First International School-Conference

Mathematics and Physics of Billiard-Like Systems –

Billiards’09

16-19 February 2009

Águas de Lindóia – São Paulo – Brazil

http://www.rc.unesp.br/igce/demac/billiards09/

2

Justification

Systems of the billiard type represent one of the most popular and best investigated class of hyperbolic dynamical systems. The billiard problem is a fascinating subject providing a fertile source of new problems such as conjecture testing in dynamics, geometry, theoretical and mathematical physics, mathematics, optics, thermodynamics, spectral theory, etc. Thus, since the early beginnings of the study of classical and quantum chaos, billiards have been used as a paradigm.

Billiards correspond to one of the best understood classes of dynamical systems that demonstrate a broad variety of behaviors including passages from integrable to mixed form and eventually to an entire chaotic dynamics. In fact, several key properties of chaotic dynamics were first observed and demonstrated for billiards. Many popular models of statistical mechanics, e.g., the Lorenz gas, the hard sphere (Boltzmann and Sinai) gas, etc., can be reduced to billiards. Moreover, examples of some formal models of the billiard type exhibit a wide class of behaviors with a number of new unexpected features. Such fundamental notions as Boltzmann ergodic hypothesis, Fermi acceleration, Maxwell's demon, Hamiltonian systems with divided phase space, the Riemann zeta-function, etc. obtain natural physical realizations.

The aim of this school-conference is to bring together renowned scientists in the field, leaving enough time and opportunity for fruitful discussions among the participants.

3

Organizing Committee

Edson Denis Leonel (edleonel@rc.unesp.br) Alexander Loskutov (loskutov@polly.phys.msu.ru) Jafferson Kamphorst Leal da Silva (jaff@fisica.ufmg.br) Makoto Yoshida (myoshida@rc.unesp.br)

Sponsors

FAPESP

CNPq

CAPES

FUNDUNESP

PROPe

PROPG

Programa de Pós-Graduação em Física – UNESP – Rio Claro

4

Schedule 15-16 February: Arrival 16 February: 08:00 - 9:00 Registration Chairman: Jafferson Silva Title: Classical results of billiard problems 9:00 - 9:15 Opening ceremony 9:15 - 10:15 Plenary lecture: Leonid Bunimovich Title: Mechanism of chaos in billiards 10:15 - 10:30 Coffee break 10:30 - 11:15 Invited lecture: Ruedi Stoop Title: Transport maps on grids of unit cells

and their relation to billiards 11:15 - 12:00 Invited lecture: Luc Rey-Bellet Title: Large deviations in hyperbolic billiards 12:00 - 12:45 Invited lecture: Orestis Georgiou Title: Survival Probability for the Stadium

Billiard 12:45 - 14:00 Lunch Chairman: Makoto Yoshida

5

14:00 - 14:45 Invited lecture: Vassili Gelfreich Title: Fermi acceleration in non-autonomous

billiards 14:45 - 15:30 Invited lecture: Serguei Popov Title: Random billiards with cosine reflection

law 15:30 - 15:45 Coffee break 15:45 - 16:30 Marina Vachkovskaia Title: Asymptotic behaviour of randomly

reflecting billiards in unbounded tubular domains

16:30 - 17:15 Invited Lecture: Silvio Thomaz de Souza Title: Noise-induced basin hopping in an

impact oscillator 17:15 - 18:00 Invited Lecture: Roberto Eugenio Lagos Title: Brownian Motors and the Long Arm of

the (Second) Law 19:00 Welcome dinner 17 February: Chairman: Leonid Bunimovich Title: Non autonomous Hamiltonian systems

and time-dependent billiards

6

9:00 - 10:00 Plenary lecture: Alexander Loskutov Title: Time-dependent billiards: Fermi

accelerations and retardation of particles 10:00 - 10:15 Coffee break 10:15 - 11:00 Invited lecture: James Howard Title: Two-Frequency Billiards 11:00 - 11:45 Invited lecture: Jafferson Silva Title: Presence and lack of Fermi acceleration

in nonintegrable billiards 11:45 - 12:30 Invited lecture: Alexandros Karlis Title: Fermi acceleration in simple

randomized driven billiards 12:30 - 14:00 Lunch Chairman: Roberto Markarian 14:00 - 14:45 Invited lecture: Roberto Venegeroles Title: Asymptotic algebraic laws in

Hamiltonian systems 14:45 - 15:30 Invited lecture: Joaquim Barroso Title: Interaction of a bouncing ball with a

sinusoidally vibrating table 15:30 - 16:15 Invited Lecture: Marcus Beims Title: Soft wall effects on interacting particles

in billiards 16:15 - 16:30 Coffee break

7

16:30 – 17:15 Invited lecture: Eugene Gutkin Title: Partial complexities for polygonal and

related billiard tables 17:15 – 18:00 Invited lecture: Edson Denis Leonel Title: Dynamical properties for one-

dimensional Fermi accelerator models: a short review

19:00 Dinner 18 February: Chairman: Alexander Loskutov Title: Generalized billiards and related topics 9:00 - 10:00 Plenary lecture: Roberto Markarian Title: Non-conservative billiards. Dominated

splitting 10:00 - 10:15 Coffee break 10:15 - 11:00 Invited lecture: José Manoel Balthazar Title: On a Nonlinear Electro-Mechanical

System Behaviour, using Transient Trajectories: Some Results and Future Perspectives on Bifurcational analysis

Short Talks: 11:00 - 11:25 Invited lecture: Mario Roberto Silva Title: Scaling properties of a sinusoidally

Corrugated Waveguide

8

11:25 - 11:50 Invited Lecture: Helio Aparecido Navarro Title: Vibrated granular beds: particle

approach simulation

11:50 - 14:00 Lunch 14:00 Free afternoon 19 February: Chairman: Esdon Denis Leonel Title: Statistical physics of many particle

systems

9:00 - 10:00 Plenary lecture: Stanislav Soskin Title: New approach to the treatment of separatrix chaos

10:00 - 10:15 Coffee break 10:15 - 11:00 Invited lecture: Genri Nornam Title: Statistical physics of many particle

systems 11:00 - 11:45 Invited lecture: Michael Zaks Title: Fluid flow across the billiard: example

of laminar disorder 11:45 - 12:30 Invited lecture: Igor Sokolov Title: Nonergodic behavior and time averages

in continuous-time random walks 12:30 - 14:00 Lunch

9

14:00 - 16:00 Posters 16:00 - 16:15 Coffee break 16:15 - 17:00 Discussions 17:00 - 17:30 Closing ceremony 19-20 February: Departure

10

Plenary Lectures

11

MECHANISMS OF CHAOS IN BILLIARDS

Bunimovich, Leonid

Georgia Institute of Technology, Atlanta, Georgia, USA

Two major discoveries of the last century were the persistence under small perturbations of chaotic and of regular behavior in dynamical systems. In typical Hamiltonian dynamical systems though these two types of behavior do coexist. To understand the phenomenon of coexistence one first should understand the mechanisms of chaos. Billiards form a class of visual Hamiltonian systems which demonstrate all the variety of possible dynamics from the completely integrable to the strongly chaotic one. Chaotic behavior in billiards is caused by hyperbolicity. We will address the following questions. 1.What are the mechanisms of hyperbolicity (chaos) in billiards? 2.Which mechanisms of chaos "allow" its coexistence with regular dynamics? 3.How smooth should be a boundary in 2D billiards in convex domain to force coexistence? 4.What are the types of coexistence of chaotic and regular dynamics?

12

TIME-DEPENDENT BILLIARDS: FERMI ACCELERATIONS AND RETARDATION OF

PARTICLES

Loskutov, Alexander

Physics Faculty, Moscow State University, Moscow 119992,

Russia - E-mail: loskutov@chaos.phys.msu.ru

Systems of billiard types with (periodically and stochastically) perturbed boundaries are considered. A generalized dispersing billiard - the Lorentz gas with an open horizon - and a focusing billiard in the form of a stadium are studied. It is analytically and numerically shown that, if the billiard possesses the property of the developed chaos, the consequence of the boundary perturbation is the Fermi acceleration. However, the perturbation of the nearly integrable billiard system leads to a new interesting phenomenon: the separation of the billiard particles in their velocities. This consists of the fact that if the initial particle velocities exceed some critical value (specific for the given billiard geometry) then the racing of the particle ensemble is observed. If the initial value is below the critical value, then the billiard particles are not accelerated. In this case a retardation of particles up to a certain level is observed. This property of a billiard system can be regarded as a billiard version of Maxwell's demon: The periodic perturbations of the boundaries result in velocity losses by slow (`cold') particles, and the acceleration of fast (`hot') particles. The dependence of the separation effect on the characteristic billiard parameters and the frequency of the boundary oscillations is described.

13

NON-CONSERVATIVE BILLIARDS. DOMINATED SPLITTING.

Markarian, Roberto

IMERL, Universidad de la República, Montevideo, Uruguay

A particle moves along straight lines inside a billiard table and when it hits one of the walls with angle A with respect to the normal line, it is reflected with angle L A (with L small than or equal to 1). We give formulas for a general class of these {\em pinball billiards}. Then we restrict the analysis: the reflection angle only depends on incidence angle (not on the boundary position). We prove that in many of billiard tables (in particular some in which classical billiard map is not hyperbolic) the dynamics has a weak form of hyperbolicity called dominated splitting (the tangent bundle splits into two invariant directions, the contractive behavior on one of them dominates the other one by a uniform factor). Joint work with Enrique Pujals and Martín Sambarino.

14

NEW APPROACH TO THE TREATMENT OF SEPARATRIX CHAOS

Soskin, Stanislav

Institute of Semiconductor Physics, Kiev, 03028, Ukraine

International Centre for Theoretical Physics, 34100 Trieste,

Italy

Separatrix chaos plays a fundamental role for Hamiltonian chaos and may be important in a broad variety of subjects in physics and astronomy [1-5]. Time-periodically perturbed 1D systems form an archetypal class of systems where it arises. One of the widely used theoretical tools for the study of chaos in them is the separatrix map (SM). On the other hand, it is well known that time-periodically perturbed systems may possess nonlinear resonances (NRs), with the corresponding slow dynamics described by the NR Hamiltonian [1-4]. We showed recently [6,7] that nonlinear resonances may be involved in the separatrix chaos, that gives rise to a significant growth of the width of the separatrix layer. This occurs in the range of logarithmically small perturbation frequencies and may be described by means of matching the SM and NR dynamics [6-8]. We have shown [7-8] that, in the context of the maximum width of the single layer, all systems are divided in two types: for the 1st type, the width exceeds the perturbation amplitude by the factor which logarithmically diverges in the asymptotic limit of a weak perturbation while, for another type, the excess factor is just numerical. We have analytically described the major peaks in the dependence of the width on the frequency, and the theory demonstrates a spectacular agreement with simulations [8]. Apart from the width in energy, our method allows to

15

describe boundaries of the layer in the Poincare section and chaotic transport within the layer. It may be hoped to be generalysed for non-resonance frequencies and for various generalizations of the separatrix map [9]. An important application of our method relates to the double-separatrix chaos. We have shown [6] that the global chaos in between adjacent separatrices is significantly facilitated if the perturbation frequency is close to certain frequencies: NRs are very wide in energy and may then connect the separatrix chaotic layers at rather small amplitudes of the perturbation. This effect may have plenty of applications, being relevant to a variety of physical systems and problems, including in particular an electron gas in a magnetic superlattice, a spinning pendulum, cold atoms in an optical lattice, a quantum electron transport in a semiconductor superlattice in magnetic and electric fields, a chaotic mixing and transport in a meandering jet flow, chaotic dynamics in billiards. [1] G.M. Zaslavsky, R.D. Sagdeev, D.A. Usikov and A.A. Chernikov, Weak Chaos and

Quasi-Regular Patterns, Cambridge University Press, Cambridge, 1991. [2] A.J. Lichtenberg and M.A. Liebermann, Regular and

Stochastic Motion, Springer, New York, 1992. [3] G.M. Zaslavsky, Physics of Chaos in Hamiltonian

systems, Imperial Colledge Press, London, 1998. [4] G.M. Zaslavsky, Hamiltonian Chaos and Fractional

Dynamics, Oxford University Press, Oxford, 2005. [5] S.S. Abdullaev, Construction of Mappings for

Hamiltonian Systems and Their Applications (Springer, Berlin, Heidelberg, 2006). [6] S.M. Soskin, R. Mannella, O.M. Yevtushenko, Phys.

Rev. E 77, 036221 (2008).

16

[7] S.M. Soskin, R. Mannella, O.M. Yevtushenko, in Chaos,

Complexity and Transport: Theory and Applications, eds. C. Chandre, X. Leoncini, and G. Zaslavsky (World Scientific, Singapore, 2008), pp. 119-128. [8] S.M. Soskin, R. Mannella, submitted to Phys. Rev. E . [9] G.N. Piftankin, D.V. Treschev, Russian Math. Surveys 62, 219 (2007).

17

Invited Lectures

18

ON A NONLINEAR ELECTRO-MECHANICAL SYSTEM BEHAVIOUR, USING TRANSIENT

TRAJECTORIES: SOME RESULTS AND FUTURE PERSPECTIVES ON BIFURCATIONAL ANALYSIS

Balthazar, Jose Manoel

UNESP, Rio Claro, SP, Brazil – E-mail:

jmbaltha@rc.unesp.br

Introduction: This work aims at a better comprehension of the features of the solution surface of a dynamical system presenting a numerical procedure based on transient trajectories. For a given set of initial conditions an analysis is made, similar to that of a return map, looking for the new configuration of this set in the first Poincaré section. The mentioned set of I.C. will result in a curve that can be fitted by a polynomial, i.e., an analytical expression that will be called initial function in the undamped case and transient function in the damped situation.Thus, it is possible to identify using analytical methods the main stable regions of the phase portrait without a long computational time, making easier a global comprehension of the nonlinear dynamics and the corresponding stability analysis of its fundamental resonance, providing a better visualization of the structure of its phase portrait. The application chosen to present this methodology is a mechanical pendulum driven ! through a crankshaft that moves horizontally its suspension point. Goals: Thus, it is possible to identify using analytical methods the main stable regions of the phase portrait without a long computational time, making easier a global comprehension of the nonlinear dynamics and the corresponding stability analysis of its fundamental resonance, providing a better visualization of the structure

19

of its phase portrait. The application chosen to present this methodology is a mechanical pendulum driven through a crankshaft that moves horizontally its suspension point. main Results: Thus, it is possible to identify using analytical methods the main stable regions of the phase portrait without a long computational time, making easier a global comprehension of the nonlinear dynamics and the corresponding stability analysis of its fundamental resonance, providing a better visualization of the structure of its phase portrait. The application chosen to present this methodology is a mechanical pendulum driven through a crankshaft that moves horizontally its suspension point. The bifurcational analysis of the crank-pendulum shows that these functions can reflect all nonlinear characteristics near its fundamental and 2nd secondary resonances (regions foreseen by the analysis with Floquet multipliers). In view of difficulties in a nonlinear analysis of a dynamical system, the initial functions provide an efficient and easy way of approaching the calculation of the solutions preserving the original characteristics of the differential equations without simplific! ations. In view of difficulties in a nonlinear analysis of a dynamical system, the initial functions provide an efficient and easy way of approaching the calculation of the solutions preserving the original characteristics of the differential equations without simplifications.

20

INTERACTION OF A BOUNCING BALL WITH A SINUSOIDALLY VIBRATING TABLE

Barroso, Joaquim J.

National Institute for Space Research (INPE) – E-mail:

barroso@plasma.inpe.br

Exploring all its ramifications, this presentation gives an overview of the simple yet fundamental bouncing ball problem, which consists of a ball bouncing vertically on a sinusoidally vibrating table under the action of gravity. The dynamics is modeled on the basis of a discrete map of difference equations, which numerically solved fully reveals a rich variety of nonlinear behaviors, encompassing irregular non-periodic orbits, subharmonic and chaotic motions, chattering mechanisms, and also unbounded non-periodic orbits. For periodic motions, the corresponding conditions for stability and bifurcation are determined from analytical considerations of a reduced map. Through numerical examples, it is shown that a slight change in the initial conditions makes the ball motion switch from periodic to chaotic orbits bounded by a velocity strip v =+-A/(1-e�), A where is the non-dimensionalized shaking acceleration and 'e' the coefficient of restitution which quantifies the amount of energy lost in the ball-table collision. Moreover, a detailed numerical discussion of the excitation of the unstable 1-periodic mode and the ensuing transition to its stable counterpart mode is also given.

21

FERMI ACCELERATION IN NON-AUTONOMOUS BILLIARDS

Gelfreich,Vassili

UK

We study Hamiltonian systems which depend slowly on time. We show that if the corresponding frozen system has a uniformly hyperbolic invariant set with chaotic behaviour, then the full system has orbits with unbounded energy growth (under very mild genericity assumptions). We also estimate the rate of the fastest energy growth. In this talk I will show how the general theory is applied to the study of a non-autonomous billiard.

22

SURVIVAL PROBABILITY FOR THE STADIUM BILLIARD

Georgiou, Orestis

University of Bristol - E-mail: maxog@bristol.ac.uk

We consider the open stadium billiard, consisting of two semicircles joined by parallel straight sides with one hole situated somewhere on one of the sides. Due to the hyperbolic nature of the stadium billiard, the initial decay of trajectories, due to loss through the hole, appears exponential. However, some trajectories (bouncing ball orbits) persist and survive for long times and therefore form the main contribution to the survival probability function at long times. Using both numerical and analytical methods, we concur with previous studies that the long-time survival probability for a reasonably small hole drops like Constant/time; here we obtain an explicit and exact expression for the Constant.

23

PARTIAL COMPLEXITIES FOR POLYGONAL AND RELATED BILLIARD TABLES

Gutkin, Eugene

Poland

We report on new techniques and recent results in this subject. Our results provide further evidence for the "polynomial complexity conjecture".

24

TWO-FREQUENCY BILLIARDS

Howard, James E.

Center for Integrated Plasma Studies, Laboratory for

Atmospheric and Space Physics, University of Colorado

Boulder, CO 80309 USA

By using a superposition of two wave forms, local chaoticity can be enhanced, destroying local invariant circles. We apply the results to the 2D Fermi map and derive a phase space island interspersal condition for maximal chaos (Howard, Lichtenberg, and Lieberman, 1983). We shall discuss ways to enhance chaoticity in the circular and stadium billiards.

25

FERMI ACCELERATION IN SIMPLE RANDOMIZED DRIVEN BILLIARDS

Karlis, Alexandros K

Department of Physics, University of Athens, GR-15771

Athens, Greece, Physikalisches Institut, Universität

Heidelberg, Philosophenweg 12, 69120 Heidelberg,

Germany

In 1949 Fermi proposed an ingenious acceleration mechanism of cosmic-ray particles, suggesting that particles would gain energy through scattering off moving magnetic inhomogeneities. Later, a conceptually simple yet intricate in its behavior mechanical model was proposed by Fermi and Ulam, in order to test the feasibility of Fermi acceleration, which comprises one fixed and one oscillating hard infinitely heavy wall and an ensemble of non-interacting particles bouncing between them. The equations defining the dynamics of the Fermi-Ulam model (FUM) are of implicit form with respect to the collision time, which complicates numerical simulations and hinders an analytical treatment. A simplification --known as the static approximation (SA)-- which treats the oscillating wall as fixed in space, yet transfer of momentum is allowed as if the wall were oscillating, has become over the time the standard approximation for studying the FUM. The SA considerably speeds-up numerical simulations and facilitates the analytical treatment of the problem, while it has been generalized to higher-dimensional billiards with time-dependent boundaries, such as the time-dependent Lorentz Gas [2], which billiard acts a paradigm allowing us to address fundamental issues of statistical mechanics such as ergodicity and mixing, as well as transport processes.

26

However, the application of the SA in the aforementioned time-dependent billiards suffers from two drawbacks. The first is that it leads to a considerable underestimation of the particle acceleration. It has been shown that the underestimation is caused by small additional fluctuations of the time of free flight due to the displacement of the oscillating wall occurring in the exact model, which are neglected within the SA [1,2]. The second has to do with the possibility of multiple consecutive collisions between the oscillating wall and the particles that the SA also does not take into account. For this reason, if a particle continues moving towards the scatterer after a collision, one has to reverse artificially the normal component of its velocity, in order to prohibit the particle from passing through the hard scatterer. However, this class of rare events is routinely neglected in the analytical treatment of the particle acceleration. This further simplification within the SA gives rise to a fundamental inconsistency in the case of the FUM: the ensemble mean of the absolute velocity obtained analytically does not change through collisions with the ``moving'' wall, despite the well-established numerical result that Fermi acceleration does take place in the phase-randomized FUM [3]. The aim of this work is twofold. Utilizing the hopping approximation introduced in Ref. [2] the transport coefficients of the Fokker-Planck equation, describing the diffusion in velocity space in the phase-randomized FUM and Lorentz gas, are calculated and the asymptotic time-evolving probability density function of particle velocities is derived. Furthermore, we propose a general scheme for treating the diffusion process in velocity space occurring in the phase-randomized FUM, providing a consistent description of the acceleration process within the framework of the SA. This is achieved by taking into account the rare

27

set of collision events characterized by an artificial reversal of particle velocities upon collision. The latter exemplifies the influence of low-probability events on the transport properties of time-dependent billiards. References: [1] Karlis A K 2006, Papachristou, P K, Diakonos F K, Constantoudis V and Schmelcher P, Phys. Rev. Lett., 97 194102. [2] Karlis A K 2007, Papachristou, P K, Diakonos F K, Constantoudis V and Schmelcher P, Phys. Rev. E, 76 016214. [3] Karlis A K 2008, Diakonos F K, Constantoudis V and Schmelcher P, Phs. Rev. E, 78 046213.

28

BROWNIAN MOTORS AND THE LONG ARM OF THE (SECOND) LAW

Lagos, Roberto Eugenio

Depto. de Física, IGCE, UNESP, Rio Claro, SP, Brazil -

Email: monaco@rc.unesp.br

Brownian Motors and the Long Arm of the (Second) Law We review the Brownian motion paradigm and present a brief survey on Brownian motors. Next, we introduce our approach to the study of charged Brownian particles. Our starting point is the celebrated Kramers equation and its asymptotic limit known as the Smoluchowski equation. Our novel approach allows for the inclusion of external magnetic and electric fields, chemical reactions and an inhomogeneous temperature profile for the thermal reservoir. Entropy and temperature are defined in a natural fashion for a general nonequilibrium regimenwithin the Brownian scheme. An hydrothermodynamic picture of charged Brownian particles follows. We present several applications, in particular hot carriers in semiconductors. In the asymptotic regime, stationary solutions are obtained and linear relations relating “forces” and “fluxes” are obtained. This Onsager like relations, nevertheless contain coefficients that are state dependent. So, among several interesting consequences, within th! e Brownian motion scheme, the approach to stationary (asymptotic) states are not in general described by a variational principle, such as the “minimum (or maximum) entropy production principle”, except of course for the terminal thermodynamical equilibrium regime in the absence of external forces.

29

DYNAMICAL PROPERTIES FOR ONE-DIMENSIONAL FERMI ACCELERATOR MODELS:

A SHORT REVIEW

Leonel, Edson Denis

DEMAC – UNESP – Rio Claro – SP. – Brazil – E-mail:

edleonel@rc.unesp.br

The main goal of this seminar is to present and discuss some results for one-dimensional accelerator models like Fermi-Ulam and Bouncer models. We consider both the conservative and dissipative cases and different external perturbations for the boundary. For the conservative case we characterise some scaling properties of chaotic seas obtaining critical exponents for the phase transition: integrability to no integrability. For the dissipative case we show the occurrence of boundary crisis and in particular a phase transition from limited to unlimited energy growth is characterised in the bouncer model.

30

VIBRATED GRANULAR BEDS: PARTICLE APPROACH SIMULATION

Navarro, Helio Aparecido

Departamento de Engenharia Mecânica, EESC, USP –

E-mail: han@sc.usp.br

A granular material consists of discrete, solid particles dispersed in a vacuum or an interstitial fluid. Granular materials are often encountered in both natural and industrial processes, so the analysis of how they behave provides information for optimization design. Discrete element model is a technique to simulate granular flows by tracking the motion of individual particles. In this approach the particles are treated individually by accounting for all the forces on each particle and the resulting momentum changes when they move and collide. Two types of models can be considered. In the rapid granular regime, where collisional interactions dominate, the hard sphere model is used. Here, particle volume fractions are low enough that binary collisions may be assumed; collisions may also be modeled as instantaneous. In contrast, dense granular flows are modeled using the soft sphere model. Here, a particle is assumed in enduring contact with several particles over extend! ed periods of time. The interaction force is typically modeled using a spring-mass-dashpot type of model for normal and tangential forces. In this work a theoretical analysis is presented. Some numerical results of vibrated granular beds are also described. For this purpose is used the Multiphase Flow with Interphase eXchanges (MFIX) code developed at National Energy Technology Laboratory (NETL, U.S. Department of Energy).

31

STATISTICAL PHYSICS OF MANY PARTICLE SYSTEMS

Norman, Genri E.

Joint Institute for High Temperatures of Russian Academy of

Sciences 13, bld. 2, Izhorskaya St., Moscow 125412, Russia

Dramatic changes of the conventional paradigm of statistical physics (cf. Landau&Lifshits course) emerged from the development of the molecular dynamics method (MMD) are considered. MMD becomes an important tool in condensed matter physics. It is based on the numerical solution of the Newtonian equations for systems of many particles, interacting with each other. Characteristics and properties of relaxation processes or equilibrium systems are extracted from the analysis of the particle trajectories calculated. MMD theory is developed: Lyapunov exponential trajectory divergence, dynamic memory time (predictability horizon), small but finite fluctuation of the total energy (break of the energy conservation law in MMD), stochastic and dynamic properties of MMD systems, irreversibility emergence. The correspondence between MMD and real systems is elucidated. Probabilistic nature of the classical statistical physics is demonstrated. Based on the theory developed requirement standards are formulated for MMD modeling of both relaxation processes and equilibrium systems (particle number choice, initial and boundary conditions, diagnostics etc). All the key items are illustrated with the examples of classical and quantum MMD applications: billiards, phase equilibrium and stability borders, and fluctuations in plasmas.

32

SOFT WALL EFFECTS ON INTERACTING PARTICLES IN BILLIARDS

Oliveira, H. A., Manchein, C. and Beims, M. W.

The effect of physically realizable wall potentials (soft walls) on the dynamics of two interacting particles in a one-dimensional (1D) billiard is examined numerically. The 1D walls are modelled by the error function and the transition from hard to soft walls can be analyzed continuously by varying the softness parameter $\sigma$. For $\sigma\rightarrow0$ the 1D hard wall limit is obtained and the corresponding wall force on the particles is the $\delta$-function. In this limit the interacting particle dynamics agrees with previous results obtained for the 1D hard walls. We show that the two interacting particles in the 1D soft walls model is equivalent to one particle inside a {\it soft} right triangular billiard. Very small values of $\sigma$ substantiously change the dynamics inside the billiard and the mean finite-time Lyapunov exponent decreases significantly as the consequence of regular islands which appear due to the {\it low-energy double collisions} (simultaneous particle-particle-1D wall collisions). The rise of regular islands and sticky trajectories Induced by the 1D wall softness is quantified by the number of occurrences of the most probable finite-time Lyapunov exponent. On the other hand, chaotic motion in the system appears due to the {\it high-energy double collisions}. In general we observe that the mean finite-time Lyapunov exponent decreases when $\sigma$ increases, but the number of occurrences of the most probable finite-time Lyapunov exponent increases, meaning that the phase-space dynamics tends to be more ergodic-like. Our results suggest that the transport efficiency of interacting particles and heat

33

conduction in periodic structures modelled by billiards, will strongly be affected by the smoothness of physically realizable walls.

34

RANDOM BILLIARDS WITH COSINE REFLECTION LAW

Popov, Serguei

USP – E-mail: serguei.popov@gmail.com

We study random billiards in general domains: a particle moves according to its constant velocity inside some domain $D$ until it hits the boundary and bounces randomly inside according to the cosine reflection law. We assume that the boundary of the domain is locally Lipschitz and almost everywhere continuously differentiable. The angle of the outgoing velocity with the inner normal vector has a specified, absolutely continuous density. We construct the discrete time and the continuous time processes recording the sequence of hitting points on the boundary and the pair location/velocity. First, we focus on the case of bounded domains. Then, we prove exponential ergodicity of these two Markov processes, we study their invariant distribution and their normal (Gaussian) fluctuations. Of particular interest is the case of the cosine reflection law: the stationary distributions for the two processes are uniform in this case, the discrete time chain is reversible though the continuous time process is quasi-reversible. Then, we consider Knudsen stochastic billiard in a random tube which stretches to infinity in the direction of the first coordinate. This random tube is stationary and ergodic, and also it is supposed to be in some sense well-behaved, containing no long dead ends or too thin bottlenecks, and having an a.e.\ continuously differentiable boundary satisfying a Lipschitz condition. Under the condition of existence of the second moment of the projected jump length with respect to the stationary measure for the environment seen from the

35

particle, we prove the quenched invariance principles for the projected trajectories of the random walk and the stochastic billiard. Using the invariant principle, we prove the equivalence of transport diffusion and self diffusion coefficients.

36

LARGE DEVIATIONS IN HYPERBOLIC BILLIARDS

Rey-Bellet, Luc

USA It is a classical result from Bunimovich, Sinai, and Chernov that the sum of the displacements between successive collisions in the periodic Lorentz gas satisfy a central limit theorem. We study the fluctuations of order 1 for such averages and prove that they satisfy a large deviation principle. We will also discuss some applications to nonequilibrium statistical mechanics (thermostatted Lorentz gas). This is joint work with Lai-Sang Young (NYU-Courant Institute).

37

PRESENCE AND LACK OF FERMI ACCELERATION IN NONINTEGRABLE BILLIARDS

Silva, Jafferson Kamphorst Leal da

Departamento de Física - UFMG

The unlimited energy growth (Fermi acceleration) of a classical particle moving in a billiard with a parameter-dependent boundary oscillating in time is numerically studied. The shape of the boundary is controlled by a parameter and the billiard can change from a focusing one to a billiard with dispersing pieces of the boundary. The complete and a simplified versions of the model are considered in the investigation of the conjecture that Fermi acceleration will appear in the time-dependent case when the dynamics is chaotic for the static boundary. Although this conjecture holds for the simplified version, we have not found evidences of Fermi acceleration for the complete model with breathing boundary. When the breathing symmetry is broken, Fermi acceleration appears in the complete model.

38

SCALING PROPERTIES OF A SINUSOIDALLY CORRUGATED WAVEGUIDE

Silva, Mario Roberto da

Universidade Estadual Paulista – E-mail:

marsilva@rc.unesp.br

Scaling properties of a sinusoidally-Corrugated Waveguide Silva, M. R.; Penalva, J,; Leonel, E. D. We study some properties of a light-ray trajectory confined in and specularly reflected by an infinity horizontal flat mirror and an infinity sinusoidally-corrugated mirror. located at unitary mean distance. Our approach consider that the direction change of the reflected ray along the corrugate mirror can be separated in horizontal and vertical components. Indeed, four combinations of possible inversions of these two components are possible and also multiple reflections of the light-ray can be observed before the ray leaves the corrugated zone. The system is described in terms of a two dimensional non-linear mapping for the discrete variables angle of the reflected ray measured counterclockwise with respect to the positive horizontal axis, and the corresponding value of the horizontal coordinate at the instant of reflection in the corrugated mirror. It is shown that the phase space is mixed exhibiting Kolmogorov-Arnold-Moser (KAM) islands, invariant spanning curves separating different portions of the phase space and chaotic seas. The chaotic sea below the first invariant spanning curve is characterized using scaling arguments revealing critical exponents connected by an analytic relationship. Our results confirm this system belongs to the same class of universality of the one-dimensional Fermi-Ulam accelerator model.

39

NONERGODIC BEHAVIOR AND TIME AVERAGES IN CONTINUOUS-TIME RANDOM WALKS

Sokolov, I.M.

Institut fuer Physikalische Humboldt-Universitaet zu Berlin

Newtonstrasse 15 D-12489 Berlin

Germany

Subdiffusion as described by continuous time random walks (CTRW) often arises in quite different physical systems and can be easily obtained in chaotic dynamics as modeled by maps and billiards. The essentially nonstationary character of motion in many variants of CTRW systems leads to considerable peculiarity of their behavior connected with weak ergodicity breaking. We investigate CTRW with an asymptotic power law distribution of waiting times lacking the characteristic mean under moving time average. We show, that contrary to what is expected, the temporal averaged mean squared displacement leads to a simple diffusive behavior at times smaller than the overall data acquisition time T. Corresponding diffusion coefficients that strongly differ from one trajectory to another. This distribution of diffusion coefficients renders a system inhomogeneous: an ensemble of simple diffusers with different diffusion coefficients. The information about the underlying anomaly can be restored by an additional ensemble average over these diffusion coefficients. References A. Lubelski, I.M. Sokolov and J. Klafter, Phys. Rev. Lett. 100, 250602 (2008)

40

NOISE-INDUCED BASIN HOPPING IN AN IMPACT OSCILLATOR

Souza, Silvio Thomaz de

We investigate some dynamical effects of parametric noise of an impact oscillator, consisting of an oscillating box containing a ball undergoing inelastic collisions with its walls. Since the noiseless dynamics is found to exhibit a complex attraction basin structure in phase space, the addition of small amount of noise provokes basin hopping, i.e., the intermittent switching between basins of different attractors.

41

TRANSPORT MAPS ON GRIDS OF UNIT CELLS AND THEIR RELATION TO BILLIARDS

Stoop, Ruedi

The transport on grids of unit cells will be reviewed from the billiard point of view. Conditions for anomalous transport are worked out and compared to results obtained from billiards. Information spread in composed, locally active, subsystems is considered and its appropriateness as a model of information transfer in the brain is investigated.

42

ASYMPTOTIC BEHAVIOUR OF RANDOMLY REFLECTING BILLIARDS IN UNBOUNDED

TUBULAR DOMAINS

Vachkovskaia, Marina

UNICAMP - marinav@ime.unicamp.br

We study stochastic billiards in infinite planar domains with curvilinear boundaries: that is, piecewise deterministic motion with randomness introduced via random reflections at the domain boundary. Physical motivation for the process originates with ideal gas models in the Knudsen regime, with particles reflecting off microscopically rough surfaces. We classify the process into recurrent and transient cases. We also give almost-sure results on the long-term behaviour of the location of the particle, including a super-diffusive rate of escape in the transient case. A key step in obtaining our results is to relate our process to an instance of a one-dimensional stochastic process with asymptotically zero drift. This is a joint work with M. Menshikov and A. Wade.

43

ASYMPTOTIC ALGEBRAIC LAWS IN HAMILTONIAN SYSTEMS

Venegeroles, Roberto

Universidade Federal do ABC – E-mail:

roberto.venegeroles@ufabc.edu.br

Hamiltonian mixed systems with unbounded phase space are typically characterized by two asymptotic algebraic laws, namely, decay of recurrence time statistics and superdiffusion. In this talk I will discuss some recent analytical results that point towards the universality of their characteristic exponents. A number of simulations and experiments by other authors that support this conjecture and perspectives for billiard systems also will be discussed.

44

FLUID FLOW ACROSS THE BILLIARD: EXAMPLE OF LAMINAR DISORDER

Zaks, Michael

Institute of Physics, Humboldt University of Berlin, D-

12489 Berlin Germany

I consider a plane steady flow of a viscous incompressible fluid past the regular array of solid obstacles. In contrast to elastic collisions of the conventional billiards, the no-slip boundary conditions demand that the fluid velocity vanishes on the border of each obstacle. Motion of a tracer along the streamline is governed by an integrable dynamical system in which there is no room for chaos. Nevertheless, for a generic geometry of the obstacle array, the Fourier spectrum of tracer velocity is neither discrete nor absolutely continuous, and the autocorrelation decays algebraically. Affinities and differences to conventional billiards are discussed.

45

Posters

46

SCALING ANALISYS OF THE HYBRID FERMI-ULAM-BOUNCER MODEL

Bizão, Rafael Amatte

UNESP - Rio Claro – E-mail: bizao@rc.unesp.br

Some dynamical properties for a one-dimensional hybrid Fermi-Ulam-bouncer model are studied under the framework of scaling description. The model is described by using a two dimensional nonlinear area preserving mapping. Our results show that the chaotic regime below the lowest energy invariant spanning curve is scaling invariant and the obtained critical exponents are used to find an universal plot for the second momenta of the average velocity.

47

SCALING INVESTIGATION OF FERMI ACCELERATION ON A DISSIPATIVE BOUNCER

MODEL

Livorati, André Luís Prando

Universidade Estadual Paulista - Campus Rio Claro –

E-mail: livorati@rc.unesp.br

The phenomenon of Fermi acceleration is addressed for the problem of a classical and dissipative bouncer model using scaling description. The dynamics of the model, in both their complete and simplified versions, is obtained by use of a two-dimensional nonlinear mapping. The dissipation is introduced using a restitution coefficient on the periodically moving wall. Using scaling arguments, we describe the behavior of average chaotic velocities on the model both as function of the number of collisions with the moving wall and as function of the time. We consider variations on the two control parameters, therefore critical exponents are obtained. We show that the formalism can be used to describe the occurrence of a transition from limited to unlimited energy growth as the restitution coefficient approaches the unity. The formalism can be used to characterize the same transition in two dimensional time-varying billiard problems.

48

FERMI ACCELERATION IN A TIME DEPENDENT OVAL BILLIARD.

Oliveira, Diego Fregolente Mendes de

Departamento de Estatística, Matemática Aplicada e

Computação - Universidade Estadual Paulista - Rio Claro –

SP. - Brazil – E-mail: dfmo@rc.unesp.br

A billiard is defined by a connected region Q � �D with

the boundary ∂Q � �D−1 separating Q from its complement. A point-like particle moves freely inside the billiard along geodesic lines until hits the boundary. In this work we revisit the problem of a classical particle bouncing elastically inside a periodically time varying Oval billiard. The problem is described using a four dimensional mapping for the variables velocity of the particle namelly: the time immediately after a collision with the moving boundary; the angle that the trajectory of the particle does with the tangent at the position of the hit; and the angular position of the particle along the boundary. Our main goal is to understand and describe the behaviour of the particle’s average velocity (and hence its energy) as a function of the number of collisions with the boundary. It was recently shown for a time dependent oval billiard that, in certain cases under the breathin! g perturbation, the particle does not exhibit unlimited energy growth. As we shall shown in our work, the breathing geometry can indeed lead the particle to experience Fermi acceleration. However, the slope of growth is rather smaller as compared to the non breathing case. The small growing exponent for the average velocity was the main reason to conclude that Fermi acceleration was not observed in the breathing case. Our results reinforce the LRA conjecture.

49

ON THE DYNAMICAL PROPERTIES OF AN OVAL BILLIARD.

Oliveira, Diego Fregolente Mendes de

Departamento de Estatística, Matemática Aplicada e

Computação - Universidade Estadual Paulista, Rio Claro –

SP. - Brazil – E-mail: dfmo@rc.unesp.br

The interest in understanding the dynamics of billiard problems becomes in earlies 1927 when Birkhoff introduced a system to describe the motion of a free particle inside a closed region with which the particle suffers elastic collisions. Inside the billiard, a point particle of mass m moves freely along a straight line until it hits the boundary. After the collision, it is assumed that the particle is specularly reflected. In our work we propose a special geometry for the boundary of a classical billiard, which we call as oval boundary. The radius of the boundary in polar coordinates is given by R(θ,p, ε)=1+εcos(pθ). It is important to say that the shape of the boundary is controlled by two relevant control parameters, namely p=integer number and ε=deformation of the boundary. We obtain and discuss some numerical results considering different possibles combination of the control parameters. In our approach, we obtained a map that describe the parti! cle's dinamics and shown there are a critical value for the parameter ε. We shown that the phase space has different structures when ε>εc and ε<εc. Finaly we obtained the positive Lyapunov Exponent reinforcing that the model has a chaotic behaviour.

50

WIGNER DISTRIBUTION FOR A CLASS OF A ISOSPECTRAL POSITION-DEPENDENT MASS

SYSTEMS

Oliveira, Juliano A. de

Universidade Estadual Paulista - UNESP - FEG – DFQ –

E-mail: juliano@feg.unesp.br

Analysis of oscillating quantum systems have been done by many authors in the last years. However, an interesting problem in quantum mechanics refers to systems where the mass of the particle depends on its position. In our studies, we consider the Schoedinger equation to a given exactly solvable potential with constant mass and, then, we construct a generalization for a class isospectral quantum systems, where the mass of the particle depends on its position. As an example, we construct a Wigner distribution of some classes of systems of this nature. We observed that there is an apparent universality in the behavior of the Wigner distribution when one change drastically the dependence of the mass in the spatial variable. Our results are related to the existence of a quantum ordering ambiguity. In the future, we intend to study the behavior of the such functions in the classical limit. This is done for on situations where the deterministic chaos is presented.

51

A BOUNDARY CRISIS IN THE FERMI-ULAM MODEL WITH TWO NONLINEARITIES

Simões, Lucas Eduardo Azevedo

Universidade Estadual Paulista "Júlio de Mesquita Filho" -

UNESP - Rio Claro – E-mail:

lucas89_chuck@yahoo.com.br

We studied in this work the one-dimensional Fermi-Ulam accelerator model with one wall moving according to an external perturbation given by a crank connecting-rod scheme. The particle experiences inelastic collisions with both walls. The model is described by using a two-dimensional nonlinear mapping for the variables v and t corresponding respectively to the velocity of the particle and time immediately after a collision with the moving wall. We assume collisions of the particle with the fixed wall are characterized by a restitution coefficient β Є [0,1], while for the moving wall is α Є [0,1]. In the case of α=β=1, all results of the conservative case are recovered. It thus implies that the phase space of the system exhibits a mixed form containing invariants spanning curves, Kolmogorov-Arnold-Moser (KAM) islands and chaotic seas. For the case of α<1 or β<1 the mixed structure of the phase space is entirely destroyed being replaced by attractors that eventually can be chaotic. We show also that changing the control parameters properly it is possible to characterize a boundary crisis.

52

SUPRESSING FERMI ACCELERATION IN THE BOUNCER

Souza, Francys Andrews de

Universidade Estadual Paulista – UNESP - Rio Claro –

E-mail: francysouz@hotmail.com

The Fermi accelerator model was proposed originally by Enrico Fermi in 1949 as an attempt to describe the cosmic particle acceleration along the interstellar way. A prototype for this model considers the interaction of cosmic particles with time varying magnetic fields. A mathematical model for this original idea consists of a classic particle of mass m confined in and suffering collisions with two walls. One wall is assumed to be fixed while the other one oscillates periodically in the time. Such a model is known as the Fermi-Ulam model (FUM). A rather similar model but in the presence of gravitational field is the so called bouncer model. It consists of a classical particle that collides against a time moving platform, under the presence of gravitational field. One of the most important property of the bouncer model, and contrary to the FUM, is that depending on initial conditions and control parameters, unlimited energy gain of the particle is observed. In this work, We study the bouncer model with the particle experiencing a drag force which is proportional to the particle's velocity and we are seeking for conditions to suppress Fermi acceleration. The dynamics of the model is described in terms of a two dimensional non-linear mapping for the variables velocity of the particle and time immediately after the collision with the moving wall. Our results confirm the introduction of damping force is sufficient to break down the Fermi acceleration.

53

OFF-EQUILIBRIUM DYNAMICS OF SPINS WITH ANTIFERROMAGNETIC INTERACTIONS ON

COMPLEX NETWORKS

Stein-Barana, A.C.M. and Yoshida, M.

UNESP - Universidade Estadual Paulista, Departamento de

Física, Instituto de Geociências e Ciências Exatas de Rio

Claro, Brazil – E-mail: alzirasb@rc.unesp.br

We investigate a antiferromagnetically coupled spin system in the one-dimensional lattice considering short and long range interactions of the spins. Each spin interacts with the nearest neighborhood one and also with another randomly picked spins of the lattice. This network of spin interactions resembles a complex network. The system shows disorder and frustration and we expect a spin-glass behavior. We consider the antiferromagnetic q-states Potts model to describe this system and the off-equilibrium dynamics was simulated by Monte Carlo method with heat-bath algorithm. We compute the two times spin-spin correlation functions as a function of the temperature. The analysis of the correlation functions in the short and long-time regimes allows us to identify the spin-glass behavior of the system.

54

THE FERMI ACCELERATOR MODEL WITH A DISSIPATION FORCE PROPORTIONAL TO A

POWER LAW ON THE VELOCITY

Tavares, Danila Fernandes

Universidade Federal do Ceará – E-mail:

dftavares@fisica.ufc.br

In this work, we study a dissipative version of the one dimensional Fermi accelerator model. The model consists of a classic particle of mass m which is confined between two rigid walls. One wall is assumed to be fixed while the other one moves periodically in time. We consider that the particle suffers elastic collisions with both walls. The dissipation force is introduced via a drag force, which we assume to be proportional to the particle’s velocity elevated to the generic exponent. The dynamics of the model is described in terms of a two dimensional non linear mapping, for the variables velocity of the particle and time immediately after a collision with the moving wall. Moreover, the mapping is obtained via the solution of the Newton’s law of motion. The results to the problem proportional to the particle's velocity and proportional to the square particle's velocity are known. In particular, the objective this work is verify if the results known can be generalized to the particle's velocity elevated to the any exponent.