibrium phase transition. Together with (10.9), the relation (10.13) demonstrates an implicit dependence of the most probable value of the scaling function both on the fragmentation probability pp, i-e. on the strength of external driving noise, and on the degree of asymmetry a of the fragmentation kernel through the dependence of T on both PF and a. Figure 10.2 shows the diagram PF — a with all fragmentation regions as well as the line r = 3/2 :

r 2 ( a + l ) r (4a + 4) PF- 2 2 a + 3 r 3 (2a + 2)

separating the two phases (10.13). A few typical multiplicity distributions for various values of r are shown in figure 10.3.

Moment Analysis of the Pragmentation-Inactivation Binary Equations 265

1.00

0.75

0.50

0.25

0.00

Evaporative regime

i i i

V T=3/2

BS regime

Cayley regime

-10 10 20

a Fig. 10.2 The diagram pp — a showing different multiplicity domains at the transition line of the fragmentation-inactivation binary process. The dashed line represents the line T = 3/2 separating two phases : z* = 0 and z* > 0, in the Brand-Schenzle (BS) fragmentation regime (from [R. Botet & M. Pioszajczak (1996c)]).

10.3.2.2 Marginal case : pp = 1/2 , a > — 1

The case j>p = 1/2 corresponds to the percolation threshold of the fragmentation-inactivation binary branching process. The leading behaviour of normalized cumulant moments :

Ka 2q

• l n ^ J V : 2«-1q

-<m) g - l N

shows divergence when N —> oo, following the divergence of the average multiplicity (10.10). Hence, the multiplicity distributions in the marginal case PF = 1/2 and a > — 1 do not belong to the class of Poisson transforms and do not obey the asymptotic first scaling law.


Fig. 10.3 Typical multiplicity distributions in the critical regime : a > — 1 , 1 /2 < pp < 1, are plotted in the first scaling variables for various values of T (from [R. Botet & M. Ploszajczak (1996c)]).

10.3.2.3 Cayley fragmentation domain : pp < 1/2, a > — 1

Fragmentation-inactivation binary process in this regime is analogous to the invasion percolation on the Cayley tree because the cut-off scale for monomers does not intervene in this case. In this regime, the sequential fragmentation process leads naturally to power law distributions in space and time and is analogous to the self-organized criticality phenomenon [R. Botet & M. Ploszajczak (1996a)].

Following (10.5) and assuming that there exists a limiting distribution : GN(U) —• G(u) when N -> oo, one may derive the following equation for the generating function :

g(u) = piu + pFQ2{u) .

Solving this equation for Q{u), one obtains the multiplicity probability distribution for the large sizes N and a > — 1 :

P(m) 2m-1

2m\ (PFPIY m J 2pF


0(z)

1.0

0.5

0.0

pp=0.45

10

Fig. 10.4 Typical multiplicity distributions in the Cayley fragmentation regime for various values of the fragmentation probability, are plotted in the first scaling variables (from [R. Botet & M. Ploszajczak (1996c)]).

All moments of P(m) tend to the finite value and neither depend on the total size N of the fragmenting system nor on the value of a. For example :

PF (m) = lirn (m)N = —

N-t-oo 1 — Zpp

This distribution is peaked at m = 1 and falls down for large values of the multiplicity (see figure 10.4). Obviously, these distributions do not belong to the class of Poisson transforms.

Even if the first scaling is meaningless since (m)jy takes a constant value for N large enough, one can consider the scaling function for the m—probability distribution. For large values of m/(m), it behaves like :

$(z) oc 2T3 /2 exp(-bz)

where the positive parameter b is given by :

b = PF

•\U[4PF(1-PF)] • l-2pF

Note the appearance of a power law distribution :

$(z) -3/2


when PF —> 1/2 .

10.3.2.4 Evaporative fragmentation domain : PF > 0 , Q: < — 1

All results presented up to now corresponded to the domain a > — 1 of fragmentation rate functions Tj^-y Many known homogeneous fragmentation kernels correspond to this domain. These include the singular kernel a = —1 in the perturbative QCD for gluons [Yu. L. Dokshitzer et al. (1991)], a = +1 in the scalar X(j>l field theory in six dimensions [I. M. Dremin (1994)], and many others. For a < — 1, the fragmentation process is dominated by the splitting (k) —> {k —1) + (1) at each step in the cascade. This process resembles the evaporation of light fragments (predominantly monomers) from a big cluster and therefore we shall call it the evaporative regime. At each step of this process, one branch of binary fragmentation dies out almost surely. The dominance of only one fragmentation branch associated with a big cluster leads to an approximate iV-independence of the average multiplicity. Hence, also in this domain the multiplicity probability distributions are neither Poisson transforms, nor they obey the first scaling, and the multiplicity anomalous dimension is equal zero when N —> oo. Typical multiplicity distributions in the evaporative fragmentation regime are shown in figure 10.5.

In the limiting case a = —oo, at each step of the fragmentation one monomer is separated from the large cluster. The generating function of the probability distribution in this case satisfies the recurrence relation :

QN(U) ~PIU + PFUGN-\(U)

which yields the exponential probability distribution :

The asymptotic average value of the multiplicity is :

W l-PF

As in the Cayley regime, one can define the scaling function $(z) writing the probability distribution (10.14) in terms of the scaled variable : z — m/(m) = m(l ~PF)- The scaling function is the exponential function :

(10.14)

$ ( 2 ) oc e


0(z)

H=0.5 -

L PF=0-9

\ PF=0.75

0 1 2 3 4 5 6

Z

Fig. 10.5 Typical multiplicity distributions in the evaporative fragmentation regime are plotted in the first scaling variables for various values of the fragmentation probability. The initial size of the system is N = 1024 (from [R. Botet & M. Ploszajczak (1996c)]).

with the positive parameter c :

In l /PF c = • .

1-PF

Meaningful analysis of scaling behavior in the phenomenological applications requires that the scaling function or its moments, as well as the value of (m) (or (m)/N), is calculated for a fixed value of the control parameter in the studied process. Experimental or numerical data obtained for different values of this parameter can then be sensibly compared. It is not always obvious what should be taken for this parameter. In the strong interaction physics, one takes often the total energy in the center of mass for a given reaction. In the nuclear heavy-ion collisions, one could use the transverse energy of emitted particles and fragments to monitor the violence of the process.


10.3.3 Structure of higher-order cumulant correlations at the transitional line

Let us briefly discuss the pattern of Aq coefficients which measure the amplification of higher order correlations and have been introduced in Section 7.6. The fragmentation-inactivation binary process, which on the transition line has the hierarchical structure of higher order cumulants, provides a physically motivated, dynamical realization of the linked pair approximation.

In the Brand-Schenzle domain of the fragmentation-inactivation binary process, one can derive the dependence of A3 on pp analytically. At pp —> 1/2, the coefficient A3 equals 3/2 independently of a, and decreases monotonously with increasing PF. For a = 0, A3 becomes negative for pp > 0.8856 and diverges to —00 when pp —>• 1. The precise value of A3 in the Brand-Schenzle fragmentation regime depends strongly on both a and pp and one may hope to learn about details of the rate functions from detailed knowledge of Aq's. In the limit pp —> 1/2 of Brand-Schenzle domain, the coefficient A3 approaches 3/2 and decreases to —00 when PF —> 1. Thus, the higher order correlations change discontinuously when passing from Brand-Schenzle domain to Cayley domain.

The marginal case pp = 1/2, a > —1, is particularly interesting because the coefficients Aq are independent of N, even though all Kq moments depend on N. In this special case, A3 equals 9/5 for a = 0.

On the contrary, the hierarchical amplitudes are strictly independent of a in the Cayley regime of fragmentation. In this domain (c.f. Section 10.3.2.3), A3 starts from 2 for pp = 0 and approaches 3 when pp —>• 1/2.

In the evaporative domain (Section 10.3.2.4), amplitudes A3 are only weakly dependent on both a and pp.

Finally, in the limiting case a = —00 (the dashed line in figure 10.6), A3 in the evaporative regime of fragmentation equals 2 independently of the value of pp. Qualitatively similar behaviour is seen also for A4. Figures 10.6 and 10.7 show the dependence of A3 and A4/A3 on the fragmentation probability pp in different fragmentation regimes on the transition line. It should be noticed nevertheless that the structure of correlations in Cayley and evaporative fragmentation domains is somewhat trivial as the mean multiplicity of fragments (m)jv is approximately independent of the system size and changes with the parameters {pp, a} only.

Binary Cascading with Scale-Dependent Inactivation Mechanism 271

A 3

3

2

1

0

-1

-2

0.0 0.2 0.4 0.6 0.8 1.0

PF

Fig. 10.6 The dependence of the hierarchical amplitude A3 on the fragmentation probability pp in different domains at the transition line of the fragmentation-inactivation binary model (from [R. Botet & M. Ploszajczak (1996c)]).

10.4 Binary cascading with scale-dependent inactivation mechanism

The dissipation is often characterized by finite and usually small length scale. It is then an open question to which extent the fragmentation processes which on one side are driven by the homogeneous scale-invariant fragmentation rate function and on other side are inactivated at a certain fixed scale by the random inactivation process, may develop scale-invariant and universal features in both the fragment size distribution hs and the fragment multiplicity distribution P(m). This question is important in view of the widespread occurrence of scale-invariant fragment mass distributions and the lack of convincing arguments for using homogeneous dissipation functions in many processes including parton cascading in the perturbative QCD or the fragmentation of highly excited atomic nuclei, atomic clusters or polymers.

For this reason, we shall consider in this section the fragmentation-

Cayley _ ^

a=-3/2

•

— a=0 — — W B ^ ^ O ^ - 1 / 2

a=0 ' ^ ^ S \ S \

V


A4/A3 6

5

4

-1

Cayley ^ —

cc=-3/2

-

ot=0

/ a=-l/2

a=0 — ^ O ^

0.0 0.2 0.4 0.6 0.8

Fig. 10.7 The same as in figure 10.6 but for A4/A3 (from [R. Botet & M. Ploszajczak (1996c)]).

inactivation binary process with the dissipation at small scales which is modelled by the Gaussian inactivation rate function :

2"

lk{c,a) = cexp 1 "2^2

fc-1 N

(10.15)

In addition, we shall consider the homogeneous fragmentation kernel : J-jtk-j = [j{k — j)]a, with two particular values of the shape exponent : a = — 1 and a = +1 .

10.4.1 First example : binary cascading with a = — 1 and the Gaussian inactivation

The upper part of figure 10.8 shows multiplicity distributions for a = —1 in the first scaling variable z\ = (m — (m))/(m) (the upper left part), and the corresponding fragment size distributions for the same parameters. One finds the first scaling for this value of a in a broad range of c, a parameters. The shape of scaling function &(z) depends on the precise value of both c


Fig. 10.8 Multiplicity probability distributions in the scaling variables and the fragment size distribution for two homogeneous fragmentation kernels and two Gaussian inactivation rate functions (from [R. Botet & M. Ploszajczak (1998a)]) : (i) (upper left part) the fragmentation kernel with a = —1 and the inactivation rate function (10.15) with c = 1 and two typical values of <j for N = 1024 (crosses) and N = 4096 (circles) is plotted in the first scaling variables; (ii) (upper right part) the fragment size distributions in a double-logarithmic scale are shown for the same parameters a,c,<r as in (i). The total size is N = 4096. Big stars represent results obtained for the same value of a, c parameters and for a much larger value of a (<x = 10), to show the independence of the scaling part of the fragment size distribution with the value of cr; (iii) (lower left part) the same as in (i) but for the fragmentation kernel with a = + 1 . These data are plotted in the second scaling variables. (iv) (lower right part) the same as in (ii) but for the same parameters a, c, <r as in the case (iii).

and a. Concerning the fragment size distributions, figure 10.8 shows the distri

butions for different parameters of inactivation rate function Ifc (c, a). For a larger than ~ 0.5, one finds the power law distribution of fragment sizes for any value of parameter c. In the studied case : a = 1, c = 1, the exponent


^ 0 . 7 6 )

0.4 | . . n

0.3

0.2

0.1

0.0 -7.0 -3.5 0.0 3.5

7 ^0.76

Fig. 10.9 Multiplicity distributions for the homogeneous fragmentation kernel (a = - 1 / 3 ) and the Gaussian inactivation (fi = 0, c = 1,<T = 1) are plotted in the A-scaling variables (A = 0.76) for systems of different sizes : N = 2 1 0 , 2 1 2 , 2 1 4 .

r equals 1.8, and its value is then smaller than 2. For a given a, the value of exponent r is remarkably independent of a but depends strongly on the value of parameter c. For a smaller value of a (a = 0.1 is shown in figure 10.8), the fragment size distribution decreases exponentially and the shape of scaling function resembles the Gaussian distribution. The precise form of this exponential distribution depends both on c and a parameters.

As a generic situation for a = - 1 , one finds the scale-invariant region of power law fragment size distributions with r < 2 for a above ~ 0.5 and the exponential region of size distributions for a less than ~ 0.5. This power law region is completely analogous to the evaporative fragmentation regime (Section 10.3.2.4) of scale-invariant fragmentation-inactivation binary model, because the multiplicity anomalous dimension :

g = r - l (0 < 5 < 1)

is identical in both cases. This relation has been verified for a broad range of c, a values. In the regime of exponential fragment size distributions g is


always equal 1 independently of the value of parameter c, i. e. this region is in the shattered phase.

10.4.2 Second example : binary fragmentation with a = + 1 and the Gaussian inactivation

In the lower left part of figure 10.8 we show typical multiplicity distributions for a = + 1 , which are plotted for different system sizes in the scaling variables of the second scaling (c.f. Section 3.5.3). The corresponding fragment size distributions are shown in the lower right part of figure 10.8. Again, the precise form of scaling function $(z) depends on the chosen set of parameters c and a.

In the studied case : a = 1, c = 1, the exponent r equals 2.8. The fragment size distributions for a = +1 and different values of a behave similarly as for the a = — 1 case, except that now for a above ~ 0.5 the power law exponent is r > 2. For all a, i. e. in both exponential and power law regions of size distribution, the multiplicity anomalous dimension is 3 = 1 and the second scaling holds. This generic situation is completely analogous to the multiplicity behaviour found in the shattering phase of the scale-invariant fragmentation-inactivation binary process.

10.4.3 A-scaling vs value of exponent r

Whenever the fragment size distribution is a power law, the first scaling of multiplicity distributions is associated with r < 2, and the second scaling of multiplicity distributions with r > 2 in both scale-invariant and scale-dependent regimes of dissipation. This clearly indicates a direct relation between the multiplicity scaling law and the fragment size distribution scaling regimes in the fragmentation-inactivation binary model.

A novel aspect of the Gaussian fragmentation-inactivation binary model is associated with properties of multiplicity scaling in the new region of exponential fragment size distributions. In this region, the second scaling law holds for a > 0 whereas the first scaling is seen for a < —1/2. In between, for —1/2 < a < 0, in a broad range of c, a parameters, the multiplicity distributions follow the A-scaling with 1/2 < A < 1 Figure 10.9 shows an example for a = —1/2 in the scaling variables for A = 0.76.


2.0

1.5

1.0

0.5

0.0 0 1 2 3 4 5 6 7 8 9 10

a Fig. 10.10 The normalized cumulant factorial moment K.2 of the fragment multiplicity distribution is plotted vs the width parameter cr of the Gaussian inactivation function (10.15) with c = 0.5, land5. The homogeneous fragmentation kernel is taken with a = - 1 .

10.4.4 Multiplicity fluctuations in different physical systems and in the binary fragmentation

The dependence of the second normalized cumulant factorial moment K2 (see definition (6.11) on the width a of inactivation rate function Xfc(c, cr) is shown in figure 10.10 for a = — 1. The multiplicity fluctuations as measured by K-2 are extremely small in the exponential region for a less than ~ 0.5. The change of K-2 when passing from the exponential to power law region is continuous but the largest variations of £2(0) appear always at a ~ 0.5. For large values of cr, the cumulant factorial moment approaches a limiting value which depends on the value of parameter c. For —1/2 < a < 0, where the A-scaling holds with A different from both 1/2 and 1, the dependence of K.2 on the width a is shown in figure 10.10. The moments K2 in this case depend on the system size, as will be discussed in the next chapter.

The experimental informations about K2 are not numerous and concern mainly charged particle multiplicities at relativistic and ultrarelativistic energies. DELPHI Collaboration [DELPHI Coll. (1991)] reported the data

Perturbative Quantum Chromodynamics Including Inactivation Mechanism 277

on hadron production in e+e~ annihilations for the center of mass energy of y/s — 91GeV finding /C2 = 0.04. In hadron-hadron collisions n+p, K+p, pp, pp for cm. energies ranging up to 1000 GeV, values of /C2 increase from about 0.05 to 0.3 as energies increase to collider values [G. Giacomelli (1989)].

Distribution of galaxy counts in the regions of sky covered by the Zwicky catalogue [F. Zwicky et al. (1961)] yields K2 ^ 0.3 [P. Carruthers & D.-V. Minh (1983)].

Independently of the question whether the first scaling holds in all those different physical systems, the measured values of £2 clearly exclude the exponential region of Gaussian fragmentation-inactivation binary process. Much more information could be extracted if in addition to the moments of the multiplicity distribution also the mass distribution would be available. In high energy lepton and/or hadron collisions for example, this would require measuring the hadron mass distribution.

10.5 Perturbative quantum chromodynamics including inactivation mechanism

In the limit of "no dissipation", the fragmentation-inactivation binary process (10.5) yields rate equations of the gluodynamics in the NNLL approximation [Yu. L. Dokshitzer et al. (1991)]. The inactivation term of fragmentation-inactivation binary model yields a unique prescription of how to obtain the generalized rate equations of perturbative QCD in the NNLL approximation which contain phenomenologically the essential elements of hadronization through the inactivation mechanism of parton cascading.

The time t appearing in (10.5), arises within the fragmentation and inactivation kernels, which themselves are probabilities per unit of t. We define then the time as : t = Tin Y, where T is a constant, Y = ki(NQ/Qo), Qo = est. Variable 0 plays the role of time, ordering the sequence of events. Assuming that all physical quantities depend only on the variable Y and not on N and 0 separately, we transform (10.5) into :

| p ( I » = J -Yo2(Y)*Zll-z[g(y + ]ogz,u)g(Y + logQ.-z),u)

-g{Y,u)}dz + n(Y,u) (10.16)


where :

n{Y,u)=l02{Y)l{Y)[u-g{Y,u)\ . (10.17)

The initial and normalization conditions are :

g(o,u) = u , £(r,o) = o

and :

2AWK, T TV I

2(Y) in equation (10.17) is the inactivation function written in the new variables. Equation (10.16) provides the generalized gluodynamics equations in the NNLL approximation if N is the initial momentum, 0 is the angular width of the gluon jet considered and T = 12Nc/(llNc — 2N/). Nc

and Nf denote the number of colors and flavors, respectively. The standard gluodynamics equations of the perturbative QCD correspond to neglecting the inactivation term lZ(Y,x) in equation (10.16).

Generalization of equation (10.16) to include both quarks and gluons is straightforward. The corresponding generating functions satisfy the coupled non-linear equations :

£ TfjSfasfMgS-jfat) - g%(u,t))

N-l

+ NfJ2 ^-<5°(^9(«.*)^-J-(«.*) - sgM)

+ 7 o2 l £ [ W - e £ ( M ) ] (10.18)

+1*i%lu-g%(u,t)}

where <?$, Qgf are the generating functions of the gluon and quark jets

respectively. Functions Tf^E? are the QCD motivated fragmentation

# M ) = 7 o 2

at ( « . * ) = 7o

Perturbative Quantum Chromodynamics Including Inactivation Mechanism 279

kernels for the splitting of the parton A into partons B, B :

*i>N-t ~ 9N { j ~1+2NJ

G^QQ = J_(f_, 3{N-j)\ j'N~j 16N \N2 N2 J

i.N-3 N y j AT V N2

The inactivation term consists now of two rate-functions : Zjy and 1°, for quark and gluon jets respectively. Equations (10.18) provide the system of discretized coupled equations of the dissipative perturbative QCD in the NNLL approximation for the generating functions of the multiplicity distributions of quarks and gluons. In the continuous form they become extension of the standard perturbative QCD equations in NNLL approximation [Yu. L. Dokshitzer et al. (1991)] :

dQG-(Y,u) = / l02{Y){$GJGG[gG{Y + logz,u)GG(Y + log(l -z),u)

-gG(Y,u)}+Nf^GtQ

zQ[GQ(Y+hgz,u)gQ(Y+\og(l-z),u)

-GG(Y,u)]}dz + KG(Y,u) (10.19)

^r{Y, u) = J' 7o2(Y){^_QzG\GQ(Y + logz, u)QG{Y + log(l - z), u)

-gQ{Y,u)]}dz + 1lQ{Y,u)

where :

KG{Y, u) = l02(Y)lG{Y)[u - gG(Y,«)]

(10.20)

1l^(Y, u) = fQ{Y)TP{Y)[u - gQ(Y, u)} .

The initial and normalization conditions are respectively :

S ( G ' Q ) ( 0 , u ) = u

g(CQ)(Y,0)=0 .


In the following, we shall restrict discussion to the dissipative gluodynamics (10.16).

10.5.1 Multiplicity distributions in the dissipative gluodynamics

In the limit of no inactivation, the rate equations of generalized QCD (10.16) for gluodynamics or (10.19) in the general case, is cut sharply at Qo, i.e. at a fixed time, and one assumes usually that the cutoff scale is of the order of the hadron masses. This is the usual procedure of solving the perturbative QCD rate equations in the NNLL approximation. One then relates parton and hadron spectra assuming the local parton - hadron duality [Ya. I. Azimov et al. (1985)]. This assumption is surprisingly successful [S. Lupia & W. Ochs (1996)] although, conceptually, it is somewhat unsatisfactory because the evolution equations do not yield the asymptotic spectra. An inclusion of the inactivation mechanism in the parton cascades is associated with one additional phenomenological function T{Y), which could be determined by fitting the data. These two ways of obtaining the final spectra are by no means equivalent. In the shattered phase, for example, equations (10.6) can be solved asymptotically (for N —> oo) and the result is that the anomalous multiplicity dimension behaves like :

j ~ 7 o ~ 1/VlnN

if the system is stopped at a fixed time (sharp low-mass cutoff), and like :

g ~ est

when the system is allowed to evolve till its final state. The dependence of g on energy has been studied in various approximations for perturbative gluodynamics and for \<j)l theory (!FJ,N-J = 6j(N - j)/N3) and many results are now available [Yu. L. Dokshitzer et al. (1991); S. Lupia k W. Ochs (1996)]. These results are obviously identical to the fixed time limiting results of the fragmentation-inactivation binary model. For 1(Y) ^ 0, various classes of the multiplicity distributions have been found at the transition line (see discussion in Section 10.3.2). The transition region between Brand-Schenzle and evaporative domains (a = — 1, 0 < pi < 1/2) is most interesting for the description of particle production in the framework of the perturbative QCD. In this case [R. Botet & M. Ploszajczak

Phenomenology of the Multiplicity Distributions in e+e~ Reactions 281

(1996c)] :

i - 2 p f

(m)jv~(lniV) " (10.21)

as a leading behaviour of the solution of the recurrent equation :

derived from the relation (10.6) for a singular kernel FjtN-j = [J{N — j ' ) ] _ 1

and constant inactivation probability p j . In this case :

5 ~ 1/lnJV .

In gluodynamics, the vector nature of massless gluons leads to the kernel which is a superposition of regular multiplicative kernels with a = 0, and a = 1 in the Brand-Schenzle domain and the singular kernel with a = — 1 in the transition region a = — 1, 0 < pj < 1/2. It turns out however, that the multiplicity distribution in this case is dominated by the singular kernel a — — 1 and the leading term of the inactivation function is Xk ~ lnfc.

10.6 Phenomenology of the multiplicity distributions in e+e~ reactions

In the absence of a quantum theory modelling transition through the non-perturbative phase of the parton evolution into the final distribution of particles, the inactivation rate-function has to be determined by fitting the experimental data. Description of this final phase in the extended perturbative QCD equations for the generating functions of the multiplicity distributions is viewed as a competition between the parton fragmentation with QCD fragmentation kernels and the parton inactivation with phenomenological kernels. The data on multiplicity distributions in the e+e~ reactions are particularly suitable for a phenomenological determination of the inactivation rate-function, as in this case the main graphs involved in the multiparticle production are of the tree-like type, similarly as in the generalized perturbative QCD approach, a variant of the generic fragmentation-inactivation binary process. Present experimental data for e+e~ reactions show that [B. Buschbeck (1995)] :


• the KNO scaling (the first scaling) is approximately satisfied • the mean multiplicity is approximately :

(m)jv ~ ao + ai In iV + a2ln2 N .

The first observation allows to locate the fragmentation domain close to the transition region a — —1, 0 < pi < 1/2 between Brand-Schenzle and evaporative regimes of fragmentation. To fit the experimental dependence of the mean multiplicity on energy for e+e~ reactions at y/s = 2NQo < 100 GeV [J. Drees (1991); G. Giacomelli (1989)], we have tried several functional forms for the inactivation probability but by far the best agreement has been found for the Gaussian inactivation rate-function :

Pi (J) =Poexp V2'

N

with a maximum at k = 1. The two parameters : po = 1/2 and c = 25, have been obtained by fitting the experimental KNO scaling function * (see figure 10.11). These data do not allow to determine po with a high precision. In fact, any value of po between 0.4 and 0.5 can be used to obtain approximately the same quantitative behaviour of the multiplicity distributions. For large parton (cluster) sizes, pi(j) decreases fast with the increasing size j , i.e. the fragmenting system at the beginning of the process is in the shattering phase. With decreasing parton (cluster) sizes, the fragmenting system approaches the transition region : a = — 1 , pi(j) —• po, between the Brand-Schenzle and evaporative domains. The influence of big partons (clusters) and, hence, of the evolution in the shattering phase on the multiplicity distribution, becomes negligible for very large initial parton size N (or y's). Hence, small deviations from the KNO scaling seen in the data for y/s < 100 GeV can be interpreted as the finite-size phenomenon related to the small size of the initial jet. Actually, for a Gaussian inactivation rate-function with the above chosen parameters po and c, the cross-over to the asymptotic region should be seen for y ' i ~ 105 GeV. It should be mentioned however, that the precise value of this cross-over energy depends strongly on the value of the parameter po in the inactivation probability pi. The choice of p0 = 1/2 leads to a particularly extended transient region because of the double-logarithmic behaviour of

•Notice that KNO scaling is a particular case of the first scaling (c.f. Section (3.5.2) and (5.5)).

Phenomenology of the Multiplicity Distributions in e+e Reactions 283

log[<m>P(m)]

J i L

' 0.0 1.0 2.0 3.0

m/<m>

Fig. 10.11 Multiplicity distributions in KNO variables for e+e~ reaction [J. Drees (1991)], which are obtained by solving equations for the generating function of the multiplicity distribution of the extended gluodynamics in the NNLL approximation with the running coupling constant. Calculations are performed for initial sizes : N = 256 (stars), 512 (triangles down), 1024 (triangles up), 2048 (diamonds), 4096 (squares) and 8192 (circles), where ^ = 2NQo and the cut-off is Qo = 270 MeV.

{m)^- For po < 1/2, the dependence of (n)jv on Ins (InJV) is algebraic (see equation (10.21) ) and the cross-over energy decreases. For po = 0.4, for example, it is seen for -y/s ~100 GeV.

In the upper most part of figure 10.12, we show the dependence on logiV of the asymptotic average multiplicity of particles (clusters) (m)jv, which should be identified with the asymptotic hadron multiplicity involving both charged and neutral particles. The cut-off scale for monomers related to an effective hadron mass is taken here to be Qo =270 MeV. We have checked that the final results depends only very weakly on the precise value of Qo- Also approximately linear dependence of (m)jv on IniV (or Ins), as seen both in the data for */s~ < 100 GeV [J. Drees (1991); G. Giacomelli (1989)] and in the figure 10.12, is a pre-asymptotic feature which is expected to be replaced by (m)jv ~ ln(lniV) (or ~ ln(lns)), when V i -> oo. Experimental finding of this transition to the asymptotic regime of the critical fragmentation, would allow to fix unambiguously the value

0.0

-1.0

-2.0

-3.0


logN

Fig. 10.12 The average multiplicity (m) and the first two cumulants of energy spectra, i.e. the average of the logarithm of energy (size) of the final parton (cluster) £ and the corresponding dispersion a2, are shown as the function of the initial cluster size N (= y/s/2Qo). For more details, see the caption of figure 10.11.

of the parameter po of the Gaussian inactivation probability pi. The asymptotic cluster-size distribution gives information about the

energy spectra of hadrons. In the lower part of figure 10.12, we show the dependence on In N of the average of the logarithm of energy of the produced hadron £ and of the corresponding dispersion a2.

Chapter 11

Fluctuations of the Order Parameter

Fluctuations in many physical processes are difficult to analyze because they develop dynamically and often keep the memory of initial conditions. This is in particular true whenever long-range correlations are present in the system. In this chapter we shall discuss features of fluctuations of the physical quantities (observables) in the finite iV-body, d-dimensional system, which is not necessarily in the thermodynamic equilibrium. With the advent of advanced detection systems, the study of large fluctuations in physical observables became accessible in 'small systems', such as formed in ultrarelativistic collisions of hadrons, leptons and nuclei, in the heavy-ion collisions at the intermediate energies or in the collisions of atomic aggregates. Let us denote by m the observable under investigation. For the reason of presentation, we shall assume that m is a scalar quantity with real and positive values. This assumption can be easily removed and one can consider e.g. \m\2 as well.

The simplest measure of fluctuations is the variance (or second cumu-lant moment) : K-I = ((m - (m))2), where brackets (• • •) represent the average over statistical fluctuations for an ensemble with fixed number N of particles. Usually, moments (m), (m2), etc., depend on the size N of the sample. It can appear, and this is the usual case, that the normalized variance K2/(m)2 still depends on N. Then one may define the characteristic size N* for which the fluctuations can be neglected, i.e. when K2/'(m)2

becomes smaller than 1. Hence, one defines a coherence length ~ N*l/d

which could be associated with the disappearance of relative fluctuations. In other words, the value of the averaged order parameter over a domain of diameter larger than JV*1/d is w e l l defined and not fluctuating.

285

286 Fluctuations of the Order Parameter

This procedure fails for self-similar systems, such as the fractal objects or the thermodynamic systems at the second-order transition point for which the characteristic length does not exist and such system look similar at any length scale. One deduces that, in this case, the normalized variance cannot depend on N. Such a behaviour has been found in a slightly different context of the linked pair approximation (in Section 7.6).

In self-similar systems, the A—scaling framework (see the Sec. 3.5) provides necessary tools to extract such generic features of the complex systems. In the present chapter, we shall discuss the general problem of the scaling of the fluctuations of the order parameter in at-equilibrium and out-of equilibrium critical systems. Various examples will be discussed in details.

11.1 Order parameter fluctuations in self-similar systems

11.1.1 The anomalous dimension

For the infinite system, one has to define an intensive order parameter tj which takes finite value for the system and satisfies the stability condition, namely that joining two infinite systems characterized by the same value of 77, the joined infinite system has still the common value of rj. For finite systems, the problem is formulated a little bit differently since, generally, the size of the system is difficult to define precisely. In this case, it is often more convenient to work with the associated extensive order parameter m = Nr], which is the observable.

The anomalous dimension for this extensive quantity can be defined as :

g = lim gN = lim (ln(m)) . (H-l)

We derived in Section 3.5.1 the value of exponent g as :

g=l-P/ud

with (3 the exponent of the singular part of the order parameter with the distance e to the critical point, and v the exponent corresponding to the divergence of the correlation length. Both must be positive, so g < 1. Some other general thermodynamic inequalities are known for critical systems at the thermodynamic equilibrium. Fisher inequality [M. E. Fisher & D. S.

Order Parameter Fluctuations in Self-Similar Systems 287

Gaunt (1964)] states that :

7<(2-ri)v (11.2)

with 7 the positive exponent which governs the divergence of the isothermal susceptibility, and r) the characteristic exponent of the correlation function. In addition, the Buckingham-Gunton inequality [J. D. Gunton & M. J. Buckingham (1968)] states that :

2~^WT,- (1L3)

One deduces from (11.2) and (11.3) that :

1 <9<1 2 _ y _

for any thermodynamic equilibrium critical system. Another exponent of importance for the present discussion comes from

the decrease of the scaling function when the probability distribution of the order parameter is written in the first scaling form at the critical point :

(m)FN(m) = $ (j^J = $(z0

where

*(«i) ~ exp {~czf+1) .

An interesting relation exists between exponents 5 + 1 and g. For the second-order critical phenomena, one knows that : a + 2/3 + 7 = 2 and hyperscaling relation states : I'd = 2 — a. Then one deduces :

5 + 1 = -!— (11.4) 1-9

and the common value is 2 + 7//S. One recovers in particular the non-Gaussian behaviour of the tail of the scaling function :

5+l>2 .

The above relations can also be used reversely. Suppose that an observed system exhibits the second order critical phenomenon, but analyzing the


data using tools of the first scaling law one finds one or several of the following results :

either : g < 1/2

or : 6 + 1 < 2

or : 6 + l?l/(l-g)

In this case, one can deduce that one is dealing with an off-equilibrium critical system. We shall see below some examples of such systems.

11.1.2 Critical cluster-size

Whenever the cluster-size in systems exhibiting the second-order phase transition can be reasonably defined, like in the case of percolation, Ising model or Fisher droplet model, the exponent r of a power law cluster-size distribution ~ l/sT satisfies additional relations [D. Stauffer & A. Aharony (1992)] :

a

r - 1 7 + 2/3=

a

what yields :

9 = TTl (1 L 5) and

S + l = ^ l • (11.6)

Since g at the second-order equilibrium phase transition is contained between 1/2 and 1, therefore allowed values of r are 2 < r < 3 and the normalized values of the size-distribution for which :

N

'^2sns=N s=l

are

ns ~ N/sT .

Order Parameter Fluctuations in Self-Similar Systems 289

11.1.3 Note about the correct order parameter

As a consequence of the results of the previous section, one can say whether a studied parameter can be the order parameter in the studied process.

Let us define, for example, the multiplicity as the total number of clusters *. The cluster multiplicity cannot be the order parameter of equilibrium second-order phase transition because with 2 < r < 3 :

N

s=l

i.e. the average multiplicity scales as the total mass of the system and, consequently, 5 = 1 .

On the other hand, the size of the largest cluster smax is a natural order parameter because :

(smax)~N^ . (11.7)

This is a direct consequence of the fact that in the average :

^ n s ~ l \smax)

i.e. there is only one largest cluster. One may notice that the relation (11.5) is correctly recovered. One should emphasize also, that the relation (11.7) is very general and its derivation does not depend on the assumption of thermodynamic equilibrium. In other words, the relation (11.7) between the anomalous dimension and the exponent r of the scale-invariant fragment-size distribution, is valid also for off-equilibrium second-order phase transitions . We shall see in Section 11.2 that the existence of power law fragment-size distribution with r < 2 does not imply automatically an existence of the second-order phase transition with fragment multiplicity as the order parameter.

*This definition of clusters includes also monomers.


11.2 Example of the non-critical model

11.2.1 The weight functions

The multi-fragmentation model proposed by Mekjian (1990b) is an equilibrium model which describes the decomposition of system into an ensemble of fragments. The statistical weights for every configuration of fragments are given explicitly in this model. If ns denotes the number of fragments of size s with the size-conservation : N = £2S sns, the weight function for the configuration {ns} is given by :

cn1bs

w K } , a ) = n n j l J n J ( a + a - 1 ) ("•*>

with a being a real positive control parameter. Many exact results can be obtained in this simple model. Here, we

are interested in the multiplicity distribution Pjy(m), where the fragment multiplicity is : m = Yl3

n« • This probability distribution could be calculated directly by summing the weight functions (11.8) over the states with constant multiplicity. But the use of the generating function is more convenient.

11.2.2 Check of the linked pair approximation

One can show that the factorial moment generating function Q (as denned in (6.6)) is given explicitly by :

m = 0 T{au)T(a + N)

from which one can deduce the following form for the probability distribution [S. J. Lee k A. Z. Mekjian (1992)] :

P „ w . . - l f l j r > i i ^

where | S ) ^ | are the signless Stirling numbers of the first kind [M. Abramowitz & I. A. Stegun (1964)]. The generating function (11.9) can

Example of the Non-Critical Model 291

be expressed as an infinite product with use of the following identity :

~ _l + a(u-l)/(j + a) 1

3=0

c(v) = TT 1 + a(u-1)/0' + a)

One then expands the logarithm of Q(l + u) as a series of u to find the cumulant factorials fq :

,,-<-,r.h-i) lg_LI5..

Only the first moment f\ = (m) diverges when the size of the system goes to infinity. For q > 1, fq's tends to a finite value :

fq = (-lY+\q-l)\a"aq,a)

where ((q, a) is the Hurwitz function :

oo

j=0

More precisely, for q = 1 one obtains the logarithmic behaviour :

N-l

(m)=a^2 ~ =alaN+(a-l)i-rl)(a)+0(l/N) . j=o 3 + a

Having these results, one can check various correlation patterns such as for example the linked pair approximation . In this case, the linking coefficients Aq (7.20) behave asymptotically as :

Aq ~ In9"2 N

what excludes the validity of the linked pair approximation in the Mekjian model.

11.2.3 Second scaling law

Making now an asymptotic development of the moment generating function M(u) = Q(exp(u)) for large N and small u, one obtains :

J2 emuPN(m) ~ JV^"-1) . m = 0


The latter approximation is known to be correct for finite values of u [R. Delannay et al. (1996)]. This means that PJV(WI) is approximately a Poisson distribution with the parameter a In N. In the leading order we have then : (m) ~ m* . Inverting (11.9) to get Pjv(m) as a Fourier transform, and making N large yields the scaling formula :

( m ) 1 / 2 P N ( m ) - - ^ e x P r -

(a-l)(7-</>(<*))

(TO — (m)) 2

2(ro)

m — (TO)

(m)

+ ° ( R ) ) • <1L10)

This is nothing else but the second scaling law for the multiplicity distribution when N becomes large enough. The second term in (11.10) is always small compared to the first one for finite a.

This result is consistent with the fact that the Mekjian model is not critical, at least with the fragment multiplicity as the tentative order parameter.

11.2.4 Note about the average size-distribution

Different fixed values of the control parameter a allow to simulate different kinds of the fragmentation process. For a < l , one has the situation of a fused system. For a ~ 0.5, the fragmentation resembles the evaporation of light fragments. The limit a > l corresponds to the complete dissociation of the mass into light fragments (monomers). Each of this situation is characterized by a different fragment-size distribution.

The case a = 1 is particular in this model since it leads to the power law size-distribution with the exponent r = 1 [S. J. Lee & A. Z. Mekjian (1992)]. Even with this self-similar feature, and even for the value r < 2, one does not see any critical behaviour in the TO-distribution. Hence, the power-law size-distribution alone does not guarantee that the system exhibits any critical behaviour.

The results of this model in the multifragmentation limit have been compared to the predictions of a Saha equation [D. D. Clayton (1968)] which describes ionization of a gas, in order to propose a phenomenological de-pendance of the parameter a with the physical parameters of a fragmenting system [A. Z. Mekjian (1990a)].

Mean-Field Critical Model: The Landau-Ginzburg Model 293

11.3 Mean-field critical model: the Landau-Ginzburg model

11.3.1 Landau-Ginzburg free energy

As an exactly solvable example of the second-order phase transition, let us consider the Landau-Ginzburg theory. The homogeneous Landau-Ginzburg free energy density is :

ffo) = erf + bV4 + • • •

where b is a positive constant. Variable rj stands for the intensive order parameter of the system. Therefore, it is expected to be exactly 0 in the disordered phase, and non-vanishing in the ordered one. In addition, in the ordered phase, its value should depend only on the physical conditions and not of the quantity of material.

The driving parameter e governs the critical behaviour, and its sign characterizes the stability domain of the system *. The most probable value of the order parameter rj in the disordered phase (e > 0) is implicitly set to 0. As soon as e becomes negative, the most probable value of rj is finite and this defines the ordered phase. The value rj = 0 is still an equilibrium point in the ordered phase, but the point is unstable.

11.3.2 Distribution of the extensive order parameter

It is more convenient to work with the corresponding extensive order parameter : m = NJ], when dealing with the finite systems, since this is typically the relevant observable in these conditions (the total mass being often difficult to estimate accurately). The probability of a state m for a given e is [J. E. Mayer & M. G. Mayer (1957)] :

P „ ( m ) = _ e x p - / ? T ( ^ — +bjp-~j . (11.11)

UBT = l//?x> a n d %N is defined by the normalization of Pjv(m). In agreement with the previous sections and without loss of generality,

we shall consider now the case where m is positive. We shall admit that N is so large that the first two terms in the free energy expansion are sufficient to study the phase transition. At the critical point (e = 0), the

"If the temperature T is the driving parameter then e = a(T/Tc — 1) with a positive constant a and the critical temperature T c .


leading term of the free energy density is just proportional to m4. Standard integrations yield the values for the partition function ZM and the average value of the order parameter (m), both proportional to N3'4. Introducing them in (11.11), one finds :

<"»>P*M = r^|yexp {-f^ ((m)

which has the form (3.5.2) of the first scaling. Note that the scaling function is :

&(zi) ~ exp(—az\) , z\ = m/(m)

and decreases very fast as one moves away from the most probable value. This result is consistent with the general analysis (11.4), and the value of the mean-field critical exponent 6 = 3.

This scaling function appearing in equation (11.12), can be entirely characterized by its moments :

< Z\ >= W 2 r « - 1 ( i / 4 ) r ( ( g +1) /4 ) .

This provides useful tool to compare experimental scaling curves to the mean-field theory of the second-order critical phenomenon.

11.3.3 First scaling at the pseudo-critical point

The pseudo-critical point is the value of e for which the finite-size thermal susceptibility reaches its maximum. Writing that the inverse of this susceptibility is the second derivative of the free energy with respect to the order parameter, one obtains :

which is correct at the first order in N~1//2. Inserting (11.13) in (11.11) leads to the scaling form of PW(TO) :

(m)Pjv(m.) = 5 e x p -

with B a normalization constant. We recover indeed the first scaling law with the exponential tail exp(—az4) for large z.

Mean-Field Critical Model: The Landau- Ginzburg Model 295

11.3.4 Gaussian first scaling in the disordered phase

Outside the critical point in the disordered phase (e > 0), the leading term of the free energy is proportional to m2 , and the probability distribution PJV(W) is essentially Gaussian. Deriving as previously (c.f. Section 11.3.3) and noticing that both Z^ and (m) behave in this case like JV1/2, we get the first scaling form (c.f. Section 3.5.2) :

4 (m)Pjv(m) = — exp

IT

A i m

•K \{m) (11.14)

but with a Gaussian scaling function reminiscent of the Gaussian fluctuations.

As at the critical point, one can define usefully the scaling function by its moments :

(zql) = ^^—T(q + l/2) .

11.3.5 Second scaling in the ordered phase

Finally, in the low temperature regime (e < 0), the most probable value of the order parameter is positive :

Developing (11.11) around this point leads to the expression :

which is no more in the first scaling form. In this case, the average value of the order parameter (m) is of same order of magnitude as its most probable value m* and one can rewrite (11.15) in the scaling form :

(m)l'2PN(m) ~ exp l-a±—

where a is a positive constant. This is the second scaling form (c.f. Section 3.5.3).


11.3.6 Correlation pattern in the Landau-Ginzburg theory

All these scalings are particular cases of the general theory of the second-order critical phenomena, as described in the Sections 3.5. Nevertheless, since one knows exactly the free energy function in the Landau-Ginzburg theory, one can derive the exact form of the scaling function. It is interesting to note that some informations can be extracted about the correlations in this simple model.

At the critical point, the generating function Q as defined in (6.6) and (6.7), can be written as :

POO I / »00

G(l+u)= e-N[0Tb^-r,Hl+u)]dri/ e-N[PrW\drl . ( 1 1 . 1 6 )

Expanding the numerator in a series of integer powers of u, one recovers the general expansion (7.21) defining the linking coefficients Aq. On the other hand, calculating differently the generating function Q (11.16), one can extract the behaviour of the Aq when q is large. For that purpose, one has to suppose that N is large. In this case, the integrals occurring in (11.16) can be estimated by the steepest-descent method. Since firbrf — 77 ln(l + u) is extremal for rf ~ u1/3, therefore the logarithm of function Q has a singular part whose dominant term for small values of u, is :

lnS(l + M)~aiVu4 /3 . (11.17)

The terms of order uc with c > 4/3 have been omitted in (11.17). The large—q behaviour of fq/q\, which according to the definition (6.7)) are coefficients of the expansion of Q(l + u) in u, is connected to the small—u behaviour of the singular part of Q(l + u). One can then use explicitly a general relation :

rs(«) = E9**n ! , dz if I \ t h e n Q, = <* -^— (11.18)

where the integration contour is around the origin. Applying this relation to Q(l + u), one first calculates the coefficients fq and then using equation (7.20) one finds the asymptotic result :

Ai-iq-iy.— . (11.19)

Example of the Critical Model: The Potts Model 297

The coefficient a in (11.19) is a constant independent of the size N of the system. This shows the deviation from the result for the Pascal distribution Aq = (q — 1)!, indicating the existence of strong correlations throughout the system at the critical point. One can also note that the suggestion of Balian and Schaeffer (1989a,1989b)) :

Aq = 2<-q~1>y-2""

is a particular case of the Landau-Ginzburg result when v = —1/2. But this case has not been investigated in details up to now.

11.4 Example of the critical model: the Potts model

A generalization of the magnetic Ising spin model has been popularized by Domb [C. Domb (1974)] after being studied in details by Potts [R. B. Potts (1952)]. In this model, one considers a system of N sites in the d-dimensional space. The magnetic state of each site i is characterized by a quantity called spin Si. Each spin is of same constant modulus : \Si\ = 1, and points in one of p equally spaced directions labelled from 0 to p — 1. The ferromagnetic short-ranged Potts Hamiltonian is then :

{id}

with 5 as the Kronecker symbol, and J the positive coupling constant. The sum is restricted to nearest-neighbour pairs.

The site percolation corresponds to the Potts model with p = 1. The ferromagnetic Ising model corresponds to the Potts model with p = 2. This model is one of the simplest non-trivial critical thermodynamic TV-body system and many exact or accurate results are known for standard values of the couple (d, p). In particular, there exists a value pc (d) such that for p < pc(d) the interacting system experiences a second-order phase transition at a finite critical temperature, whereas for p > pc(d) the transition is of the first-order kind*.


* ( Z i )

Fig. 11.1 Scaled m-distribution for the three-states Potts model on two-dimensional L x L square lattice at the critical temperature. Three sizes are shown : L — 32 (stars), L = 48 (circles) and L = 64 (squares) (from [R. Botet & M. Ploszajczak (2000)]).

11.4.1 Scaling laws for the order-parameter distribution

Let us consider now the case of the second-order phase transition . All scalings described above should hold. We have first to define the order parameter for the system. If for a given configuration of the system, we call Nk the number of sites in the state k with k varying from 0 to p — 1, then the order parameter m is given by :

m = q(Nk/Nmax) - 1

(11.20)

where the quantity Nmax is defined as the maximum of all the Nk. Figure 11.1 shows for (d,p) = (2,3) Potts model the m-distribution in the first-scaling form at the critical temperature. Of course, the scaling is recovered very precisely even for such small system sizes as N = 32. Note also the complicated shape of the scaling curve. The sharp decrease of the positive tail is ~ exp(-z j + 1 ) , with 6 = 14 [F.Y. Wu (1982)].

We can discuss this scaling here in a slightly different context of the

•For example, for d = 2 one finds p c = 4 and {PTJ)C = ln(l + y/p) on the square lattice.

Example of the Critical Model: The Potts Model 299

correlated random variables [J.-P. Bouchaud k. A. Georges (1990)]. Let us consider for simplicity the Ising model, i. e. (d, p) = (d, 2) case of the Potts model. The extensive order parameter is just the sum of TV correlated variables : M = 5^5j. When the system is disordered, the spins are correlated on the short distance £ (£/TV —>• 0 in the thermodynamic limit) and their mean value is zero. The Central Limit Theorem tells then that the distribution of the random variable M/y/N is Gaussian when TV becomes large, with zero mean and finite variance. This can also be expressed by the asymptotic law :

which is correctly under the first scaling form, with a Gaussian shape and the trivial anomalous exponent g = 1/2 (see (11.14)).

On the other hand if the system is in the ordered phase, the average value of the individual spins is finite : {Si) = m, and the same reasoning can be used for the variable (M — Nm)/-\/N. This variable is of zero mean, finite variance and short-ranged correlated. So, its fluctuations are Gaussian and can be put in the second scaling form :

Of course, the most interesting case corresponds to the critical temperature . At this point, the spins are correlated throughout the whole system, and the magnetization cannot be evaluated by the Central Limit Theorem. Instead, we can remark that the spin-correlations are power law :

{sPoSP+Po) ~ ^ 3 ^

with r/ a critical exponent whose value is between 2 — d and 2. Looking at the total magnetization as the sum of TV correlated variables one gets :

(M2) = Y,(SiSi) = J2(Si) + N I {S5SP)rd-Hr i,j i J l

with L ~ TV1'd the typical macroscopic length of the system. This means that the leading behaviour is :

{M2) ~ T V 1 + ^


This non-trivial anomalous exponent :

1 2-77 Q = - H y 2 2d

with values in between 1/2 and 1 is here the sign of the criticality. The first scaling law should hold in this for the (d,p) = (2,3) Potts model discussed above, but the scaling function should be different since it depends on the precise form of the interactions.

11.5 Reversible aggregation: example of the percolation model

The percolation model can be defined as follows. In a box, i.e. a part of the regular lattice, each site corresponds to a monomer and a proportion p of active bonds is set randomly between sites. This is the so-called bond percolation model.

Such a network results in a distribution of clusters defined as ensemble of occupied sites connected by active bonds. For a definite value of p, say pcr, a giant cluster almost surely spans a whole box. The reversible sol-gel transition corresponds to the appearance of an infinite cluster (get) at a finite time. Infinite in this context means that the gel contains a finite fraction of the total mass of the system.

The sol-gel transition in finite systems can be suitably studied using moments of the number-size distribution n s , i.e. the number of finite clusters of size s :

M'g= J2 s<ln° ( 1 L 2 1 )

where the summation is performed over all clusters with the exception of the largest cluster s = smax- The superscript ' recalls this constraint on summation in equation (11.21). The mass of the largest cluster is then N -M[ with :

N =^sna . all s

In infinite systems, one works with normalized moments of the

Reversible Aggregation: Example of the Percolation Model 301

concentration-size distribution cs, i.e. the concentration of clusters of size s :

m'q= Yl s"c* ( 1 L 2 2)

where the summation in (11.22) runs over all finite clusters. Generally, concentrations are normalized such that :

cs = hm — . N-yoo N

The probability that a monomer belongs to the infinite cluster (gel) is equal to 1 — m\ with :

M' TO' = lim - r f .

Therefore, we get :

TOj = 1 for p < pcr

and

m'i < 1 for p > pcr .

Moreover, TO'J is a decreasing function of the occupation probability. This typical behaviour is commonly and incorrectly called the "failure of mass conservation". But as stated before, m^ is more simply the probability for a vertex to belong to some finite cluster.

11.5.1 Order parameter in the percolation on the Bethe lattice

The bond percolation on the Bethe lattice with integer coordination number z, has been solved by Fisher and Essam (1961). It corresponds to the mean-field theory of the percolation on the d-dimensional regular lattice with d = z/2.

The main result we are interested in is the concentration-size distribution :

C°-Z((z-2)s + 2)\S\P [1 P)


and the first normalized moment : 2z-2

1 P 4 (11.23) l-p*.

p* in equation (11.23) is the smallest solution of equation :

p*{i-P*y-2 = p{i-py-2 . (n.24)

Let us define :

Pcr - z - 1 '

We suppose in the following that pcr < 1, i.e. z > 3. Therefore, the marginal case z = 2 which corresponds to the linear-chain case, is excluded.

For p < pcr, the only solution of equation (11.24) is p* = p, but when p is larger than pcr then there exists a smaller non-trivial solution which behaves as pcr — \p — Pcr\ near pcr. Above this threshold, the moment m'^ is smaller than 1 and behaves approximately as :

m1 ~ 1 2(P ~ Per)

1 ~Pcr

Coming back to the concentrations, we can see that for large values of the size s, the following Stirling approximation holds :

cs ~ s 5'2exp(—as)

with a given by :

a = In / i \ 2 - 2 '

p f l-p pcr \1~ pc

(11.25)

~ (z - 2) ln(l -p)<0

For this model, a power law behaviour of the concentrations : cs ~ s~T

with r = 5/2, is seen at the threshold pcr. Outside of the threshold, an exponential cut-off is always present [D. Stauffer (1979)].

This sort of critical behaviour at equilibrium is analogous to the thermal critical phenomena, and in particular, there exist two independent critical exponents, for example r and a. The latter one is the exponent of the mean cluster-size divergence. Together, those two critical exponents, whose


respective values are : r = 5/2 and a = 1, describe completely the critical features.

This singular behaviour is due to the appearance of a giant cluster, called the percolation cluster, at the transition point. More precisely, at the thermodynamic limit (the infinite system), the probability for a given site to belong to this infinite cluster is zero below the critical threshold pcr and positive above it. This probability is non-analytical at the critical point. Because of this behaviour, the extensive order parameter defined for finite systems is just the size of the largest cluster smax. As discussed in Section 11.1.3, the corresponding finite-size order parameter scales as :

Even though the system experiences the second-order critical phenomenon, fluctuations of the multiplicity distribution remain small and the first scaling (KNO scaling in this particular case) does not hold. Of course, m'0 is not in this case an order parameter since r > 2, even though there is some irregularity in its behaviour at the threshold. This non-analyticity can be illustrated by the exact result for the bond percolation on the Bethe lattice. In this mean-field case, the normalized moment of rank zero is :

with : e = p — pcr, and e <C 1. There is a jump of the first derivative of m'0

with respect to p from

-z/2 for p -s- p~r

to

( 4 - 3 z ) / 2 for p->p+r .

Another analogy is interesting to note as well. If we impose for the large cluster that smax < oo even in the thermodynamic limit, then the bond occupation p corresponding to smax is smaller than pcr and the limiting value po exits and equals pcr. The limiting probability po in percolation is analogous to the limiting temperature T0 in the statistical bootstrap model [R. Hagedom (1965); S. Frautschi (1971)]. Both p0 and T0 follow from the consistency conditions in the two models and describe the asymptotic size-distribution. Obviously, the limiting value po/^o of the control parameter


p/T does not have to correspond to the critical point in the equilibrium system. For this to be true, the exponent r of the size distribution must be contained in between 2 and 3.

11.5.2 The three-dimensional percolation model

11.5.2.1 Multiplicity distributions

The multiplicity distribution in the three-dimensional bond percolation model on the cubic lattice at the infinite-network percolation threshold exhibits the second scaling [R. Botet et al. (1997)]. Hence, the fragment multiplicity is not related to the order parameter in this process. Note also that the multiplicity distributions in figure 11.2c are plotted in semi-logarithmic form to show clearly the Gaussian behaviour of the multiplicity distribution.

11.5.2.2 Order-parameter distribution

The intensive order parameter in the percolation model is the normalized mass of the gel-phase, i.e. the mass of the largest cluster divided by the total mass of the system smax/N. Different probability distributions Pjv(smox /N) for different system sizes N can be all compressed into a unique characteristic function (see figure 11.2a) :

(smaJN)PN(smax/N) = $ (s<™*-(8™*)\ \ \°max) J

which is correctly in the first scaling form.

11.5.2.3 Shifted order parameter

Figure 11.2b shows the A—scaling for the shifted order parameter :

M[=N- smax .

The value of A (A=0.8), is consistent with the value of the anomalous dimension (11.4) (g — 0.8435) for the accepted values of critical exponents /?,7 in the three-dimensional percolation [D. Stauffer & A. Aharony (1992)]. One should also remember, that A has been extracted from small size percolation network calculations at the infinite-network percolation threshold. This explains a small difference (here about 5%) between the value for A


0(zA)

max

M[

M0

^ A

Fig. 11.2 (a) The first scaling of ^max-distributions at the percolation threshold (j>cr — 0.2488) of the three-dimensional bond percolation for lattices of different sizes : N = 143

(diamonds), N = 203 (squares), N = 323 (circles). (b) The A—scaling of the distributions of M\ = N — smax, for the same parameters as in (a). (c) The second scaling of the multiplicity distributions, for the same parameters as in (a) (from [R. Botet & M. Ploszajczak (2000)]).

from the scaling analysis and the expected value A = g in the infinite network.

11.5.2.4 Outside of the critical point

According to the results derived above for the second-order phase transition, the second scaling should hold outside the critical point. This is correctly


OfeA)

H 0 ~ ^ 1 S -^_

Mi

Mn

A=l/2 • • (a)

(b) A=l/2 ^

9 •

- T * - # ! * — . 1 1 p-»»^«((

A=l/2 .• \ (c) J

_ i i i , i [_•

Fig. 11.3 (a) The second scaling of smaa:-distributions above the percolation threshold (p = 0.35) of the three-dimensional bond percolation for lattices : AT = 143 (diamonds), N = 203 (squares), N = 323 (circles). (b) The second scaling of the distributions of Afj = N — smax for the same conditions as in (a). (c) The second scaling of the multiplicity distributions for the same conditions as in (a) (from [R. Botet & M. Ploszajczak (2000)]).

realized with the three variables smax, M[ and M0 for large or small values of the probability p. Figure 11.3 shows such results for the value p = 0.35 of the bond activation probability.


O(zi)

p=0.252

/ /

i

^

(a)

, ^ ~

p = 0 . 2 4 8 8 , ^ ^

•

• •

% •

+ p=0.245

A •

•

* « % •

#

V \

(b)

(c)

^ . i * 1 ^ n a g

-1.5 0.0 1.5

Fig. 11.4 sma2;-distributions for lattices AT = 143 (diamonds), N = 323 (circles) are plotted in the first scaling form for parameters p close to the percolation threshold of the three-dimensional bond percolation : (a) p = 0.252 ; (b) p = pcr = 0.2488 and (c) p = 0.245 (from [R. Botet & M. Ploszajczak (2000)]).

11.5.2.5 Close to the critical point

Finally, it is instructive to see how the first scaling is disappears when the value of p is slightly shifted away from its critical value. Figure 11.4 illustrates the deviations from the first scaling for the values of parameter p close to pcr, on both sides of pcr. Even very close to the critical point, these deviations are quite significant and can be easily seen in this representation.


11.6 Irreversible aggregation: example of the Smoluchowski kinetic model

The irreversible sol-gel transition can be modelled using the coupled nonlinear differential equations in variables cs [M. von Smoluchowski (1917); R. L. Drake (1972)] :

—— = — y ^ AijCiCj — y ^A-sjCgCj . {11.2b) i+j=s j

Coefficients Aij represent the probability of aggregation per unit of time between two clusters of respective sizes i and j . Smoluchowski equations are derived from the master equation in the mean-field approximation [A. H. Marcus (1968)] :

(ciCj) = (Ci)(Cj-) .

The time t includes both diffusion and reaction times. Equations (11.26) suppose irreversibility of the aggregation, i.e.- the cluster fragmentation is excluded. One should notice that the sum over j in (11.26) does not include the infinite cluster (gel) because :

Cj=co = l / o o = 0 .

Experimentally known aggregation kernels Aij are homogeneous functions [S. Simons (1986)] :

with a being the homogeneity index. Perhaps the simplest physically relevant example of a homogeneous kernel is : Aij — (y)M . It has been shown in this case that if n is larger than 1/2, then there exists a time tcr

(tcr < oo) such that m\ < 1 for t > tcr [F. Leyvraz & H. R. Tschudi (1981); R. M. Ziff et al. (1981)].

11.6.1 Basic behaviour of the order parameter

Let us consider now the case : Aij = (ijY with fi — 1 in more details. Leyvraz and Tschudi (1981) have shown in this case that the critical gelation time is : tcr = 1, and the solution for size-distribution of the Smoluchowski equations with the monodisperse initial condition is [J. B. McLeod

Irreversible Aggregation: Example of the Smoluchowski Kinetic Model 309

(1962)] :

„ « - 2

cs = ——e^H3-1 for t<\ s\

(11.27)

ss~2 e-s\ t

The asymptotic solution for large s is :

for t > 1

CS ~ _ J= s -5 /2 e - . ( t - l+ In t ) for t < 1

tV2TT

cs r —s~5/2 for t > 1

(11.28)

Note that the power law behaviour is present for t > 1 and not only at the threshold. The whole distribution of finite-size clusters evolves self-similarly and the appearance of a power law behaviour is not here a sign of the critical behaviour but a specific characteristics of the gelation phase.

The solution for the first normalized moment is :

m[ = 1 for t<\

(11.29)

m'i = - for t > 1 .

With those asymptotic forms of cs one can calculate the gel fraction in the infinite system before and after the critical point :

mo = 0 for t < 1

(11.30)

mc = 1 for t > 1

It has been shown [E. M. Hendriks et al. (1983)] that gelation is analogous to a dynamical critical phenomenon with :

mG = Jim -rz(smax)

as the order parameter. For one realization, smax corresponds to the mass of the gel above tc = 1. For finite sizes, one makes the usual assumption

Fluctuations of the Order Parameter

* ( Z i )

Fig. 11.5 First scaling of sm ax-distributions in the Smoluchowski kinetic model with the kernel A.ij = ij at the critical time t = tcr = 1. The calculations are performed for two system sizes : N = 2 1 0 (diamonds) and N = 2 1 4 (circles) (from [R. Botet & M. Ploszajczak (2000)]).

that there exists a characteristic size diverging at the transition :

J V c ~ | t - l | - 1 / f f w

such that for the mass of the gel in a finite system :

1 N

{Smax) ~(t-l)f(N/Nc) for i > l .

In particular, at the gelation time one has :

(Smax) ~ N1-*"

with crjv = 1/3, because r = 5/2 following equation (11.7). The average value of the order parameter (smax) increases logarithmically for t < 1 and is a finite proportion of the system size when t > 1.

11.6.2 Scalings of the order-parameter distributions

The illustration of the above discussion is shown in figures 11.5 and 11.6. Figure 11.5 shows the distribution of smax in the first scaling variables for


-1/2

Fig. 11.6 Second scaling of sm ax-distributions in the Smoluchowski kinetic model with the kernel Aij = ij above the critical time t = 2tcr = 2. The calculations are performed for two system sizes : JV = 2 1 0 (diamonds) and N = 2 1 4 (circles) (from [R. Botet & M. Ptoszajczak (2000)]).

systems of different sizes. The results have been obtained in the Smoluchowski model with the kernel: Aij = ij, at the critical time : t = tcr = 1. Fluctuation properties of smax outside of the critical time : t = 2tcr, are shown in figure 11.6. The remaining parameters of the Smoluchowski calculations are the same as used in the calculations shown in figure 11.5. In this case, the data for different system sizes collapse into the universal curve in the scaling variables of the second scaling. One should keep in mind that the fragment-size distribution in both cases is a power law with r = 5/2 (see (11.28)).

11.6.3 Tails of the scaling functions

The relation between the form of tail of the scaling function and the anomalous dimension (11-4) was derived in Section 11.1.1 for the equilibrium systems at the second-order phase transition. For non-equilibrium systems, we do not know equally rigorous derivation. On the other hand, one may expect that the relation between the ./V-dependence of the average value


log[ <&(*,)] 0

-1

-2

-3

-4

Fig. 11.7 The plot of the large—Z\ tail of the decimal logarithm of the scaling function log<f>(zi) against z\ for N = 2 1 2 . The solid line shows the dependence : $ ( z i ) ~ exp(—zj1), which is expected from the value of the anomalous dimension g = 2 /3 (from [R. Botet & M. Ploszajczak (2000)]).

of the order parameter, i.e. between the anomalous dimension g, and the asymptotic form of the scaling function in the limit N —> oo, is related to the asymptotic stability of the limit distributions. Actually, there is a very close connection of the renormalization group ideas and the limit theorems in the probability theory [P. M. Bleher & Ya. G. Sinai (1973)]. If true, then the relation (11.6) could be valid in a more general framework than the one provided by the equilibrium statistical mechanics. To check this assertion, we plot in figure 11.7 the logarithm of the scaling function $(zi) (see figure 11.5) versus zf for large values of z\. If the relation (11.6) is valid also for the non-equilibrium sol-gel second-order phase transition , then : $(zi) ~ exp(—azf), and the tail of the scaling function should be a straight line in figure 11.7. This is indeed the case.

11.6.4 Scaling for the shifted order parameter

Figures 11.8 and 11.9, show the A—scaling for the shifted order parameter variable : Ml = N — smax. Results of the Smoluchowski calculations with


•^•0.67

Fig. 11.8 A—scaling of the distributions of shifted order parameter (M[) in the Smoluchowski kinetic model with kernel Aij = ij at the critical time t — tcr = 1. Two system sizes are considered : iV = 210 (diamonds) and N = 214 (circles) (from [R. Botet & M. Ploszajczak (2000)]).

the kernel Aij — i j , are shown at t — tcr (see figure 11.8) and at t = 2tcr

(see figure 11.9). One sees that the A^-distribution exhibits a qualitative change while going from the critical time t = tcr where A = 0.67, to t = 2tCT

for which A = 1/2. At t = tcr, the value of A obtained by superposing different M1-distributions in the scaling plot (3.29) agrees perfectly with the value of the anomalous dimension g (=2/3).

11.6.5 Origin of fluctuations in non-equilibrium aggregation

11.6.5.1 Argument of Van Kampen

^-expansion is a systematic expansion of the master equations in powers of 1/JV [N. G. Van Kampen (1981)]. Lushnikov (1978) was the first to express the generating functions as the contour integrals for quantities like the moments M'k. Then Van Dongen and Ernst [P. G. J. Van Dongen & M. H. Ernst (1987)] used fi - expansion to calculate explicitly these integrals


Z\I2

Fig. 11.9 Second scaling of the shifted order parameter (M{) distributions of in the Smoluchowski kinetic model above the critical time (t = 2tcr) for two system sizes : N = 2 1 0 (diamonds) and N = 2 1 4 (circles) (from [R. Botet & M. Ploszajczak (2000)]).

for the moments in some simple cases like : Aij = ij. For example, the result for M[ can be expressed in terms of the generating function for the distribution ?N{smax) as :

£ P j r ( w , ) e - - » J ^ e - ^ ^ e x p ' N

£ .9=1

Ca(ze U)q Q(N-q)/(2N)

lfq-1

(11.31)

Using then the identity :

gn exp J2ai ,qu = £ n!

l!a iai! . . .n!a" an!

On

e

E a ^ l e " u

. 9

*p(r£<v*tt

\ a


where the sum runs over the sets {a\, ...an} with the constraint :

ai + ... + a „ = n

and the particular result for the case Aij = ij :

exp

' N

E .9=1

C.,Z s*~ eq(N-q)/(2N) fifq-1

N

9=1 H

qt (1-q/N)

one finds :

c)e° exp + u

+ 0(1/N)

(11.32)

. (11.33)

11.6.5.2 Gelling systems

With the help of the fi-expansion of the generating function, we can conclude about the scaling at the gelation point. Moments m' of the size-distribution for infinite systems close to the gelation time are known to diverge at the transition [R. M. Ziff et al. (1983)] :

mq ~ \t 1 |3-2?

This means, using crjv = 1/3, that for finite systems :

m ~ \t l\-2q+3fq{N\t-l\z)^N^ (11.34)

at the gelation time. We have then the asymptotic result :

Nm'q ~ aq(smax)q

with some positive constants aq. Using (11.33), we have been able to show that the generating function depends on a single variable : (smax)u. According to the discussion in Section 3.5.2, this is a sufficient condition for the validity of the first scaling law.

11.6.5.3 Scaling of the second moments for gelling systems

We can also extract informations on scaling features for other moment distributions. Cl - expansion leads to the results :


(M'q) = Nm'q (11.35)

for the values of q when all the quantities are defined. At the transition (t = 1), the relation (11.34) allows to calculate the reduced moments m'. The results can be written under in a compact form :

(M>2) - (M>)2

(Mq)^

; values of exponent A :

A = 1/2 for

A = 2q/3 for

A = 1 for

~ est

<Z<3/4

3/4 < q < 3/2

3/2 <q

These are clear indications for the A—scaling according to the remarks in Section 7.6.3. More precisely : the moments of order q < 3/4 are not critical (the second scaling law), the moments of order q between 3/4 and 3/2 exhibits the A—scaling with A = 2q/3. In particular, for q = 1, one recovers the correct value : A = g = 2/3, corresponding to the general argument of the shifted order parameter (3.27) with aj = a<i — 1. At last, when the value of q is larger than 3/2, we obtain the first scaling law for the distribution of the moments Mq. This is also a consequence of the shifted order parameter argument since in these cases :

(M'q) ~ (sLJ •

Far from the critical point, all reduced moments m'q are independent of N since the correlation size :

1

i - l + lni

(see (11.28)) is finite. Then, for any value of q, the second scaling law holds, as expected from the general theory.

The latter results about A—scaling for various moments of the size-distribution in the Smoluchowski model with kernel Aij = ij, are not complete since the arguments involve only the second cumulant moment of M moments. In principle, as shown in Section 3.5.4, all cumulants should be investigated. But even though many exact results are known in this model, the complete analytical solution is not yet available.

Off-Equilibrium Fragmentation 317

11.6.5.4 Non-gelling systems

The same study as for the gelling systems, can be performed also for the non-gelling systems. Let us take : Aij — i + j , as an example of the non-gelling system. In this case, the size-distribution is a power law with the exponent r = 3/2 and, following the discussion in Section 11.1.3, the cluster multiplicity may be the order parameter. One can derive analytically, that the multiplicity distribution is binomial :

PN(m,t)= (^Z\) (l-e-m)N~me-(m-VNt

and can be approximated for N —> oo and for a finite value of {m)/N by :

X 6 X P V 2(1 -e-M) (m) J ( 1 L 3 6 j

what corresponds to the second scaling. One may notice, that this binomial distribution is exactly equivalent to those obtained for the bond percolation on a Bethe lattice (Section 11.5.1) with the occupation probability :

p = l - e - m .

In spite of self-similar features in the fragment-size distribution at the infinite time, one does not see any critical behaviour in the cluster multiplicity distribution at any time in the non-gelling aggregation systems. This confirms the observation made before in the Mekjian equilibrium model (Section 11.2.4), that the power law size-distribution alone does not guarantee that the system exhibits the critical behaviour.

11.7 Off-equilibrium fragmentation

As a last example, let us consider the fragmentation-inactivation binary model which exhibits the second-order shattering phase transition [E. D. McGrady & R. M. Ziff (1987); Z. Cheng & S. Redner (1988); R. Botet & M. Ploszajczak (1994)]. This model has been discussed in Chapter 10, but some remarks could be useful in the present Section.


* & A )

0.4

0.0

0.4

0.0

Fig. 11.10 Order parameter fluctuations in the fragmentation-inactivation binary model for initial sizes : N = 2 1 0 and N — 2 1 4 , are plotted in the A—scaling variables : (a) (upper left plot) the critical process {pp = 0.875, ct = 0) which corresponds to the anomalous dimension g = 0.75; (b) (upper right plot) fluctuations of m = N — m are plotted in the A—scaling variables with A ~ 0.73; (c) (lower left plot) critical process (pp = 0.7, a = 0) which corresponds to the anomalous dimension g = 0.4; (d) (lower right plot) shattering process (a = 0, b = — 1, T = 4).

As seen in Section 10.1.1, the order parameter is related to the reduced cluster multiplicity m/N or the reduced monomer multiplicity, both of them closely interrelated. The cluster mass independence of PF (k) at any step until the cut-off scale characterizes the critical transition region. Fragmentation-inactivation binary process is self-similar in this regime. For PF > 1/2, the anomalous dimension (11.1) varies from 0 to 1, what is different from the limits on g in the equilibrium systems for which g takes values between 1/2 and 1.

Off-Equilibrium Fragmentation 319

Most of the interesting physical applications correspond to homogeneous fragmentation functions, such as Fj^-j ~ \j(N — j)]a. For the homogeneous inactivation rate-function : Ik ~ k13, the critical transition region in fragmentation-inactivation binary model corresponds to the line : P = 2a+l. Such homogeneous rates J-j,k-j and Ik, will be used in examples shown in figure 11.10.

On the contrary, the shattered phase is the region where 0 < 2a + 1. In this phase, the average multiplicity is : (m) ~ N, the cluster-size distribution is a power law with T > 2 and the anomalous dimension is 5 = 1. In this phase, pp is an increasing function of cluster mass k and the fragmentation-inactivation binary cascade is not self-similar.

Figure 11.10a (the upper left plot) shows the scaling function 3>(zi) of critical fragmentation-inactivation binary process for pp = 0.875, a = 0, what yields the anomalous dimension g — 0.75. The asymmetry of 3?(zi) about Z\ = 0, is common in the critical sector of fragmentation-inactivation binary model and the symmetric solution corresponds only to PF = 0.75 for which T = 1.5 and, hence to g = 1/2. This sector and its characteristic first scaling extends to the domain 0 < g < 1/2, which is excluded in the equilibrium models. In this domain (g < 1/2), the most probable value of $(zi) is found at z\ = — 1, whereas for g > 1/2 the most probable value is close to z^ = 0. This can also be seen in figure 11.10c (the lower left plot) which exhibits the scaling function ${z\) of critical fragmentation-inactivation binary process for PF = 0.7, a = 0, for which g = 0.4.

What happens if instead of P(m) one plots P(JV —m), is shown in figure 11.10b (the upper right plot). Similarly as in the three-dimensional percolation, ¥{N — m) is scaling with the non-trivial, numerically determined exponent A ~ 0.73, which is close to the value A = g — 0.75 obtained using analytical arguments.

The order parameter distribution in the shattered phase (a = 0,(3 = — 1,T = 4), is shown in figure ll.lOd (the lower right plot) in the scaling variables of the second scaling law. Again, one finds an analogy to the situation in the non-critical phase of percolation.

Hence, one can conclude that the off-equilibrium fragmentation-inactivation binary model shows the identical relation between criticality and scaling of order parameter fluctuations as it has been derived analytically for the second-order equilibrium transitions. This is related to the underlying self-similarity which is common to both equilibrium and


off-equilibrium realizations of the second-order phase transition . In that sense, the scaling laws are the salient features of any system exhibiting the second-order transition and the scaling function $ ( ^ A ) is a fingerprint of the system and its transition. The function $ ( « A ) is also a basic quantity from which one can derive the thermodynamics for off-equilibrium processes. Its logarithm coincides with the free-energy of the equilibrated system.

The universal features of the order parameter fluctuations do not depend on whether the studied process is an equilibrium or an off-equilibrium process. In the latter case, the arsenal of available tools to characterize statistical properties of the system is very limited, and the universality of order parameter fluctuations may be a valuable tool in many phenomenological applications. The precise relation between the scaling functions $ ( ^ A ) > the nature of order parameter, and the critical exponents, yield a new tool for determining the combinations of critical exponents even in small systems and for learning about the nature of critical phenomenon.

Chapter 12

Universal Fluctuations in Nuclear and Particle Physics

In this chapter, we shall present experimental results concerning fluctuations in the hot hadronic matter, produced in violent collisions of leptons, hadrons or nuclei. For nucleus-nucleus collisions, we shall discuss fluctuations in two different regimes : (i) collisions at ultrarelativistic energies, and (ii) collisions at intermediate energies (25 • A MeV < E < 150 • A MeV, where A is the number of nucleons in the projectile nucleus). In the first case, discussion concerns the spectra of produced hadrons, whereas in the latter case the spectra of produced nuclear fragments. Hot hadronic system in these two different decay regimes are separated in excitation energy by many orders of magnitude. Nevertheless, the same mathematical tools and physical concepts, discussed earlier in this book, can be applied. Actually, the problematic of scaling laws of fluctuations are not a 'chasse gardee' of strong interaction physics, even though in this branch of physics excellent data are available. Event-by-event fluctuations of small complex systems obey the same laws in problems ranging from a distribution of visible matter in the Universe, polymerization, or innumerable kinds of aggregation phenomena in soft matter physics, up to the economic and social systems.

As we have discussed earlier, there are two generic classes of dynamical critical phenomena, which are characterized by different "relevant observ-ables". The first one is the sequential cluster fragmentation, where the average cluster size decreases and the cluster multiplicity increases during the process. The cluster multiplicity is an order parameter for this class of critical phenomena. The second one is the sequential cluster formation (cluster aggregation), where the average cluster size increases and the cluster multiplicity decreases during the process. The size of the largest cluster

321

322 Universal Fluctuations in Nuclear and Particle Physics

is here an order parameter. In the first class, one finds for example the shattering phase transition in the fragmentation-inactivation binary process as well as in the dissipative perturbative QCD process (c.f. Section 10.5). The second class contains for example the liquid-gas phase transition (Fisher model), the percolation transition (equilibrium reversible aggregation) or the sol-gel phase transition in the irreversible kinetic aggregation . Which of them is realized in the multiparticle production process or in mul-tifragmentation process, can be discovered by studying the A-scaling law, the tail of the scaling function, and the anomalous dimension, as described in previous chapters.

12.1 Phenomenology of high energy collisions in the scaled factorial moments analysis

The experimental results on the scaled factorial moment and correlation integral analysis and on the short correlations measurements in the high energy collisions have been reviewed recently [E. A. De Wolf et al. (1996); P. Bozek et al. (1995)]. In this section, we shall list some important observations of the large experimental survey in this domain. Before presenting experimental results, let us however shortly discuss some non-ideal effects which are important in the phenomenological studies.

12.1.1 Nonsingular parts in the correlations

In the mathematical sense, fractal dimensions and scaling indices are always defined in the limit of the scale going to 0. Of course, in the experiment we are limited by the resolution of the apparatus. Also in the theoretical models of physical systems, one expects the presence of some limiting scale at which the self-similarity is broken. Although for the case of weak-intermittency one could in principle go with the size of the momentum scale to 0 without self-consistency problems, we know, however, that the production of physical particles is limited by some infrared cut-offs. In the experiment it is even more problematic to isolate a scaling behaviour since often we have only a limited range of accessible scales. Thus, although the nonsingular or less-singular terms are irrelevant for the mathematical limiting definition, nevertheless these terms are extremely important in the experimental analysis to define precisely the fitting form of the correlations

Phenomenology of High Energy Collisions 323

in the limited range of finite scales. This problem was at the origin of the discussion whether one should expect the self-similarity in the cumulants or in the densities [P. Carruthers & I. Sarcevic (1989)]. In general, any nonsin-gular term in the correlations is important, not only the one coming from the lower order correlations. Obviously, the two approaches give different results in the fits for the intermittency exponents [K. Fialkowski (1992)]. The standard approach consist of removing the nonsingular terms and approximating their dependence on the scale by a constant. This means that the two-particle densities have not only the trivial 1 coming from the uncorrected particles, but also they could have some long range non-singular correlations. Thus, a description of the experimental data should include both the non-singular, long-range correlations and the power law component in the cumulants [K. Fialkowski (1991)].

12.1.2 Choice of the variables

Similarly as in the discussion of the problem of nonsingular parts in the correlations, the formal limit when the resolution goes to 0 does not depend on the choice of the variables for the singular part of the correlations. However, at finite resolution the form of the correlations depends on the choice of the variables. Then the possible fits and results for the values of the intermittency indices depend on the variables. Several variables can be studied and naturally, one such set of variables is the rapidity (y), transverse momentum (PT) and the azimuthal angle (ft). But even for this set of variables important changes are induced by the definition of the rapidity axis according to the beam axis or to the axis defined by the symmetry of the event. Other variables should be preferred if the expected correlations depend on the square of the momentum difference (Pi—P2)2 [A. Bialas &; J. Seixas (1990)] or on the covariant four-momentum square difference Q2 = - ( p i - P2)2 [P- Bozek & M. Ploszajczak (1990)]. The covariant type of variables is important for the spatio-temporal intermittency due to the causality constraint, which makes two space-time points which are not causally connected uncorrelated. Also sometimes the preferred variable is the invariant mass which is important e.g. for the correlated e+e~ pairs. Finally, the QCD cascade has simple correlation functions in the relative angle variables [W. Ochs Sz J. Wosiek (1992); Yu. L. Dokshitzer & I. M. Dremin (1993)].

Another possibility is the anisotropy of the correlation functions, which


may have different intermittency indices in different variables, and this both for factorizing and non-factorizing correlations. This type of correlations is expected in the Regge field theory and also in some scenarios of the fractal source in the Bose-Einstein correlations, if the space-time development is anisotropic. Certainly several components of the correlation functions have different important variables. The microscopic covariance induces the Q2

variable, but many models have a preferred longitudinal direction corresponding to the rapidity and transverse variables. Generally, one should perform the analysis in several sets of variables, trying to identify the different sources of the correlations.

12.1.3 General phenomenology and experimental results

Different experimental groups analyzed the correlations between particles produced in high energy collisions. First the one-dimensional and two-dimensional data were analyzed in the scaled factorial moment method. Also the three-dimensional data were studied, but the main results of this analysis are obtained using the correlation integral method.

The e+e~ annihilation was first analyzed by the TASSO Collaboration (1989). The analysis was performed in the one-dimensional rapidity (y) distribution along the sphericity axis and in the two-dimensional rapidity-azimuthal angle (y-<f>) distributions. At LEP energies, DELPHI (1990), ALEPH (1991) and OPAL (1991) Collaborations performed the intermittency analysis in one-dimensional, two-dimensional or three-dimensional distributions. Figure 12.1 shows the results of the DELPHI Collaboration on the intermittency analysis of the particles produced in the e+e~ annihilation. The second factorial moment for the one-dimensional analysis in rapidity, two-dimensional analysis in rapidity-azimuthal angle, and the full three-dimensional analysis are shown. The effect of the dimensional projection, which reduces the three-dimensional correlations if observed one-dimensional or two-dimensional, is clearly visible. In this analysis, the LUND model predictions were found to be consistent with the data. The CELLO Collaboration (1991) analyzed the three-dimensional intermittency signal in e+e~ annihilation and also found a good agreement with the LUND model. A characteristic feature of the three-dimensional data is the strong correlation between the like-charge particles. These correlations are much stronger than those between the unlike-charge particles [DELPHI Coll. (1990a); DELPHI Coll. (1992)]. This means that the two and


J /_L / / /

7-/ s

/ _/ / ^3-/ ^^

* X^ / / / / / T * 2

/ / / / */ __» =*=-" /j£- _=&= ~~~

/ / „ ' * '

/ / -9*-'

//,'' r' , — _ f t = = ^ =

— i

•

--=•=

i

0 4 6 \og2M/d

Fig. 12.1 The dependence of the second scaled factorial moment on the total number of cells in the one-dimensional, two-dimensional and three-dimensional analysis of the e+e~ reaction (from [DELPHI Coll. (1990a)]).

many particle correlations are mainly due to the Bose-Einstein interference mechanism. However, some part of the correlations in three-dimensional was identified as due to the resonance decay and cascade structure of the production mechanism. The charge correlations in one-dimensional and two-dimensional, look very much the same, but this could be due to the strong effect of the dimensional projection.

The ir+/K+p collisions were extensively analyzed by the NA22 Collaboration [NA22 Coll. (1990); NA22 Coll. (1991a); NA22 Coll. (1991b); NA22 Coll. (1993)]. Like in the other reactions, a strong dimensional projection effect is observed, leading to the suppression of the fluctuations in the lower dimensional spectra. In three-dimensional analysis, however, the power law also does not hold and the data are seen to bend upward. This deviation from exact scaling could result from an anisotropy of occupied phase-space.

The data of the UA1 Collaboration for pp collisions at 630 GeV indicated an increase of the intermittency signal for the low multiplicity samples [UA1 Coll. (1990)]. The three-dimensional analysis of the pp collisions by


a 1

10;

101

" T -njiiiiirnrnriniBijiiBiaiaaMttiMWII j = 2

10 -2

Q

Fig. 12.2 Density strip integrals Ci ( i = 2, 3, 4,5 ) in three-dimensional analysis for all-charged (open circles) and same charged (black circles) combinations in pp collisions at 630 GeV by the UA1 Collaboration (from [UA1 Coll. (1993)]).

the UAl Collaboration confirms the observation that the main part of the correlations at small scales are due to the correlations between like-sign particles. As can be seen in figure 12.2, the dominance of the Bose-Einstein correlations was identified in the correlation integrals up to order 5 [UAl Coll. (1993)].

The NA22 Collaboration (1991b) performed the intermittency analysis


Fig. 12.3 The dependence of the second correlation integral of like-sign (solid line) and unlike-sign (dashed line) particles on the number of cells in the three-dimensional analysis of the NA35 Collaboration for the following reactions : (a) p + Au, (b) O +Au, (c) S + S (from [NA35 Coll. (1993)]).

of the particles produced in 7r+/i('+-nucleus interactions, using the same experimental setup as for the n+/K+-p collisions studied earlier by this group [NA22 Coll. (1990); NA22 Coll. (1991a); NA22 Coll. (1991c)]. The results show weaker intermittency signal for larger targets. Generally, the three-dimensional correlations seem to be of the Bose-Einstein type, and have a power-component in the two-particle correlations [UA1 Coll. (1993)]. Of course, the signal is always reduced in the one-dimensional and two-dimensional analysis by the dimensional projection effect.

The proton-nucleus and nucleus-nucleus collisions were first analyzed by the KLM Collaboration (1989a,1989b) in one-dimensional and two-dimensional distributions. The intermittency signal decreased for larger projectiles, but this decrease was smaller than expected from the increase of the mean multiplicity in the collision. It was impossible to reproduce this nonlinear dependence on the multiplicity by the models of independent collisions. The EMU01 Collaboration (1990,1991) performed also the intermittency analysis for different nuclear projectiles and targets and found a similar dependence as the KLM Collaboration. Generally, the intermittency signal decreases rapidly with increasing incident energy (14.6, 60 and 200 GeV).

The NA35 Collaboration (1993) analyzed the multiparticle production


in the nuclear collisions also in the three-dimensional momentum space. In figure 12.3, the results for the correlation integral of like-sign and unlike-sign particles in different reactions are shown. Once more the correlations are much stronger between the like-sign particles. The strength of the correlations does not decrease as the inverse of the mean multiplicity of the reaction, indicating that we deal with a collective effect such as the Bose-Einstein correlations.

In summary, all studied high energy processes show an increase of fluctuations with increasing resolution. However, the scaled factorial moments dependence on the bin size flattens for small bins in one-dimensional analysis. An intermittency signal was found stronger in two-dimensional and even stronger in three-dimensional analysis, where no sign of flattening is seen. The rise of the factorial moments at small scales could be parameterized by a singular power law component of the two-particle correlation function. Alternatively, as we shall discuss in this chapter, the scaling in three-dimensional factorial moments can be reconstructed in the class of self-affine scale transformations. It is now clear beyond any reasonable doubt that the strong correlations at small scales are present in the samples of like-charge particles. This means that the Bose-Einstein correlations are mainly responsible for increasing factorial moments.

12.1.4 Self-similarity or self-affinity in multiparticle production?

The self-similarity may well be only approximately satisfied due to the well-known anisotropy of the phase space in the multiparticle production [L. Van Hove (1969)]. For example, the longitudinal momenta are bounded only by the energy - momentum conservation and can be large, whereas the transverse momenta are small with an average value of 0.3 - 0.5 GeV/c. This could be the reason of different scaling properties in the longitudinal and transverse directions * and the plausible explanation for non-ideal effects seen in three-dimensional analysis [Y. Wu & L. Liu (1993)].

*It is interesting in this context to notice an analogy with the fractal landscapes in the presence of gravity which leads to different scalings of vertical and horizontal variations of fractal landscapes [B. B. Mandelbrot (1991)]. Consequently, the anomalous scaling of fractal landscapes is self-affine in the vertical plane, and self-similar in the horizontal plane. In the multiparticle production scenario, one could imagine the self-affinity of the phase space structures in the plane consisting of longitudinal and transverse directions and the self-similarity of the phase space structures in the transverse plane.


The self-affine analysis of the three-dimensional phase space {Pi,Pj,Pk} corresponds to the transformations :

dpi -» Spi/Xi, Spj ->• 5pj/\j, Spk -> Spk/Xk

with different scaling factors : Xi,\j,\k- Under this transformation the scaled factorial moments are expected to have the scaling property :

Fq{6Pi, 6Pj, 5Pk) = xf'xf^X^F^XiSpi, XjSpj, Xk5pk) (12.1)

The anisotropy of the system, i.e. the ratios Aj,Aj,Afc, are characterized by the Hurst exponents :

Hab=1^ (a,b = i,j,k; a^b; Xa < Xb) (12.2)

with 0 < Hab < 1. For : Hab = 0, A0 = 1, A& = A 1, one finds scaling only in the 6-direction. In the other limit : Hab = 1, Aa = A& = A, the self-affine transformation reduces to a self-similar transformation, i.e. the system is isotropic in a and b directions. For 0 < Hab < 1, it exists the non-trivial self-affine fractality in the (a, b) plane and the fluctuations are anisotropic.

12.1.4.1 Self-affine analysis of n+/K+p data

The Hurst exponents have been extracted from the data [NA22 Coll. (1996a); NA22 Coll. (1998b)] by fitting three saturation curves of different one-dimensional second scaled factorial moments [W. Ochs (1990)] :

F^a) = aa - {3aMr* , a = i,j,k

where Ma = Apa/Spa is the number of intervals in a-direction. Apa and Sa

are the initial interval size and the final interval size respectively. aa, j3a, j a are the fit parameters. Then :

Hab= - ° , a,b = i,j,k; a^b . (12.3) 1 + 7 6

With so determined Hurst exponents, one can perform the self-affine analysis and see whether the exact scaling is observed in three-dimensions.

The self-affine three-dimensional analysis of n+ /K+p data at 250 GeV/c is shown in figure 12.4. In spite of large statistical errors, it seems that the scaling is observed in self-affine (open circles) rather than in self-similar (full


lnF2

0.9

0.7

0.5

0.3

0.1 -0.2

lnMv

Fig. 12.4 The three-dimensional self-afflne analysis (open circles) of l n i ^ vs. I n M for real M is compared with the self-similar analysis (full circles) (from [NA22 Coll. (1998b)]).

circles) analysis. The scaling exponent deduced from this three-dimensional analysis is [NA22 Coll. (1998b)] : v = 0.061 ± 0.01. The Hurst exponents could be deduced using both three-dimensional data and the corresponding three different one-dimensional projections. The results for the Hurst exponents : Hypt = 0.48±0.06, Hyip = 0.47±0.06 and HPt4> = 0.99±0.01, are compatible with the picture of anisotropy between longitudinal directions (Hya — 0.5, a ^ y) and isotropy in the transverse plane (Hab ~ 1, with a,b both in the transverse plane). Using the same Hurst exponents, the (y,Pt) and (y,(j)) two-dimensional self-affme phase space projections have been analyzed. In both cases, l n i ^ increases similarly, what could indicate that influence of dimensional projection in self-affine analysis is weaker.

12.2 A-scaling in pp collisions?

In pp collisions, KNO scaling of the multiplicity distributions of produced particles, holds approximately up to ISR energies [V. V. Amonosov et al.

A-Scaling in pp Collisions? 331

(1972); C. M. Bromberg et al. (1973); A. Breakstone et al. (1984); S. Barish et al. (1974); A. Firestone et al. (1974); J. Whitmore et al. (1974); H. B. Bialkowska et al. (1976); W. M. Morse et al. (1977)] (see also [P. Carruthers k C. C. Shih (1987)]). At higher energies, the UA5 Collaboration reported the distributions at -y/s=200 and 900 GeV which show significant deviations from the KNO scaling [UA5 Coll. (1987)]. In particular, the shape of the distribution in full space at 900 GeV , when plotted in the KNO variables, is narrower in the peak region than at 200 GeV (see figure 12.5). Moreover, the maximum of the distribution at 900 GeV is

Fig. 12.5 Multiplicity distributions for inelastic, non single-diffractive events at 200 (triangles) and 900 (circles) GeV in full space are plotted in the first scaling variables (KNO variables ) (from [UA5 Coll. (1987)]).

higher and shifted to lower zi-values than at 200 GeV. Also a characteristic large—z tail appears, what signifies a departure from the fit using the Pascal distribution. Much work has been done to separate soft and semi-hard processes in pp collisions and to use a superposition of Pascal distributions for separate classes of processes to fit the data [A. Giovannini & R. Ugoc-cioni (1999)]. The signification of this procedure is difficult to assess since the generic origin of Pascal distribution in the multiparticle production


is still debated (c.f. Section 5.2.5). Moreover, this distribution does not appear naturally in perturbative QCD [Yu. L. Dokshitzer et at (1991); R. Botet & M. Ploszajczak (1996c); R. Botet & M. Ploszajczak (1997b)]. In figure 12.6 we plot the same multiplicity distribution as in figure 12.5 but

O(Z0.9)

1.5

1.0

0.5

0.0 - 1 0 1 2 3 4

^0.9

Fig. 12.6 The same as in figure 12.5 but in the A-scaling variables for A = 0.9.

in the A-scaling variables for A = 0.9. Even if uncertainty on the average multiplicity is probably large, the scaling looks convincing. Tail of scaling function is a benchmark of the fluctuations. In the scaling variables of the A-scaling, the logarithm of multiplicity distributions is plotted versus z\ 9

in figure 12.7. We see that the tails are linear in this plot, proving that tails of the multiplicity distributions are essentially Gaussian, similarly as in the KNO scaling regime at lower energies. The Gaussian tail of the multiplicity distributions and the tendency of a gradual decrease of from A « 1 up to ISR energies to A < 0.9 at CERN energies, signifies a gradual change of the disordered phase and the deviation from the particle multiplicity as the order parameter. It remains to be seen at much higher bombarding energies whether this tendency remains. In this case, exponent of the A-scaling of the multiplicity distributions is expected to approach 1/2 asymptotically.

* A • A

• A

• A •

• %

A A

•. ft •

\

• ft \

\

1 -

v A ^ s * » « _ _ —

A-Scaling in pp Collisions? 333

$ ( Z 0 . 9 )

10u

10 -1

10 -2

Fig. 12.7 Logarithm of scaled multiplicity distributions [UA5 Coll. (1987)] shown in figure 12.6 is plotted versus z^ g (zo.9 > 0).

This would signify a radical departure from the particle multiplicity distributions as the relevant observables characterizing phases of the hadronic matter. This would also signify 'death of fragmentation scenario' of particle production, characteristic for the perturbative QCD scenario, and birth of aggregation-like scenario, typical for extended blob of matter, with its proper phase diagramme.

12.2.0.2 Aggregation scenario for pp and AA collisions?

Correlation studies have disclosed the deficiency of the Lund Monte Carlo program PYTHIA, containing jets, resonance production and the Bose-Einstein correlations as final state interactions, for the description of the differential density correlation function [B. Buschbeck et al. (1994)]. The correlations are both overestimated at high PT (high multiplicity) and underestimated at low px (low multiplicity) by PYTHIA [B. Buschbeck et al. (1994); UAl Coll. (1990)]. Significant improvement can be gained assuming existence of clusters of low pr [B. Buschbeck et al. (1994); UA5 Coll. (1987a)], as included in the non-diffractive event generator


GENCL [UA5 Coll. (1987a)]. The picture of hadron-hadron collisions which emerges from these phenomenological studies shows the necessity of a combined description of hard scattering processes followed by the string fragmentation (the perturbative QCD regime) and the formation and decay of clusters (the non-perturbative QCD regime).

Another indication comes from Pb+Pb collisions at 158 • A GeV, where the multiplicity distributions for narrow centrality bins are perfectly Gaussian [WA98 Coll. (2001)]. Again this underlines the inadequacy of the fragmentation scenario in those complicated collisions leading to the formation of an extended blob of hot matter.

The arguments put forward in favour of the cluster picture refer often to the statistical bootstrap model [R. Hagedorn (1965); S. Frautschi (1971)]. In this model, the number of species of possible constituents of the fireballs and the number of species of fireballs grow asymptotically like ~ m-Texp(m/T 0 ) , with r = 3 [W. Nahm (1972)]. As discussed in Section 11.5.1, this value of r excludes a possible identification of To with the critical temperature of the thermodynamical quark-gluon phase transition. Another family of models refers to the spontaneous breakdown of the chiral symmetry of the QCD vacuum at a critical temperature. In the approximation mu « rrid sa 0, ms —• oo, the chiral QCD phase transition is of second order and the order parameter is given in terms of the sigma and pion field condensates [F. Wilczek (1992); K. Rajagopal & F. Wilczek (1993)]. The critical point belongs to the universality class of a three-dimensional Ising model and is located on a line of non-zero baryonic density [F. Wilczek (2000)]. Critical baryonic effects in heavy-ion physics have been studied by Antoniou (2001). In general, we get the guidance in designing appropriate models simulating the critical behavior in finite temperature QCD from the lattice QCD calculations (see e.g. [F. Karsch (1998)]). In all these models, the proper characterization of the soft phase of QCD should involve an understanding of features of the largest cluster (fireball) distribution.

To design the relevant experimental observable in pp and AA collisions is by far the most difficult task. At the same time, this objective is the most important challenge of soft physics which if unsolved will jeopardize the analysis of different phases of hot hadronic and quark-gluon matter. The hadron (baryon) multiplicity distribution may contain useful information only if the studied system happens to be found at the critical point, what

Universal Fluctuations in Excited Nuclear Matter 335

obviously limits the relevance of this observable. In this case, the study of scale dependence of correlations by means of scale dependence of the factorial (cumulant) moments of the density distribution, could reveal the underlying fractal structure of the largest cluster [N. G. Antoniou et al. (1998)]. This is for the moment largely a theoretical possibility because the exact reconstruction of the geometry of the largest cluster may be even a harder experimental task than finding its mass event by event [R. C. Hwa k Y. Wu (1999)].

12.3 Universal fluctuations in excited nuclear matter

In nucleus-nucleus collisions at intermediate energies, one observes the transition from the particle evaporation regime at low excitation energies to the explosion of the hot source at about 5-10 MeV/nucleon, which is associated with a copious production of intermediate mass fragments. At still higher excitation energies, the hot source vaporizes into light particles. Possibility of the critical behavior associated with this transition cannot be excluded, though most probably this transition happens in out of equilibrium environment.

The transition takes place in a small non-equilibrium system with strong long-range correlations. Coulomb interaction, which is an essential ingredient in this process, enables to define the thermodynamic limit in a usual sense for this transition. Such system is explicitly non-extensive, and usual tools of Boltzmann-Gibbs thermodynamics cannot be applied [C. Tsallis (1988); C. Tsallis (1995)]. It has been shown in the non-extensive statistical multifragmentation model (SMM) that even small non-extensivity may have important consequences on the transition itself changing many of its aspects [K. K. Gudima et al. (2000)]. The signatures in the transition region have been found to be fragile, strongly depending on even small deviation from the Boltzmann-Gibbs equilibrium thermodynamics. This effect has been explained in the non-extensive spherical spin model (i.e. the non-extensive generalization of Berlin-Kac model [T. H. Berlin &: M. Kac (1952)]) by the appearance of a new phase of weak-paramagnetism [R. Botet et al. (2001)], in addition to the two phases known before in the extensive Berlin-Kac model : the ferromagnetic phase and the paramagnetic phase. This new phase, which competes with the two latter phases and exists also in the thermodynamic limit of the non-extensive spherical spin


* ( * A )

10°

10"

10" -0.7

Fig. 12.8 Zmax —distributions in the first scaling variables for central Xe + Sn collisions a t Eiab/A = 25 (asterisks), 32 (crosses), 39 (triangles), 45 (diamonds) and 50 (circles) MeV (from [R. Botet et al. (2001)]). The centrality condition is associated with the conditions on the angle Qjiow between the beam direction and the main emission direction of matter in each event.

model, appears solely as a result of non-extensive long-range correlations. Consequences of those exact results may be far going. They may signify that the transition seen in excited, electrically charged small systems formed dynamically in the collision, like heavy atomic nuclei or atomic aggregates, should not be associated with the well-known liquid-gas phase transition of equilibrium thermodynamics. Moreover, an interpretation of experimental data may be strongly assumption/model dependent because most of theoretical tools applied for its understanding borrow heavily from the arsenal of standard tools of equilibrium physics. For that reason, the analysis of data from the point of view of universal characteristics of fluctuations of observables in finite systems is very attractive since it does not assume that the system is at equilibrium.


12.3.1 A-scaling in nucleus-nucleus collisions in the Fermi energy domain

What makes the study of multifragmentation of heavy-nuclei in the intermediate energy collisions particularly interesting is that the order parameter for generic scenarios of aggregation and fragmentation are measurable. The size of the cluster can be identified with the number of protons of the nuclear fragment (the fragment charge) since neutrons do not form self-bound fragments. Analogously, the fragment multiplicity is given by the multiplicity of charged fragments. So we can analyze in these collisions how the total charge of excited nuclear system formed in the collision of heavy-ions splits into fragments*. Both the charge of the largest fragment (the fragment-size) and multiplicity of charged fragments including protons (the fragment multiplicity) are measurable in nuclear collisions with high precision.

It is commonly expected that the nuclear multifragmentation process is controlled by the amount of thermalized excitation energy in the fragmentation residue after emission of pre-equilibrium particles and fragments. Unfortunately, this quantity is not directly measurable, so several selection criteria have been proposed to monitor the violence of the collision. These criteria lead to the selection of ensembles of events corresponding to different dynamical features of the fragment production, such as e.g. the degree of equilibration.

Figures 12.8-12.10 show the A-scaling analysis of Zmax— distributions for Xe -(- Sn collisions at 25 MeV < Eiab/A < 50 MeV, where events have been selected by the following centrality condition : complete events (i.e., more than 80% of the total charge and momentum is detected) and ®fiow > 7r/3. The latter quantity is a global observable defined as the angle between the beam direction and the main emission direction of matter in each event, which is determined from the energy tensor. It has been shown for the reactions in the Fermi energy domain that events with small Qfiow are dominated by binary dissipative collisions [J. Cugnon & D. L'Hote (1983); INDRA Coll. (1994); INDRA Coll. (1997)]. For events with little or no

*At present, neutrons and fragment masses cannot be detected event-by-event with a sufficient precision to allow for the study of nuclear mass fragmentation. Numerical tests using generic models like the fragmentation-inactivation binary model or the Ising model indicate however that the signatures of the scaling behaviour are preserved even in the absence of monomers (neutrons).


*(*A)

* x*

X

+ X

+ X

X

1/2

^

m-*+

-4 0 4

Fig. 12.9 Zmax —distributions in the second scaling variables for central Xe + Sn collisions at Etab/A = 25 and 32 MeV (from [R. Botet et al. (2001)]). The centrality is defined as in figure 12.8.

memory of the entrance channel, ®fiow is isotropically distributed [R. Botet et al. (2001)]. Figure 12.8 shows that Zmax—distributions for Eiab/A = 39,45,50 MeV can be compressed into a single curve in the scaling variables of the first scaling. The distributions for 25 and 32 MeV, which show strong deviations with respect to this scaling curve both near the maximum and in the tail for large z&, can be compressed into another single curve in the variables of the second scaling, as shown in figure 12.9. This perfect compression of data for 25 and 32 MeV in the variables of the second scaling is somewhat surprising if one recalls an example of the Landau-Ginzburg model , where the scaling function in the ordered phase of this model depends explicitly on the control parameter (c.f. Section 11.3.5). On the contrary, the scaling function in the disordered phase of the Landau-Ginzburg model is independent of the control parameter and hence provides a unique characterization of the whole disordered phase.

A global indicator of scaling features is provided by the cumulant moments. In case of the A-scaling, normalized cumulant moments : Kg = Kq/(Ki)qA , are independent of the system size (m). The logarithm


ln(K2)

4.5

3.5

2.5 5 6 7

l n « ) Fig. 12.10 The normalized second cumulant K.% (= K^/KI2) of Zmax—distribution is plotted as a function of \n{Zmax)2 (= ln/c^) together with the statistical error bars. Symbols for data at different energies are same as in figure 12.8 (from [R. Botet et al. (2001)]).

of the normalized cumulant moment K = K.\ of Zmax—distribution is plotted in figure 12.10 versus the logarithm of (Zmax)

2, i.e. versus the logarithm of Kj. The data for different (Zmax), i.e. for different bombarding energies, should lie on a straight line if the scaling holds with a fixed value of A. The slope of this line gives then the value of A. It is clear that the higher energy branch (Eiab/A = 39,45,50 MeV) follows the line A = 1 (the solid line), in agreement with the collapse of distributions in the first scaling variables which are seen in figure 12.8 for the same bombarding energies. The points for Eiab/A — 25 and 32 MeV lie clearly off this line. This behaviour is consistent with the figures 12.8 and 12.9, which show clearly that the collisions at these energies belong to the case A = 1/2. In this "near-crossing" case, one has to investigate higher order moments, or better, the whole probability distribution, in order to be able to detect some critical behaviour.

The pattern of charged fragment multiplicity distributions P(n) does not show any significant evolution with the bombarding energy (figure


10-1

10"2

-2.5 0.0 2.5 *A

Fig. 12.11 Multiplicity distributions of charged fragments for central Xe + Sn collisions are plotted in the second scaling variables. Symbols for data at different energies are same as in figure 12.8 (from [R. Botet et al. (2001)]).

12.11), and the da ta is perfectly compressible in the scaling variables of

the second scaling *. This means tha t the multiplicity fluctuations are

small (nq ~ ( K I ) 9 ^ 2 ) in the whole studied range of bombarding energies.

The scaling features of experimental Zmax— and n— distributions in fig

ures 12.8-12.10 and 12.11 are complementary and allow to affirm tha t the

fragment production in central heavy-ion reactions follows the aggregation

scenario and exhibits the structural change at Eiab/A > 32 MeV between

the two phases of excited nuclear mat ter with distinctly different pa t te rns

of

Zmax —fluctuations.

For more peripheral collisions, the scaling pa t te rn of P(Zmax) follows

a similar general evolution with the bombarding energy (the excitation

energy) as seen in central collisions, except tha t the branch A = 1 in semi-*As mentioned before, this feature is not generic and many models which exhibit the second order critical phenomenon yield a weak dependence of the scaling function on the control parameter in the ordered phase. The appearance of the unique scaling function in the ordered phase is a salient feature of nuclear fragmentation which is not yet fully understood.

"s

I

bo 3

^ T t A=l/2

i %

x


«>(ZA)

10°

io-'

io-2

-0.7 0.0 0.7 ZA

Fig. 12.12 Zmax— distributions for semi-central {Eiaf,/A = 45 (full diamonds) and 50 MeV (full circles)) and central (Eiai,/A = 39 (open triangles), 45 (open diamonds) and 50 MeV (open circles)) Xe + Sn collisions are plotted in the first scaling variables (from [R. Botet et al. (2001)]).

central collisions starts at higher energies than in central collisions. This is perfectly understandable if we suppose that the fragmentation regime is controlled by the source excitation energy which for the same bombarding energy varies with the centrality of the collision. Figure 12.12 presents all ^max— distributions which in semi-central and in central collisions belong to the respective first scaling branches. It is non-trivial that the probability distributions corresponding to different selection criteria collapse into the unique scaling curve.

Another conditions for most violent collisions correspond to selecting events according to either the large total transverse energy of charged particles and fragments Etrans, or the large total transverse energy of light charged particles (protons and heliums) Etransi2- The first criterion weights strongly intermediate mass fragments and in this respect is somewhat to the selection of central collisions by &fiow, though one could expect that the selection by large Etrans is less sensitive to event-by-event equilibrium characteristics of the energy tensor. The second criterion, according to maximal


value of Etransii, weights in particular light charged particles and as such accepts events corresponding to strongly off-equilibrium fragment production. Maximizing excitation energy carried by light charged particles, this selection minimizes amount of available excitation for the production of intermediate mass fragments. Hence for the same bombarding energy, the excitation energy of heavy reaction residue contributing to a production of intermediate mass fragments in ensemble of events selected by Qfiow — or £Jtrans—centrality conditions will be higher than those in the ensemble of events selected by Etransi2—centrality condition. In the aggregation scenario of nuclear multifragmentation, which is clearly favoured by the data (see figures 12.8 - 12.12), one expects that the transition from ordered phase to disordered phase will be seen at higher bombarding energies in the sample of events selected by Etransi2-

Figure 12.13 shows the logarithm of the normalized cumulant moment K\ ~ ' of Zmax—distribution versus the logarithm of KI 2 , for the ensemble of events selected by the largest total transverse energy of light charged particles Etransi2- The pattern is similar to those in figure 12.10, with one notable difference, namely, the transition to the first scaling happens at higher bombarding energies [Eiab/A ~ 50 MeV) than for events selected by Qfiow centrality condition. Comparing figures 12.13 and 12.10, one may notice that the width of the scaling function in the disordered phase depends on whether events have been selected by Qfiow or by Etransi2- This means that the reconstructed scaling function is sensitive to the difference in dynamics of the fragmentation process in the disordered phase.

Complete characterization of the nuclear fragmentation in the scaling regime is provided by the scaling function. Form of these functions in ordered and disordered phases is largely independent of the criteria of selecting events. The scaling function of the multiplicity distribution is nearly Gaussian with the variance (z^,2) - (zi/2)2 — 0.338±0.020 for central collisions selected by Qfiow. The variance and skewness of the universal scaling function of Zmox-distribution in the disordered phase are 0.087 ±0.008 and 1.29 ± 0.03, respectively. These values are much smaller than in the disordered phase of the Landau-Ginzburg model (c.f. Section 11.3.4) for which one finds n/8 and 49/ y/2, respectively.

The characteristic feature of critical behavior is the anomalous tail of the scaling function with v = 6+1 > 2. Using all P(Zmax) shown in figure 12.12, one can fit the scaling function $(21) for z\ > 0 by aexp(—b(zi — zl)"), where zj" is the estimate of the most probable value of the distribution


ln(K2)

5

4

3

2

1

0 3 4 5 6 7 8

ln(Kj) Fig. 12.13 The normalized second cumulant K.2 (= K2/K12) of Zmax —distribution is plotted as a function of l n ( Z m o x ) 2 (= In K 2 ) together with the statistical error bars. The line A = 1 is shown to guide the eyes. Events are selected according to the largest total transverse energy of light charged particles (from [A. Chbihi & A. Mignon (2001)]) Full circle, square, diamond, triangle and inverted triangle correspond to Xe + Sn collisions at Elab/A = 25, 32,39,45 and 50 MeV measured by INDRA Coll. at GANIL/Caen. Open inverted triangle, triangle, diamond, square and circle correspond to collisions at Eiab/A = 50,65,80,100 and 150 MeV measured by INDRA at GSI/Darmstadt.

and a,b,v are the fitting parameters. One finds : v = 1.6 ± 0.4, what is incompatible with typical values (3.5 < v < 6) in the critical region for aggregation scenarios. Hence, the existing data do not show a critical behavior in the transition region between ordered and disordered phase. One should also notice that the scaling function in the transition region has no two-hump structure which is expected in case of the first-order phase transition.

The present analysis shows that the fragment production in central heavy-ion collisions at around the Fermi energy is governed by the aggregation scenario with (Zmax) (size of the largest fragment) as the order parameter. The change of the regime of Zmax—fluctuations from the sec-


ond scaling at low energies to the first scaling at higher energies, with the Gaussian tail of the scaling function in both regimes, is compatible with the transition from the ordered phase to the disordered phase. The nature of this phase change is yet unknown, i.e. more data are necessary to distinguish between a possibility of the phase transition in a finite-system and the cross-over phenomenon.

Chapter 13

Final Remarks

There is still a long way to understand the multihadron-production processes and, in particular, those aspects of these processes which are related to the strong-coupling long-distance regime of QCD. Experimental studies of correlations in small domains of the phase-space, posed the problem of the information contained in the multiparticle correlations of produced hadrons and, more generally, in the fluctuation pattern of observables. In this book we have stressed at several places how limited is the information contained in the multiparticle total cross section. The crucial phase information is hidden in the correlation data, particularly in the Bose-Einstein correlations. We have shown that fluctuations (correlations) of observables in complex multiparticle systems have universal properties even in small finite systems, independently of whether the studied process is classical or quantal, equilibrium or non-equilibrium, dissipative or conservative, whether the underlying microscopic variables are strongly correlated as in critical phenomena or, on the contrary, are mutually independent, etc. We dispose a complete characterization of the convergence (scaling) of probability distributions in finite systems of different sizes. In this sense, these general results and associated mathematical tools can be applied in any physical, economical, social, etc. systems which can appear at different sizes.

The information contained in the particle density and correlations can be extracted by determining : (i) the scaling features (the A—scaling law) of the measured probability distributions of the observable, in particular, the asymptotic properties of tail of the scaling function, (ii) the anomalous dimension, and (iii) the scaling features of multiparticle correlations

345

346 Final Remarks

(clustering) in small phase-space cells. In this way, the relevance of the observable can be tested and the constraints on the conjectured reaction mechanism can be searched for in experimental results.

Much of the experimental search for the scaling features in the multi-hadron distributions has been motivated by the prediction of the KNO scaling as an ultimate symmetry of the 5-matrix in ultrarelativistic collisions. This scaling is only a particular case of the universal A—scaling law which has been found recently.

A crucial problem in a phenomenological analysis is the determination of the relevant observable for each multihadron-production process, which could exhibit different phases through the non-trivial fluctuation pattern and the evolution of this pattern. Whereas the hadron multiplicity distribution seems to be relevant for e+e~ collisions, which exhibit many features of the fragmentation scenario, this observable seems to be of secondary importance in pp, pA and AA collisions. In these latter processes, it is plausible that the aggregation scenario dominates and hence, the clustering measures have to be studied. The development of new methods of event by event determination of large correlated phase-space structures (clusters) decaying into hadrons is the challenging urgent problem which will determine the future evolution of the soft physics and the multiparticle correlations.

The experimental analysis of the ratio of the average charged multiplicity to the dispersion in e+e~ —> qq events shows that the multiplicity distributions obey the first-scaling law [B. Buschbeck (1995)]. Unfortunately, the tail of the multiplicity probability distribution has not been studied, so it is unknown at present whether e+e~ —> hadrons is the critical process at presently available energies. Nevertheless, the fact that such a simple observable as the hadron multiplicity is relevant, allows to hope that future studies will solve the enigma of possible off-equilibrium shattering phase transition (fragmentation scenario) in lepton - antilepton annihilation reaction. This transition in the perturbative regime of the QCD, is not related with the quark-gluon phase transition. The latter transition is dominated by the strong-coupling regime of QCD (the aggregation scenario).

The experimental analysis of multifragmentation in nuclear collisions at intermediate bombarding energies (20 MeV < Eiab/A < 150 MeV) is perhaps easier to grasp because the relevant observables for two generic scenarios of the fragment production are in principle measurable. These are the size of the largest cluster (proton/baryon number of the biggest fragment)

Final Remarks 347

for the aggregation scenario and the multiplicity of fragments for the fragmentation scenario. The recent experimental studies have demonstrated the change in the fluctuation pattern of the largest nuclear fragment and the transition to the disordered aggregation phase of nuclear fragmentation which is characterized by the universal scaling function. Different aspects of dynamical evolution and equilibration can be investigated using various criteria of event selection.

In this book we have emphasized as much as possible the intrinsic properties of classes of probability distributions in order to see clearly those features which are not tied to a specific dynamical model. We believe this aspect is crucial because whichever model of multi-hadron and multi-fragment production is conjectured, it will produce characteristic distributions which can be compared with the experimental distributions. We have especially emphasized the ambiguities in order to discourage premature conclusions which too often accompany studies of complex multiparticle systems with finite and usually small number of elementary components (microscopic variables). The microscopic dynamics of any single component of such systems is unpredictable. However even a small collection of these generically unstable components show mysterious structural stability, which is reflected in the universal probability laws for the frequency of different (rare) behaviours, repeating themselves at each scale until the ideal of the asymptotic limit is reached. This structural stability of complex phenomena, which enables, e.g. learning, reasoning and intelligence in absurdly changeful environment of elementary appearances of the world, is not less surprising to us today than it was to the ancients :

"They do not apprehend how being brought apart it is brought together with itself: there is a back-stretched connexion, as in the bow and the lyre." [Heraclit (VI B.C.)]

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Index

a-model, 225, 235, 239, 242, 243 additive quantities, 30 additive stochastic process, 130 aggregation kernel, 308 aggregation probability, 308 algebraic singularity, 62 anomalous diffusion, 111 anomalous diffusion law, 87 anomalous dimension, 67, 72, 286,

304, 312, 318 anomalous exponent, 66, 67, 299 anomalous multiplicity dimension,

280 apparent Renyi dimension, 229 APW formalism, 215 arc-sine law, 39 Arrhenius formula, 93 asymmetric random walk, 84 asymmetric stable distribution, 20 asymptotic freedom, 246 asymptotically stable laws, 27 attraction domain of stable law, 90 autocatalytic birth process, 131 average cluster size, 321

Balian-Schaeffer model, 185, 297 Bernoulli's game, 84 Bessel-K distribution, 118 Bethe lattice, 301 binary process, 253

binomial distribution, 82, 115, 123, 154, 317

block random variable, 55 bond occupation probability, 303 bond percolation, 305, 307, 317 bond percolation model, 300, 304 bond percolation on Bethe lattice,

301, 303 Bose-Einstein condensation, 207, 212 Bose-Einstein correlation, 216, 219 Bose-Einstein distribution, 115, 123,

129 Bose-Einstein interference, 199 Bose-Einstein transform, 135 branching generating function, 232,

235, 238 branching process, 258 Brand-Schenzle regime, 263, 270 Brownian motion, 75, 100, 106 Buckingham-Gunton inequality, 70,

287 Buda-Lund parameterization, 203 bunching parameter, 144, 150

Cantor set, 125 Carleman's condition, 148 Cauchy asymmetric distribution, 21 Cauchy distribution, 18 Cauchy expansion, 23 Cauchy symmetric distribution, 21

363

364 Index

Cayley regime, 263, 270 Cayley tree, 258, 266 Central Limit Theorem, 8, 31, 82, 83,

299 chaoticity parameter, 201 characteristic function, 13, 15, 27, 85,

91, 151 characteristic function of probability

distribution, 69 characteristic function of stable

distribution, 26, 90 characteristic length, 286 characteristic size, 310 chiral phase transition, 334 clan, 155 clan model, 125 closed horizon, 96 coherence length, 55, 107, 285 combinants, 144, 150, 153 composed-particle first moment, 255 concentration-size distribution, 301 conditional probability, 154 conformal invariance, 184 conformal symmetry, 187 connectivity constant, 108 conservative process, 251 continued-fraction expansion, 148 continuous Levy nights, 89 control parameter, 303 convolution theorem, 14 correlated Gaussian process, 47, 50 correlated random variables, 39, 54 correlation function, 48 correlation integral, 180, 216, 327 correlation integral method, 180 correlation size, 316 correlation time, 79 covariance exponent, 59 covariance function, 46, 48, 49, 64,

77, 100 Cracow algorithm, 212, 214 critical class, 69 critical gelation time, 308 critical point, 304, 309

critical process, 318 critical system, 66 critical temperature, 299 criticality in finite systems, 73 cumulant branching generating

function, 236 cumulant correlator, 179 cumulant factorial moment, 291 cumulant generating function, 140,

149, 158 cumulant moment, 141 cut-off parameter, 247 cut-off scale, 318 cut-off time, 127

A-scaling, 30, 65, 71, 73, 123, 186, 276, 305, 313, 316, 318, 332

A-scaling variable, 72 D4 wavelet, 194, 196 damped anharmonic oscillator, 132 Debye theory, 41 density matrix, 208 dielectric constant, 41 dielectric relaxation, 41 difference function, 193 differential factorial correlation, 182 diffusion constant, 76, 78, 82, 83, 100 diffusion distance, 83 diffusion equation, 76, 80 diffusion on disordered fractal, 111 dilatation equation, 193 dimensional projection, 174 directed percolation, 258 directed self-avoiding random walk,

259 disordered phase, 257, 293 dissipative gluodynamics, 278 dissipative perturbative QCD, 279,

322 domain of attraction, 28, 59 domain of attraction of normal law,

10 domain of attraction of stable law, 29 double logarithm approximation, 246

Index 365

dynamical critical phenomenon, 309 Dyson model, 52

e-expansion, 57 equilibrium phase transition, 289 erraticity, 190 erraticity index, 190 Euler theorem, 62 evaporative regime, 268, 270 excluded-volume constraint, 104 exclusive distribution, 165 extensive order parameter, 286, 293,

299, 303

factorial cumulant generating function, 186

factorial cumulant moment, 142, 145, 150, 187

factorial moment, 119, 141, 149, 168 factorial moment generating function,

119, 141, 149, 155, 167, 290 father function, 193 ferromagnetic phase, 335 Fick's law, 75 Fick's representation, 75 field conjugate to order parameter, 68 finite-size scaling, 67, 69 first scaling law, 69, 73, 125, 136, 275,

282, 294, 303, 305, 307, 310, 315, 316, 319

first scaling variables, 310, 331, 336 Fisher inequality, 286 Fisher model, 322 fixed-point, 64 fixed-point equation, 57, 58, 63, 65 fixed-point of stability condition, 15 fixed-point solution, 56 Flory approach, 102 Fokker-Planck equation, 79, 100 Fokker-Planck representation, 79 Fox function, 117 fractal, 61, 63, 286 fractal dimension, 63, 108 fractional Brownian motion, 99, 101

fragment size, 253, 337 fragment-size distribution, 292, 311,

317, 319 fragmentation function, 229, 230 fragmentation kernel, 255, 256, 319 fragmentation probability, 255 fragmentation-inactivation binary

automaton, 259 fragmentation-inactivation binary

cascade, 261, 319 fragmentation-inactivation binary

model, 253-255, 257, 318, 319 fragmentation-inactivation binary

process, 322 free energy, 68, 71, 73 freeze-out time, 127

Gamma distribution, 125 Gaussian density matrix, 209 Gaussian distribution, 47, 144 Gaussian first scaling law, 295 Gaussian Levy flight, 81 Gaussian process, 48, 49, 64, 77 Gaussian random walk, 81 gel, 300, 301, 308 gelation phase, 309 gelation point, 315 gelation time, 315

Gell-Mann - Low transformations, 54 generalized Bose-Einstein

distribution, 121 generalized fractal dimension, 228,

229 generalized Gamma distribution, 117,

128, 129, 132, 264 generalized isothermal susceptibility,

70 generating function, 74, 124, 135, 315 generating function leading

singularity, 126 generating function of cluster

production, 156 generating function of probability

distribution, 314

366 Index

generating functional, 165, 167, 247 GHP topology, 181, 182 gluodynamics, 277 Gumbel distribution, 37 Gumbel law, 37

Haar wavelet, 194, 195 Haar wavelet difference function, 195 Haar wavelet scaling function, 195 hadronization scale, 240 hadronization time, 127 HBT correlations, 202, 206 HBT effect, 199, 201, 219 HBT measurement, 199 Hegyi's equivalence hypothesis, 137 Hermite rank of function, 49 high-viscosity phase, 259 Holtsmark problem, 39 homogeneity index, 62, 308 homogeneity relations, 137 homogeneous fractal, 228 homogeneous function, 62, 308 horizontal scaled factorial moment,

171, 172 Hurst exponent, 61, 62, 64, 66, 110,

329 Hurwitz function, 291 hyperscaling relation, 287

in-out picture, 205, 206 inactivation kernel, 255, 256, 319 independent particle production, 209 infinite cluster, 300, 301, 308 infinite-cluster phase, 255, 257 infinitely divisible distribution, 151,

152 inhomogeneous fractal, 228 instantaneous fragmentation, 251 intensive order parameter, 286, 293 interconnection effect, 214 intermediate dissipation range, 240 intermittency, 170 intermittency exponent, 170, 178, 323 intermittency exponent of scaled

factorial correlator, 177 intermittent behaviour, 170 invasion percolation, 266 irreversible kinetic aggregation, 308,

322 Ising model, 299 isothermal susceptibility, 287 Ito-McKean interpretation, 80

joint probability, 154

Kesten distribution, 95 Kesten variable, 34 KNO scaling, 134, 135, 282, 303 KNO variables, 331 Kronecker distribution, 141

Levy distribution, 86 Levy ensemble, 87 Levy flight, 83, 101, 108 Landau-Ginzburg free energy, 293 Landau-Ginzburg model, 293, 296 Langevin equation, 77, 133 Laplace transform, 119 Laplace transform of Poisson weight,

188 largest fragment distribution,

305-307, 310, 311 leading-log approximation, 126 limit distribution, 8, 16 limit fractal, 63 limiting temperature, 303 Lindeberg's theorems, 9 linked pair approximation, 183, 184,

291 linking coefficient, 183, 184, 291, 296 liquid-gas phase transition, 322 log-normal distribution, 32, 145, 188 loopless fractal, 111 Lund model, 213

macroscopic random variable, 7, 52 magnetic Ising model, 297 Markovian process, 79, 126

Index 367

Marr wavelet, 197 master equation, 80, 82, 247, 261,

308, 313 Maxwell distribution, 125 mean free path, 78 mean-field percolation, 258, 301 mean-field percolation threshold, 263 Mekjian model, 251, 290, 292 microscopic random variable, 7, 68 moment generating function, 140,

148, 149, 158, 260, 291 moments of stable distribution, 26 monofractal, 64, 228 monomer, 255 Morlet wavelet, 196 mother function, 192 multifractal, 111, 228 multifractal exponents, 238 multifractal formalism, 236 multifractal singularity spectrum,

227, 228 multiparticle correlation, 166 multiparticle correlation function,

166, 183 multiparticle cumulant, 166 multiparticle distribution function,

164 multiparticle inclusive cross section,

165 multiplicative Gaussian noise, 131 multiplicative stochastic process, 130 multiplicity, 115, 289, 321, 337 multiplicity anomalous dimension,

274 multiplicity distribution, 164, 264,

280, 303-306, 317, 331, 332, 340 multiresolution analysis, 193 multiscaling transformation, 242 multiscaling variables, 242, 243, 248

NNLL approximation, 126 non-conservative process, 251 non-equilibrium aggregation, 313 non-equilibrium sol-gel transition, 312

non-extensive Berlin-Kac model, 335 non-extensive correlations, 336 non-extensive spherical spin model,

336 non-extensive statistical

multifragmentation, 335 non-linear wave mixing, 131 non-Markovian process, 79 non-perturbative QCD, 334 normal distribution, 9-11

fi-expansion, 313, 315 off-equilibrium critical system, 288 off-equilibrium phase transition, 289 open horizon, 97 order parameter, 67, 293-295, 298,

303, 321 order-parameter distribution, 298,

319 ordered phase, 257, 293, 295, 299 ordinary moment, 140 Ornstein-Uhlenbeck representation,

77

p-model, 194, 235, 238 Pade approximants, 140 parabolic cylinder function, 118 paramagnetic phase, 335 partition function, 68, 294 Pascal distribution, 115, 117, 121,

123, 125, 129, 155, 157, 159, 297, 331

Paul wavelet, 196 percolation cluster, 303 percolation model, 300 percolation threshold, 305 percolation transition, 322 perturbative QCD, 126, 246, 268,

277, 332, 334 perturbative QCD cascade, 245, 246 Poisson distribution, 119, 123, 124,

148, 152, 157, 160, 188 Poisson transform, 114, 115, 155, 159,

265

368 Index

Poisson weight, 159 Pollard expansion, 24 Potts model, 297, 298 power spectral density function, 64 principle of common ancestry, 233 product of uncorrelated variables, 30 pseudo-critical point, 294

QCD cascade, 245, 248 QCD jets, 126 QCD scale, 246

Raman scattering, 131 random variables

strongly correlated, 46 weakly correlated, 46

random walk, 75, 83, 92 random walk with Gaussian memory,

97 random walk with memory, 97 random walk without memory, 81 Rayleigh distribution, 38 reciprocity relation for stable

distributions, 24 reduced cluster multiplicity, 318 reduced correlation function, 166, 185 reduced cumulant, 179 reduced density, 172 relaxation in disordered system, 41 relaxation time distribution, 42 renormalization group, 45, 59, 63 renormalization group theory, 52, 54,

66 Renyi dimension, 228 resolution scale, 192 reversible aggregation, 300, 304, 322 Rosenblatt's model, 50

same-lineage cumulant, 232, 233, 235 scale-dependent dissipation, 271 scale-invariance, 66 scale-invariant dissipation, 256 scaled cumulant moment, 74, 158, 265 scaled factorial correlator, 176, 177

scaled factorial cumulant, 178, 179 scaled factorial cumulant moment,

143, 184 scaled factorial moment, 121, 129,

142, 150, 169-171, 173, 224, 227, 329

scaling fields, 60 scaling function, 192, 312, 319, 320 scaling theory, 66 Schaeffer model, 185 second scaling law, 70, 275, 291, 295,

299, 304-306, 311, 316, 319 second scaling variables, 338 second-order phase transition, 106,

186, 288, 298, 312, 320 self-affine analysis, 330 self-affine process, 61 self-affine transformation, 329 self-avoiding walk, 103, 109 self-organized critical state, 259 self-organized criticality, 253, 266 self-similar analysis, 330 self-similar probability distributions,

61 self-similar process, 61, 62, 66, 109 self-similar transformation, 329 self-similarity law, 64 semi-stable law, 17 sequential cluster formation, 321 sequential fragmentation, 251, 321 shattered phase, 255, 275, 280, 319 shattering process, 318 shattering transition, 255 shifted order parameter, 304, 313 short-correlated random noise, 80 Sinai billiard, 96 singularity strength, 237 singularity strength correlation, 237 site percolation, 297 size of largest cluster, 289, 303 size-distribution, 303 sleepy ladder problem, 34 Smoluchowski kinetic model, 308,

310, 311, 313

Index 369

snake topology, 181, 182 sol-gel transition, 300, 308, 322 spatio-temporal intermittency, 323 spin, 297 split-bin correlator, 178 splitting cumulant, 233, 235 St. Petersburg paradox, 84 stability equation, 61 stability problem, 13, 14, 16, 55 stable asymmetric distribution, 24 stable distribution, 15, 19 stable law, 17, 65, 91 stable symmetric distribution, 22 Stacy distribution, 117, 127 star cluster topology, 182 stationary process, 46 statistical bootstrap model, 303, 334 statistical correlation factor, 52 Stieltjes series, 147 Stieltjes transform, 140 Stirling approximation, 302 Stratonovich interpretation, 80 stretched-exponential relaxation, 41 strip integral, 326 strong intermittency limit, 225 strong intermittency phase, 226, 229 sum of uncorrelated variables, 13 superdiffusive behaviour, 107 support dimension, 229 symmetric stable distribution, 19 symmetry breaking, 60

tail of scaling function, 311 Taqqu's reduction theorem, 49, 51, 61 thermodynamic limit, 303 transition probability, 76 tree parameter, 231 two-particle correlation, 157, 217

two-particle correlation function, 166, 183, 211, 220

two-particle intensity correlation function, 200, 202

two-particle reduced correlation, 248 two-point correlation function, 106,

108

unbiased random walk, 81

Wg-moment , 151 vertical scaled factorial moment, 172 Virial Theorem, 62 viscous cut-off, 240 void distribution, 191 void probability, 116, 165, 188 void scaling function, 189

wavelet, 193 wavelet amplitude, 194 wavelet basis, 192 wavelet correlation density, 194 wavelet correlations, 194 wavelet representation, 192 wavelet transform, 192 wavelet transform density, 194 wavelet transformation, 194 weak intermittency limit, 224 weak intermittency phase, 224, 226,

229 weak-paramagnetic phase, 335 Weibull distribution, 37, 38 Widom's hypothesis, 63 Wiener-Ito integral, 50 Wilson transformations, 54 Wintner expansion, 23, 24

YKP parameterization, 203

Fluctuations

The main purpose of this book is to present, in a comprehensive

and progressive way, the appearance of universal limit

probability laws in physics, and their connection with the

recently developed scaling theory of fluctuations. Arising

from the probabil i ty theory and renormalization group

methods, this novel approach has been proved recently to

provide efficient investigative tools for the collective features

that occur in any finite system.

The mathematical background is self-contained and is

formulated in terms which are easy to apply to the physical

context. After illustrating the problem of anomalous diffusion,

the book reviews recent advances in nuclear and high energy

physics, where the limit laws are now recognized as being

able to classify different phases of a system undergoing the

pseudo-critical behaviour. A new description of the hadronic

matter in terms of the fluctuation scaling is appearing as a

consequence of this approach.

World Scientific www. worldscientific. com 4916 he

ISBN 981-02-4898-9

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