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We present a pedagogical introduction to self-organized criticality (SOC), unraveling its connections with nonequilibrium phase transitions. There are several paths from a conventional critical point to SOC. They begin with an absorbing-state phase transition (directed percolation is a familiar example), and impose supervision or driving on the system two commonly used methods 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-Sneppen model, and selforganized directed percolation, are also examined. We review the status of experimental realizations of SOC in light of these observations. I IntroductionThe label \self-organized" is applied indiscriminately in the current literature to ordering or pattern formation amongst many i n teracting units. Implicit is the notion that the phenomenon of interest, be it scale invariance, cooperation, or supra-molecular organization (e.g., micelles), appears spontaneously. That, of course, is just how the magnetization appears in the Ising model but we don't speak of \self-organized magnetization." After nearly a century of study, w e've come to expect the spins to organize the zero-eld magnetization below T c is no longer a surprise. More generally, s p o n taneous organization of interacting units is precisely what we seek, to explain the emergence of order in nature. We can expect many more surprises in the quest to discover what kinds of order a given set of interactions lead to. All will be self-organized, there being no outside agent on hand to impose order! \Self-organized criticality" (SOC) carries greater speci city, because criticality usually does not happen spontaneously: various parameters have t o b e t u n e d to reach the critical point. Scale-invariance in natural systems, far from equilibrium, isn't explained merely by showing that the interacting units can exhibit scale invariance at a point in parameter space one has to show how the system is maintained (or maintains itself) a t t h e critical point. (Alternatively one can try to show t h a t there is generic scale invariance, that is, that criticality appears over a region of parameter space with nonzero measure 1, 2 ].) \SOC" has been used to describe spontaneous scale invariance in general this would seem to embrace random walks, as well as fractal growth 3], diffusive annihilation (A + A ! 0 and related processes), and nonequilibrium surface dynamics 4]. Here we r estrict the term to systems that are attracted to a critical (scale-invariant) stationary state the chief examples are sandpile models 5]. Another class of realizations, exempli ed by the Bak-Sneppen model 6], involve extremal dynamics (the unit with the extreme value of a certain variable is the next to change). We will see that in many examples of SOC, there is a choice between global supervision (an odd state of a airs fo...

We present generic scaling laws relating spreading critical exponents and avalanche exponents ͑in the sense of self-organized criticality͒ in general systems with absorbing states. Using these scaling laws we present a collection of the state-of-the-art exponents for directed percolation, dynamical percolation, and other universality classes. This collection of results should help to elucidate the connections of self-organized criticality and systems with absorbing states. In particular, some nonuniversality in avalanche exponents is predicted for systems with many absorbing states. ͓S1063-651X͑99͒06205-4͔PACS number͑s͒: 05.40. Ϫa, 05.65.ϩb, 05.70.Ln Directed percolation ͑DP͒ is broadly recognized as the paradigmatic example of systems exhibiting a transition from an active to an absorbing phase ͓1,2͔. DP critical behavior appears in a vast array of systems, among others chemical reaction-diffusion models of catalysis ͓3͔, the contact process ͓4,1͔, damage spreading transitions ͓5͔, pinning of driven interfaces in random media ͓6͔, roughening transitions in one-dimensional systems ͓7͔, and Reggeon field theory ͓8͔. This universality class has proven very robust with respect to the introduction of microscopic changes, and many apparently different systems share the same critical ''epidemic'' or ''spreading'' ͓9͔ and ''bulk'' exponents ͓1,2͔. Nevertheless, examples of a system exhibiting a transition to an absorbing state outside the DP class have been identified in recent years. Some examples follow.͑1͒ Systems with two symmetric absorbing states or, what is equivalent in many cases, systems in which the parity of the number of particles is conserved ͓10,11͔.͑2͒ Systems with an infinite number of absorbing states, which exhibit nonuniversal spreading exponents ͓12,13͔.͑3͒ Systems in which the dynamics is limited to the interface between active and absorbing regions. These are in the class of the exactly solvable voter model ͓14͔, and compact directed percolation ͑CDP͒ ͓15͔.͑4͒ Some models of epidemics with immunization ͑no reinfection͒ ͓16͔. These belong to the so-called dynamic percolation class; the final set of immune sites at criticality is a percolation cluster.Recently, connections between self-organized criticality ͑SOC͒ and systems with absorbing states have attracted much attention. For example, there has been a debate on whether the extremal Bak-Sneppen model for punctuated evolution ͓17͔ and certain variants are related to DP ͓18͔. It has also been argued that sandpile models ͓19͔ share a number of features with systems having many absorbing states ͓20͔, and certain self-organized forest-fire models are related to dynamical percolation ͓21͔.In self-organized models the so-called avalanche exponents are customarily determined. Surprisingly, in spite of their obvious similarities, the general connections between spreading and avalanche exponents have not, to the best of our knowledge, been given explicitly for general systems with absorbing states. Establishing the general scaling laws relating avalan...

For a large class of processes with an absorbing state, statistical properties of the surviving sample attain time-independent values in the quasi-stationary (QS) regime. We propose a practical simulation method for studying quasi-stationary properties, based on the equation of motion governing the QS distribution. The method is tested in applications to the contact process. At the critical point, our method is about an order of magnitude more efficient than conventional simulation.

We explore the connection between self-organized criticality and phase transitions in models with absorbing states. Sandpile models are found to exhibit criticality only when a pair of relevant parameters -dissipation ⑀ and driving field h -are set to their critical values. The critical values of ⑀ and h are both equal to zero. The first result is due to the absence of saturation ͑no bound on energy͒ in the sandpile model, while the second result is common to other absorbing-state transitions. The original definition of the sandpile model places it at the point (⑀ϭ0,hϭ0 ϩ ): it is critical by definition. We argue power-law avalanche distributions are a general feature of models with infinitely many absorbing configurations, when they are subject to slow driving at the critical point. Our assertions are supported by simulations of the sandpile at ⑀ϭhϭ0 and fixed energy density ͑no drive, periodic boundaries͒, and of the slowly driven pair contact process. We formulate a field theory for the sandpile model, in which the order parameter is coupled to a conserved energy density, which plays the role of an effective creation rate. ͓S1063-651X͑98͒08805-9͔

We use a phenomenological field theory, reflecting the symmetries and conservation laws of sandpiles, to compare the driven dissipative sandpile, widely studied in the context of self-organized criticality, with the corresponding fixed-energy model. The latter displays an absorbing-state phase transition with upper critical dimension d c 4. We show that the driven model exhibits a fundamentally different approach to the critical point, and compute a subset of critical exponents. We present numerical simulations in support of our theoretical predictions. [S0031-9007(98) [3], and the depinning of interfaces in quenched disorder [4]. When driven continuously, such systems may exhibit stick-slip instabilities, or broadly distributed avalanches, commonly associated with self-organized criticality (SOC) [5,6].SOC sandpiles [5] possess an infinite number of absorbing configurations (i.e., from which the system cannot escape), and are placed, by definition, at the critical point in a two-dimensional parameter space [7,8] resembling that of directed percolation (DP) [3] or contact processes [9][10][11].Under an external drive (i.e., input of particles at rate h), the system jumps among absorbing configurations via avalanchelike rearrangements. Close to the absorbingstate phase transition, a slow drive induces avalanches whose size distribution decays as a power law-the hallmark of SOC. What distinguishes the sandpile from other models with absorbing states is a conservation law: avalanche dynamics conserves the number of grains of "sand," and the order parameter is coupled to this conserved field [8].In this Letter we use a phenomenological field theory of sandpiles to show how conservation alters the phase transition. The critical behavior for h°! 0 (the SOC limit) differs from that for h ϵ 0. In particular, when driving and dissipation are absent, the sandpile shows an absorbing-state phase transition (with d c 4). Our approach clarifies the effect of driving on dynamic phase transitions, and resolves several long-standing issues regarding sandpiles, such as the upper critical dimension, the effect of conservation on critical exponents, and universality classes [12][13][14][15][16]. We perform extensive simulations to check our theoretical predictions.In sandpiles [5], each site i of a d-dimensional lattice bears an integer variable z i $ 0, which we call energy.When a site reaches or exceeds a threshold z c it topples: z i°! z i 2 z c , and z j°! z j 1 1 at each of the g nearest neighbors of i. Energy is fed into the system at rate h, and is dissipated at rate e during toppling [7]. At each time step, each site has a probability~h to receive an energy grain; in each toppling, a grain is lost with probability~e. These rules generalize the original Bak, Tang, and Wiesenfeld (BTW) sandpile automaton [5], which is recovered in the limit h°! 0 and e°! 0 [7,8]. While the BTW model restricts dissipation to the boundaries, we focus on (conceptually simpler) bulk dissipation; most conclusions apply to the boundary dissipation...

We study critical spreading dynamics in the two-dimensional contact process (CP) with quenched disorder in the form of random dilution. In the pure model, spreading from a single particle at the critical point λ c is characterized by the critical exponents of directed percolation: in 2+1 dimensions, δ = 0.46, η = 0.214, and z = 1.13. Disorder causes a dramatic change in the critical exponents, to δ ≃ 0.60, η ≃ −0.42, and z ≃ 0.24. These exponents govern spreading following a long crossover period. The usual hyperscaling relation, 4δ + 2η = dz, is violated. Our results support the conjecture by Bramson, Durrett, and Schonmann [Ann. Prob. 19, 960 (1991)], that in two or more dimensions the disordered CP has only a single phase transition.

We study sandpile models as closed systems, with the conserved energy density playing the role of an external parameter. The critical energy density c marks a nonequilibrium phase transition between active and absorbing states. Several fixed-energy sandpiles are studied in extensive simulations of stationary and transient properties, as well as the dynamics of roughening in an interface-height representation. Our primary goal is to identify the universality classes of such models, in hopes of assessing the validity of two recently proposed approaches to sandpiles: a phenomenological continuum Langevin description with absorbing states, and a mapping to driven interface dynamics in random media.