Self-organized critical state of sandpile automaton models

Inversion symmetry and exact critical exponents of dissipating waves in the sandpile model

By an inversion symmetry, we show that in the Abelian sandpile model the probability distribution of dissipating waves of topplings that touch the boundary of the system shows a power-law relationship with critical exponent 5/8 and the probability distribution of those dissipating waves that are also last in an avalanche has an exponent of 1. Our extensive numerical simulations not only support these predictions, but also show that inversion symmetry is useful for the analysis of the two-wave probability distributions.

Crossover phenomenon in self-organized critical sandpile models

We consider a stochastic sandpile where the sand grains of unstable sites are randomly distributed to the nearest neighbors. Increasing the value of the threshold condition the stochastic character of the distribution is lost and a crossover to the scaling behavior of a different sandpile model takes place where the sand grains are equally transferred to the nearest neighbors. The crossover behavior is analyzed numerically in detail; especially we consider the exponents which determine the scaling behavior.

Absorbing-state phase transitions in fixed-energy sandpiles

We study sandpile models as closed systems, with the conserved energy density zeta playing the role of an external parameter. The critical energy density zeta(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.

Proof of breaking of self-organized criticality in a nonconservative abelian sandpile model

By computer simulations, it was reported that the Bak-Tang-Wiesenfeld (BTW) model loses self-organized criticality (SOC) when some particles are annihilated in a toppling process in the bulk of system. We give a rigorous proof that the BTW model loses SOC as soon as the annihilation rate becomes positive. To prove this, a nonconservative Abelian sandpile model is defined on a square lattice, which has a parameter alpha (>/=1) representing the degree of breaking of the conservation law. This model is reduced to be the BTW model when alpha=1. By calculating the average number of topplings in an avalanche <T> exactly, it is shown that for any alpha>1, <T><infinity even in the infinite-volume limit. The power-law divergence of <T> with an exponent 1 as alpha-->1 gives a scaling relation 2nu(2-a)=1 for the critical exponents nu and a of the distribution function of T. The 1-1 height correlation C11(r) is also calculated analytically and we show that C11(r) is bounded by an exponential function when alpha>1, although C11(r) approximately r(-2d) was proved by Majumdar and Dhar for the d-dimensional BTW model. A critical exponent nu(11) characterizing the divergence of the correlation length xi as alpha-->1 is defined as xi approximately |alpha-1|(-nu(11)) and our result gives an upper bound nu(11)</=1/2.

Bubbling and large-scale structures in avalanche dynamics

Using a simple lattice model for granular media, we present a scenario of self-organization that we term self-organized structuring where the steady state has several unusual features: (1) large-scale spatial and/or temporal inhomogeneities and (2) the occurrence of a nontrivial peaked distribution of large events which propagate like "bubbles" and have a well-defined frequency of occurrence. We discuss the applicability of such a scenario for other models introduced in the framework of self-organized criticality.

Moment analysis of the probability distribution of different sandpile models

We reconsider the moment analysis of the Bak-Tang-Wiesenfeld and the stochastic sandpile model introduced by Manna [J. Phys. A 24, L363 (1991)] in two and three dimensions. In contrast to recently performed investigations our analysis reveals that the models are characterized by different scaling behavior, i.e., they belong to different universality classes.

Boundary spatiotemporal correlations in a self-organized critical model of punctuated equilibrium

In a semi-infinite geometry, a one-dimensional, M-component model of biological evolution realizes microscopically an inhomogeneous branching process for M-->infinity. This implies a size distribution exponent tau(')=7/4 for avalanches starting at a free, "dissipative" end of the evolutionary chain. A bulklike behavior with tau(')=3/2 is restored by "conservative" boundary conditions. These are such as to strictly fix to its critical, bulk value the average number of species directly involved in an evolutionary avalanche by the mutating species located at the chain end. A two-site correlation function exponent tau(')(R)=4 is also calculated exactly in the "dissipative" case, when one of the points is at the border. Together with accurate numerical determinations of the time recurrence exponent tau(')(first), these results show also that, no matter whether dissipation is present or not, boundary avalanches have the same space and time fractal dimensions as those in the bulk, and their distribution exponents obey the basic scaling laws holding there.

Scaling of waves in the bak-tang-wiesenfeld sandpile model

We study probability distributions of waves of topplings in the Bak-Tang-Wiesenfeld model on hypercubic lattices for dimensions D>/=2. Waves represent relaxation processes which do not contain multiple toppling events. We investigate bulk and boundary waves by means of their correspondence to spanning trees, and by extensive numerical simulations. While the scaling behavior of avalanches is complex and usually not governed by simple scaling laws, we show that the probability distributions for waves display clear power-law asymptotic behavior in perfect agreement with the analytical predictions. Critical exponents are obtained for the distributions of radius, area, and duration of bulk and boundary waves. Relations between them and fractal dimensions of waves are derived. We confirm that the upper critical dimension D(u) of the model is 4, and calculate logarithmic corrections to the scaling behavior of waves in D=4. In addition, we present analytical estimates for bulk avalanches in dimensions D>/=4 and simulation data for avalanches in D</=3. For D=2 they seem not easy to interpret.

Modified renormalization strategy for sandpile models

Following the renormalization-group scheme recently developed by Pietronero et al. [Phys. Rev. Lett. 72, 1690 (1994)] we introduce a simplifying strategy for the renormalization of the relaxation dynamics of sandpile models. In our scheme, five subcells at a generic scale b form the renormalized cell at the next larger scale. Now the fixed point has a unique nonzero dynamical component that allows for a great simplification in the computation of the critical exponent z. The values obtained are in good agreement with both numerical and theoretical results previously reported.

Critical states in a dissipative sandpile model

A directed dissipative sandpile model is studied in two dimensions. Numerical results indicate that the long time steady states of this model are critical when grains are dropped only at the top, or everywhere. The critical behavior is mean-field-like. We discuss the role of infinite avalanches of dissipative models in periodic systems in determining the critical behavior of same models in open systems.

Self-organized criticality in a nutshell

In order to gain insight into the nature of self-organized criticality (SOC), we present a minimal model exhibiting this phenomenon. In this analytically solvable model, the state of the system is fully described by a single-integer variable. The system organizes in its critical state without external tuning. We derive analytically the probability distribution of durations of disturbances propagating through the system. As required by SOC, this distribution is scale invariant and follows a power law over several orders of magnitude. Our solution also reproduces the exponential tail of the distribution due to finite size effects. Moreover, we show that large avalanches are suppressed when stabilizing the system in its critical state. Interestingly, avalanches are affected in a similar way when driving the system away from the critical state. With this model, we have reduced SOC dynamics to a leveling process as described by Ehrenfest's famous flea model.

Dynamical real space renormalization group applied to sandpile models

A general framework for the renormalization group analysis of self-organized critical sandpile models is formulated. The usual real space renormalization scheme for lattice models when applied to nonequilibrium dynamical models must be supplemented by feedback relations coming from the stationarity conditions. On the basis of these ideas the dynamically driven renormalization group is applied to describe the boundary and bulk critical behavior of sandpile models. A detailed description of the branching nature of sandpile avalanches is given in terms of the generating functions of the underlying branching process.

Highly optimized tolerance: a mechanism for power laws in designed systems

We introduce a mechanism for generating power law distributions, referred to as highly optimized tolerance (HOT), which is motivated by biological organisms and advanced engineering technologies. Our focus is on systems which are optimized, either through natural selection or engineering design, to provide robust performance despite uncertain environments. We suggest that power laws in these systems are due to tradeoffs between yield, cost of resources, and tolerance to risks. These tradeoffs lead to highly optimized designs that allow for occasional large events. We investigate the mechanism in the context of percolation and sand pile models in order to emphasize the sharp contrasts between HOT and self-organized criticality (SOC), which has been widely suggested as the origin for power laws in complex systems. Like SOC, HOT produces power laws. However, compared to SOC, HOT states exist for densities which are higher than the critical density, and the power laws are not restricted to special values of the density. The characteristic features of HOT systems include: (1) high efficiency, performance, and robustness to designed-for uncertainties; (2) hypersensitivity to design flaws and unanticipated perturbations; (3) nongeneric, specialized, structured configurations; and (4) power laws. The first three of these are in contrast to the traditional hallmarks of criticality, and are obtained by simply adding the element of design to percolation and sand pile models, which completely changes their characteristics.

Universality and self-similarity of an energy-constrained sandpile model with random neighbors

We study an energy-constrained sandpile model with random neighbors. The critical behavior of the model is in the same universality class as the mean-field self-organized criticality sandpile. The critical energy E(c) depends on the number of neighbors n for each site, but the various exponents are independent of n. A self-similar structure with n-1 major peaks is developed for the energy distribution p(E) when the system approaches its stationary state. The avalanche dynamics contributes to the major peaks appearing at E(p(k))=2k/(2n-1) with k=1,2,...,n-1, while the fine self-similar structure is a natural result of the way the system is disturbed.

Self-organized criticality: sandpiles, singularities, and scaling

We present an overview of the statistical mechanics of self-organized criticality. We focus on the successes and failures of hydrodynamic description of transport, which consists of singular diffusion equations. When this description applies, it can predict the scaling features associated with these systems. We also identify a hard driving regime where singular diffusion hydrodynamics fails due to fluctuations and give an explicit criterion for when this failure occurs.

Complexity, contingency, and criticality

Complexity originates from the tendency of large dynamical systems to organize themselves into a critical state, with avalanches or "punctuations" of all sizes. In the critical state, events which would otherwise be uncoupled become correlated. The apparent, historical contingency in many sciences, including geology, biology, and economics, finds a natural interpretation as a self-organized critical phenomenon. These ideas are discussed in the context of simple mathematical models of sandpiles and biological evolution. Insights are gained not only from numerical simulations but also from rigorous mathematical analysis.