Brazilian Journal of Physics, vol. 30, no. 1, Marco, 2000 27 Paths to Self-Organized Criticality 1 2 3 4 Ronald Dickman , Miguel A. Munoz ~ , Alessandro Vespignani , and Stefano Zapp eri 1 Departamento de Fsica, ICEx, UniversidadeFederal de Minas Gerais, Caixa Postal 702, 30161-970 Belo Horizonte, MG, Brazil 2 Institute Carlos I for Theoretical and Computational Physics and Departamento de Electromagnetismo y Fsica de la Materia 18071 Granada, Spain. 3 The Abdus Salam International Centre for Theoretical Physics (ICTP) P.O. Box 586, 34100 Trieste, Italy 4 PMMH - Ecole de Physique et Chimie Industriel les, 10, rue Vauquelin, 75231 Paris CEDEX 05, France Received 15 Octob er 1999 We present a p edagogical intro duction to self-organized criticality (SOC), unraveling its connections with nonequilib riu m phase transitions. There are several paths from a conventional critical p oint to SOC. They b egin with an absorbing-state phase transition (directed p ercolation is a familiar example), and imp ose sup ervision or driving on the system; two commonly used metho ds are extremal dynamics, and driving at a rate approaching zero. We illustrate this in sandpiles, where SOC is a consequence of slow driving in a system exhibiting an absorbing-state phase transition with a conserved density. Other paths to SOC, in driven interfaces, the Bak-Snepp en mo del, and self- organized directed p ercolation, are also examined. We review the status of exp erimental realizations of SOC in light of these observations. I Intro duction how the system is maintained (or maintains itself)atthe critical p oint. (Alternatively one can try to showthat The lab el \self-organized" is applied indiscriminately in there is generic scale invariance, that is, that criticality the current literature to ordering or pattern formation app ears over a region of parameter space with nonzero amongst manyinteracting units. Implicit is the notion measure [1,2].) \SOC" has b een used to describ e sp on- that the phenomenon of interest, b e it scale invariance, taneous scale invariance in general; this would seem to co op eration, or supra-molecular organization (e.g., mi- embrace random walks, as well as fractal growth [3], dif- celles), app ears sp ontaneously. That, of course, is just fusive annihilation (A + A ! 0 and related pro cesses), how the magnetization app ears in the Ising mo del; but and nonequilibrium surface dynamics [4]. Here were- we don't sp eak of \self-organized magnetization." Af- strict the term to systems that are attracted to a criti- ter nearly a century of study,we've come to exp ect the cal (scale-invariant) stationary state; the chief examples spins to organize; the zero- eld magnetization b elow are sandpile mo dels [5]. Another class of realizations, T is no longer a surprise. More generally,spontaneous exempli ed by the Bak-Snepp en mo del [6], involve ex- c organization of interacting units is precisely what we tremal dynamics (the unit with the extreme value of seek, to explain the emergence of order in nature. We a certain variable is the next to change). We will see can exp ect many more surprises in the quest to discover that in many examples of SOC, there is a choice b e- what kinds of order a given set of interactions lead to. tween global sup ervision (an o dd state of a airs for a All will b e self-organized, there b eing no outside agent \self-organized" system), or a strictly lo cal dynamics in on hand to imp ose order! which the rate of one or more pro cesses must b e tuned to zero. \Self-organized criticality" (SOC) carries greater sp eci city, b ecause criticality usually do es not happ en The sandpile mo dels intro duced byBak,Tang and sp ontaneously: various parameters havetobetuned Wiesenfeld (BTW) [5], Manna [7], and others have at- to reach the critical p oint. Scale-invariance in natural tracted great interest, as the rst and clearest examples systems, far from equilibrium, isn't explained merely by of self-organized criticality. In these mo dels, grains of showing that the interacting units can exhibit scale in- \sand" are injected into the system and are lost at the variance at a p oint in parameter space; one has to show b oundaries, allowing the system to reach a stationary 28 Ronald Dickman et al. state with a balance b etween input and output. The of the eld are not discussed. input and loss pro cesses are linked in a sp ecial wayto the lo cal dynamics, which consists of activated, conser- II A simple example vative, redistribution of sand. In the limit of in nitely slow input, the system displays a highly uctuating, We b egin with a simple mo del of activatedrandom scale-invariantavalanche-like pattern of activity.One walkers (ARW). Eachsitej of a lattice (with p erio dic may asso ciate rates h and , resp ectively, with the ad- b oundary conditions) harb ors a number z =0; 1; 2::: of dition and removal pro cesses. Wehave to adjust these j random walkers. (For purp oses of illustration the ring parameters to realize SOC: it app ears in the limit of h + 1;:::;L will do.) Initially, N walkers are distributed and ! 0 with h= ! 0[1,8,9, 10]. (The addition randomly amongst the sites. Eachwalker moves inde- and removal pro cesses o ccur in nitely slowly compared p endently, without bias, to one of the neighb oring sites to the lo cal redistribution dynamics, which pro ceeds (i.e., from site j to j +1 or j 1, with site L +1 1 at a rate of unity. Loss is typically restricted to the and 0 L), the only restriction b eing that an isolated b oundaries, so that ! 0 is implicit in the in nite-size walker (at a site with z = 1) is paralyzed until such j limit.) time as another walker or walkers joins it. The active Questions ab out SOC fall into two categories. First, sites (with z 2) followa Markovian (sequential) dy- j Why do es self-organized criticality exist? What are the namics: each active site loses, at a rate 1, a pair of walk- conditions for a mo del to have SOC? Second, the many ers, which jump indep endently to one of the neighbors questions ab out critical b ehavior (exp onents, scaling of site j .(Thus in one dimension there is a probability functions, p ower-sp ectra, etc.) of sp eci c mo dels, and of 1/2 that each neighb or gains one walker, while with whether these can b e group ed into universality classes, probability 1/4 b oth walkers hop to the left, or to the as for conventional phase transitions b oth in and out right.) of equilibrium. Answers to the second typ e of question The mo del wehavejustde nedischaracterized by come from exact solutions [11], simulations [12], renor- d the numb er of lattice sites, L ,andthenumber of par- malization group analyses [13], and (one mayhope) ticles, N .Ithastwo kinds of con gurations: active, in eld theoretical analysis. Despite these insights, asser- which at least one site has two or more walkers, and tions in the literature ab out sp ontaneous or parameter- absorbing, in whichnositeismultiply o ccupied, ren- free criticalityhave tended to obscure the nature of the d dering all the walkers immobile [17]. For N>L only phase transition in sandpiles, fostering the impression active con gurations are p ossible, and since N is con- that SOC is a phenomenon sui generis, inhabiting a dif- d served, activitycontinues forever. For N L there ferentworld than that of standard critical phenomena. are b oth active and absorbing con gurations, the latter In this pap er weshow that SOC is a phase transition to representing a shrinking fraction of con guration space an absorbing state, a kind of criticality that has b een d as the density N=L ! 1. Given that we start well studied, principally in the guise of directed p ercola- in an active con guration (a virtual certaintyforan tion [14]. Connections b etween SOC and an underlying initially random distribution with >0and L large), conventional phase transition have also b een p ointed will the system remain active inde nitely, or will it fall out by Narayan and Middleton [15], and by Sornette, into an absorbing con guration? For small it should Johansen and Dornic [16]. b e easy for the latter to o ccur, but it seems reasonable Starting with a simple example (Sec. I I), wewillsee that for suciently large densities (still < 1), the like- that the absorbing-state transition provides the mech- liho o d of reaching an absorbing con guration b ecomes anism for SOC (Sec. I I I). That is, we explain the exis- so small that the walkers remain active inde nitely.In tence of SOC in sandpiles on the basis of a conventional other words, we exp ect sustained activity for densities critical p oint. In Sec. IV we discuss the transforma- greater than some critical value , with < 1. tion of a conventional phase transition to SOC in the c c contexts of driven interfaces, a sto chastic pro cess that A simple mean- eld theory provides a preliminary repro duces the stationary prop erties of directed p erco- check ofthisintuition. Consider activated random lation, and the Bak-Snepp en mo del. We nd that criti- walkers in one dimension. For a site to gain parti- cality requires tuning, or equivalently, an in nite time- cles, it must have an active(z 2) nearest neighb or.