Paths to self-organized criticality

Abstract

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...

On singular probability densities generated by extremal dynamics

“…If we regard the density of active sites, Prob[x ∈ D], as the order parameter ρ, then the BS model and its variants are seen to realize the 'SOC limit' [4,5] ρ → 0 + . Correlations between site variables x i , x j , ..., x n are zero unless one or more of these values falls in D.…”

Self-organized criticality induced by quenched disorder: Experiments on flux avalanches inNbHxfilms

“…Recently however, there have been a number of experimental observations of criticality in both two- [5,6] and three dimensional systems [7][8][9]. However, the critical ingredients to obtain SOC in an experimental system still remain obscure.Recent theoretical advancements have studied the nature of the criticality in SOC and made a link with phase transitions describing how a moving object comes to rest [10,11]. Such absorbing state phase transitions are closely related to the roughening of an elastic membrane in a random medium [12].…”

Conserved Sandpile with A Variable Height Restriction

“…Despite apparent differences, it is important to recall that previous studies have revealed that simulations of large systems (20 000 sites or larger) are needed to obtain reliable values of critical exponents for this class [28,30]. For example, studies using smaller system sizes overestimated the value of the critical exponent β in the conserved Manna sandpile, in both its restricted and unrestricted versions [3,18]. We study a height-restricted fixed-density version of the Oslo sandpile in one dimension.…”

“…Sandpile models are paradigmatic examples of self-organized criticality (SOC) [1,2], a control mechanism that forces a system with an absorbing-state phase transition to its critical point [3][4][5], without explicit tuning of control parameters [6].…”

“…SOC in a slowly driven sandpile corresponds to an absorbing-state phase transition in a model with the same local dynamics, but a fixed number of particles [3,[7][8][9][10][11], so-called conserved sandpiles [10,[12][13][14]. Absorbing-state phase transitions arise in the context of spatial stochastic models, and correspond to a transition between an active, fluctuating phase, and an absorbing one, which allows no escape [24,25,27].…”