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Volume 72, Number 11 PHYSICAL REVIEW LETTERS 14 March 1994

Renormalization Scheme for Self-Organized Criticality in Sandpile Models

L. Pietronero, A. Vespignani, and S. ZapperiDipariimemo di Fisica. Universiia di Roma "La Sapienza" Piazzale A. Mow 2. 00185 Roma, hah

(Received I October 1993)We introduce a renormalization scheme of novel type that allows us to characterize the critical state

and the scale invariant dynamics in sandpile models. The attractive fixed point clarifies the nature ofself-organization in these systems. Universality classes can be identified and the critical exponents canbe computed analytically. We obtain r = 1.253 for the avalanche exponent and - -1.234 for the dynamical exponent. These results are in good agreement with computer simulations. The method can be naturally extended to other problems with nonequilibrium stationary states.

PACS numbers: 64.60.Ak. 02.50.-r. 05.40.+J

Sandpile models, introduced by Bak and co-workers[l], represent an interesting case of dynamically drivensystems that evolve spontaneously in a stationary criticalstate. This is an example of self-organized criticality(SOC) in which a system with many interacting degreesof freedom and with short range couplings self-stabilizesin a marginally stable state in which a disturbance canlead to avalanches of all sizes. The distribution ofavalanches is characterized by a power law P(s)—s ~r inwhich .<r is the number of sites involved in a relaxationprocess. A second exponent (r) characterizes the dynamics by linking time and linear extension in a singleavalanche: t—l:. With respect to usual statisticalmechanics problems these systems are characterized by anonlinear dynamics and by the fact that the critical stateis reached spontaneously without the fine tuning of anycritical parameter. Fractal growth phenomena [2], likediffusion limited aggregation (DLA), are another example of SOC, in the sense that they lead spontaneously tocomplex fractal patterns. Various authors [3] believe,however, that both DLA and the sandpile models posequestions of a new type for which it would be desirable todefine a common theoretical scheme. The attempts to apply usual renormalization schemes to these models havebeen problematic because self-criticality implies the absence of "relevant" critical parameters from which usualscaling can be defined. For DLA-like problems we haveintroduced a new theoretical framework, the fixed scaletransformation (FST) [4], that is based on two elements:One is the stability with respect to the dynamical evolution and the second is the identification of the scale invariant dynamics. This second element correspondsessentially to a renormalization scheme in which, however, the critical exponents are related directly to the fixedpoint properties of the dynamics, instead of their derivative with respect to the relevant critical parameter, that inthese SOC problems does not exist.

In this Letter we follow the same reasoning for thesandpile model and develop a renormalization scheme forits dynamics. This permits us to clarify the SOC natureof the process, to identify the universality classes, and tocompute analytically the critical exponents r and r.

There are several sandpile models; the first class is the

one introduced by Bak and co-workers El] in which theinstability is defined by the critical height. A second classof models is defined with respect to the critical slope andit appears to belong to a different universality class [5,6].For these models there have been extensive studies bycomputer simulations [5-9] and some exact results havebeen derived by Dhar and co-workers [10] with grouptheoretical arguments. These models have also been generalized to the case in which the local conservation lawsare violated to simulate dissipation [l 1.12]. This leads toa relevant modification of the SOC properties. In generalthese results are discussed in terms of scaling arguments[1.5,13,14]. SOC properties have also been identified inmodels related to invasion percolation for which a fieldtheory approach has recently been proposed [15].

Our discussion will refer to the critical height models.These models are cellular automaton defined on a lattice.A variable (energy) is assigned to each site; energy isthen added randomly on the system. When the energy ofa site reaches a critical value, its entire energy is releasedto the neighboring sites in various ways. These may become unstable in their turn and so on. The system hasopen boundary conditions that allow the dissipation of energy outside. In full generality we can characterize in allthese models three classes of sites (Fig. I):

(i) White sites are those whose energy (height) is farfrom the threshold value. This implies that the additionofa "quantum" of energy will not induce relaxation.

(ii) Black sites are those whose energy is critical in thesense that the addition of a "quantum" of energy will introduce relaxation.

(iii) Encircled sites are unstable sites that will relax atthe next time step according to the specific rules assigned.An example of this relaxation is shown in the lower partof Fig. 1. In this case the relaxation of the central site induces a redistribution of the energy over two of the neighbors, as in the model of Manna [8], and these becomecritical.

We should now extend the characterization of the static and dynamical properties of the system at a genericscale b, by considering coarse grained variables. A cellof size b relaxes if subrelaxation processes span the celland transfer energv to some neighbors. Independently of

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volume 72, Number

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PHYSICAL REVIEW LETTERS 14 March 1994

The process shown refers to the probability that the unstable subcell relaxes towards the other critical subcell[Fig. 4(b)]. This occurs with probability (1/4)/? f*' wherethe index (k) refers to relaxation processes at scale b/2and (A:+ 1) will refer to relaxation processes at scale b.At this point we consider the probability that the next relaxation event at scale b/2 involves two neighboring sites

of size b/2. one inside and one outside the original cell ofsize b [Fig. 4(c)]. This occurs with probability (2/3)p_*-\This series of relaxations at scale b/2 contributes to theprobability p\k + ]) that characterizes the relaxation processes at scale b [Fig. 4(d)].

By summing over all the processes that lead to p\one obtains for a =2

.(* + !)

+ (ipik)+7pik)Hiplk)+JPik)+7pik)). (3)In a similar way one can also write the expressions forPi(A +1) _ (A + 1 ) (A + I)p\i ', and pr u. Finally, these probabilitiesare normalized for each configuration a. This calculationshould then be repeated also for the configurations a = 3and 4. The configuration a = I is not included in the renormalization of the dynamics because it cannot lead torelaxation processes that span the whole cell of size b.The calculation is rather laborious but straightforwardand it will be reported in detail in a long paper [161.

The renormalization of the relaxation processes alonewould lead to trivial fixed points because the system simply decays. In order to describe the dynamics of the critical slate it is necessary to couple the dynamics to a sta-tionarity condition similar to the one used in Ref. [14] todefine the average energy of the critical state. Thismeans that, given a cell of size b. on average, the energythat goes in the cell should be equal to the energy thatgoes out. If we assume to inject a "quantum" of energy8E(b), the probability that the cell relaxes is pu + 1) and,once this happens, it can occur with the four possible situations we have discussed previously (Fig. 2). This leadsto the equilibrium condition

8E(b)=p{k + ])[5E(b)p\k + i) + 25E(b)p±k + {)+ 38E(b)p{ik + i)+45E(b)p<ik + "] (4)

that couples the dynamical properties to the static onesand provides a feedback mechanism that is an essentialelement for the process of self-organization [17] and forthe definition of the nonlocal properties of the dynamics.

The structure of our renormalization scheme is therefore the following. Given (pik);pik)) one defines thetransformation F„ for the dynamics at the scale (k+\)for each configuration a.

Pn( k + \ ) ( \ _

(a)=F„[{p}k)}] (/.„.' = 1,4) (5)

These probabilities are then averaged over the configurations \a} whose weight W(a) is defined by plk\

pJA + ,)=<pnu + l)(a)> = I W(a)p!k + *Ha). (6)o-I,3

The equilibrium condition Eq. (4) provides then the re-normalized density of critical sites at scale (k + \). independent of the definition of 5E(b). The approximations involved in this scheme are due to the specific im

plementation of the spanning condition and to closure ofthe renormalization equations [16].

Given this scheme we can jiow start from a small scalestate characterized by (p(0);/>(0)) and study how this willevolve under scale transformation. If we start with asmall scale dynajnics characterized by the model of Manna [8] we have Pl0) = (0,1,0,0). Considering a low density of critical sites p(0)=0.1, the evolution under scaletransformation is shown in the upper part of Table I. Inthe lower part we show instead the evolution of the BTW[l] four-state model, starting from a large density p(0)=0.9.

Both models lead to the same asymptotic (/.-—■«>)fixed point dynamics (p*;P*), so they belong to the sameuniversality class. In addition we have checked that allmodels of this type also belong to this universality class.

TABLE I. Renormalization transformation for the static (p)and dynamic {p-l properties for two sandpile models. The indexk refers to a change of scale. The limit k —■ <» identifies the attractive fixed point (p*;P*) that is the same for both models.

Number ofiterations

t t ) p P\ P_ Pi PAManna two-state model

o.:1 0.612 0.436 0.495 0.068 0.0012 0.575 0.405 0.463 0.118 0.0133 0.542 0.362 0.456 0.158 0.0244 0.518 0.324 0.434 0.188 0.033

0.468

0.9

0.240 0.442 0.261 0.057

BTW four-state model0 0 0

1 0.252 0 0 0.033 0.9672 0.308 0 0.012 0.726 0.2623 0.353 0.030 0.261 0.553 0.1524 0.388 0.090 0.357 0.437 0.116cc 0.468 0.240 0.442 0.261 0.057

Volume 72, Number PHYSICAL REVIEW LETTERS 14 March 1994The fixed point is attractive and il allows us therefore tounderstand the self-organized nature of the critical stationary state.

The fixed point parameters (p*:/5*) provide a complete characterization of the static and dynamical behavior of the system at large scale and can be tested by suitable computer simulations. In addition one can deriveanalytically the critical exponents.

The avalanche exponent r can be obtained as follows.The existence of a nontrivial fixed point with respect tothe scale transformation guarantees that the properties oflhe system will be described by power law distributions.Therefore we can assume a power law avalanche distribution and relate the exponent r to the fixed point properties (p iP ) shown in Table I. The first step is lo transform the avalanche distribution in a distribution for thesize r of the clusters. This is related to the total numberof sites 5 by S — rD. It is easy to show with FST methods[18] that sandpile clusters in two dimensions are not fractal but compact and therefore D=2 in agreement withsimulations [1,6,7]. The size distribution is therefore

fr(iT P([)dr=sr 2T)dr- B>' using discrete length scalesb -2 x_>0. we define the parameter K as the probability that an active relaxation process is limited between thescales b{k " and b(k) and it does not extend further.This can be expressed as

K-f9 '*-»PM*/ f~*-aP(r)dr- \ -22(," '>. (7)Note that this is different from the quantity (I -p(k))that includes the ^situation in which no relaxation occurseven at scale b{k x). The parameter K is defined by thecondition that a relaxation event occurs at scale bik) withthe dynamics defined by {pik)} Or —1,4), and that thisprocess should stop and not affect the neighboring cells.Asymptotically (k — co) we can therefore" express K interms of our fixed point parameters in the following way:

A ' = p r ( i - p * ) + p ; * ( i - p * ) 2 + / . * ( , - p * ) 3+ p * ( l - p * ) 4 = 0 . 2 9 6 . ( g )

Using Eqs. (7) and (8) the exponent r is then given byI ln( I -A ' )r = l -

ln2= 1.253. (9)

This value of r is in good agreement with the computersimulations on large systems that give r = l 22 [7] andr = 1.28 [8].

It should be noted that the method we have used tocompute the critical exponent r is analogous to the FSTapproach to fractal growth [19] and it is related directlyto the fixed point parameters. In usual critical phenomena exponents are derived instead from the derivative ofthe RG transformation with respect to the "critical" parameter. In SOC models this is impossible because thefixed point is attractive and the exponents that are relatedto the distance from the critical point do not exist. Along

the same lines it is possible to compute also the dynamical exponent r = 1.234, in good agreement with Refs.16.7.10]. Details of these calculations will be reported inRef. [16].

The introduction of a dissipation parameter y can alsobe considered in our renormalization scheme and thisshould be also renormalized in its turn. The main resultis that dissipation introduces a length scale into the system and destroys the SOC properties [16]. In ourscheme, in fact, the introduction of y turns the fixed pointinto a trivial one. This is in agreement with the simulation on large systems [l l] of the type that we have discussed here. Instead, the dissipative mode) of Ref. [12]seems to belong to a different universality class.

It is a pleasure to thank P. Bak and D. Dhar for interesting discussions.

[2]

[3]

[4]

[5]

[6]17][S](9]

10]

11]

12]

[19]

P. Bak, C. Tang, and K. Wiesenfeld, Phvs. Rev Lett 59381 (1987); Phys. Rev. A 38. 364 (1988); P. Bak and M."Creutz. in Fractals and Disordered Systems, edited by ABunde and S. Havlin (Springer-Verlag. Heidelberg1 9 9 3 ) . V o l . I I . *T. Vicsek. Fractal Growth Phenomena (World Scientific.Singapore. 1992).L. P. Kadanoff. Physica (Amsterdam) 163A, I (1990);see also Phys. Today 44, No. 3, 9 (1991).L. Pietronero. C. Erzan, and C. Evertsz, Phys. Rev. Lett.61. 861 (1988). For a recent review, see A. Erzan, L.Pietronero. and A. Vespignani. "The Fixed Scale Transformation Approach to Fractal Growth" (to be published).L. P. Kadanoff. S. R. Nagel. L. Wu, and S. Zhou, PhysRev. A 39. 6524 (1989).S. S. Manna. Physica (Amsterdam) I79A, 249 (1991).S. S. Manna. J. Stat. Phys. 59. 509 (1990).S. S. Manna. J. Phys. A 24. L363 (1991).P. Grassberger and S. S. Manna, J. Phys. (Paris) 511077 (1990).D. Dhar, Phys. Rev. Lett. 64, 1613 (1990); D. Dhar andR. Ramaswamy, Phys. Rev. Lett. 63, 1659 (1989); D.Dhar and S. N. Majumdar, J. Phys. A 23, 4333 (1990).S. S. Manna, L. B. Kiss, and J. Kertesz, J. Stat. Phys 61 "923 (1990).Z. Olami, H. J. S. Feder, and K. Christensen, Phys RevLett. 68, 1244 (1992); K. Christensen and Z Olami.Phys. Rev. A 46, 1829 (1992).Y. C. Zhang, Phys. Rev. Lett. 63, 470 (1989).L. Pietronero, P. Tartaglia, and Y. C. Zhang, Physica(Amsterdam) 173A, 22 (1991).P. Bak and K. Sneppen, Report No. BNL 49030 (to bepublished); M. Paczuski, S. Maslov, and P. Bak (to bepublished).A. Vespignani, S. Zapped, and L. Pietronero (to be published).D. Somette, J. Phys. I (France) 2, 2065 (1992).L. Pietronero and W. R. Schneider, Phvs. Rev Lett 662336 (1991).R. Cafiero. L. Pietronero, and A. Vespignani, Phys RevLett. 70, 3939 (1993).

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