On self-organized criticality in nonconserving systems

Emergence of self-organized criticality and phase transition in the Olami-Feder-Christensen model with a single defect

We examine the conditions for the emergence of self-organized criticality in the Olami-Feder-Christensen model by introducing a single defect under periodic boundary conditions. Our findings reveal that strong localized energy dissipation is crucial for self-organized criticality emergence, while weak localized or global energy dissipation leads to its disappearance in this model. Furthermore, slight dissipation perturbations to a system in a self-organized criticality reveal a novel state characterized by a limit cycle of distinct configurations. This newly discovered state offers significant insights into the fundamental mechanisms governing the emergence of self-organized criticality.

Aftershocks in a frictional earthquake model

Inspired by spring-block models, we elaborate a "minimal" physical model of earthquakes which reproduces two main empirical seismological laws, the Gutenberg-Richter law and the Omori aftershock law. Our point is to demonstrate that the simultaneous incorporation of aging of contacts in the sliding interface and of elasticity of the sliding plates constitutes the minimal ingredients to account for both laws within the same frictional model.

New approach to Gutenberg-Richter scaling

We introduce a new model for an earthquake fault system that is composed of noninteracting simple lattice models with different levels of damage denoted by q. The undamaged lattice models (q=0) have Gutenberg-Richter scaling with a cumulative exponent β=1/2, whereas the damaged models do not have well defined scaling. However, if we consider the "fault system" consisting of all models, damaged and undamaged, we get excellent scaling with the exponent depending on the relative frequency with which faults with a particular amount of damage occur in the fault system. This paradigm combines the idea that Gutenberg-Richter scaling is associated with an underlying critical point with the notion that the structure of a fault system also affects the statistical distribution of earthquakes. In addition, it provides a framework in which the variation, from one tectonic region to another, of the scaling exponent, or b value, can be understood.

Age distribution of trees in stationary forest system

A statistical theory for the age distribution of spatially dominant trees in a stationary forest system is developed. The result depends whether or not mortality is spatially correlated, as well as whether or not the stand boundaries are pre-determined. In the case of spatially non-correlated mortality, the tree age distribution is an exponential with survival rate as the base. In the case of spatially correlated mortality within a stand with pre-determined boundaries, the age distribution within the stand is an exponential with natural base. For a small stand, the median life span of the stand is inversely proportional to the number of trees (n); the median life span in relation to stand closure time is inversely proportional to nln(n). For a large stand, the stand life does not extend to the closure time. The behaviour of a forest system without fixed stand boundaries depends on the dimensionality of the system. In the case of a one-dimensional system, the longevity distribution is exponential, most of the trees however having the same longevity. Consequently, the probability density of tree age is constant. However, the probability mass of size of catastrophe destroying a particular tree is evenly distributed. This is due to trees being rapidly born on empty areas in the beginning of the life cycle, and clusters rapidly growing into larger ones close to the end of tree life.

Olami-Feder-Christensen model with quenched disorder

We study the Olami-Feder-Christensen model with quenched disorder in the coupling parameter alpha . In contrast to an earlier study by Mousseau [Phys. Rev. Lett. 77, 968 (1996)], we do not find a phase diagram with several phase transitions, but continuous crossovers from one type of behavior to another. The crossover behavior is determined by the ratio of three length scales, which are the system size, the penetration depth of the boundary layer, and the correlation length introduced by the disorder.

Pattern formation in a metastable, gradient-driven sandpile

With a toppling rule which generates metastable sites, we explore the properties of a gradient-driven sandpile that is minimally perturbed at one boundary. In two dimensions we find that the transport of grains takes place along deep valleys, generating a set of patterns as the system approaches the stationary state. We use two versions of the toppling rule to analyze the time behavior and the geometric properties of clusters of valleys, also discussing the relation between this model and the general properties of models displaying self-organized criticality.

Crossover behavior in the event size distribution of the Olami-Feder-Christensen model

The avalanche size distribution and supercritical toppling value distribution in the Olami-Feder-Christensen model are examined, demonstrating that there exists a crossover value alpha(X) approximately 0.14 for the conservation parameter in the model. We have further confirmed the location of this crossover by identifying upper and lower bounds for alpha(X). For levels of conservation below alpha(X) the asymptotic behavior, in the limit of both infinite-system-size and infinite-precision arithmetic, consists only of avalanches of size 1 with all sites toppling exactly at the threshold value. For larger levels of conservation the probability of finding avalanches of size 2 or bigger remains nonzero in the asymptotic limit.

Nonequilibrium phase transition in a self-activated biological network

We present a lattice model for a two-dimensional network of self-activated biological structures with a diffusive activating agent. The model retains basic and simple properties shared by biological systems at various observation scales, so that the structures can consist of individuals, tissues, cells, or enzymes. Upon activation, a structure emits a new mobile activator and remains in a transient refractory state before it can be activated again. Varying the activation probability, the system undergoes a nonequilibrium second-order phase transition from an active state, where activators are present, to an absorbing, activator-free state, where each structure remains in the deactivated state. We study the phase transition using Monte Carlo simulations and evaluate the critical exponents. As they do not seem to correspond to known values, the results suggest the possibility of a separate universality class.

Complex scaling behavior of nonconserved self-organized critical systems

The Olami-Feder-Christensen earthquake model is often considered the prototype dissipative self-organized critical model. It is shown that the size distribution of events in this model results from a complex interplay of several different phenomena, including limited floating-point precision. Parallels between the dynamics of synchronized regions and those of a system with periodic boundary conditions are pointed out, and the asymptotic avalanche size distribution is conjectured to be dominated by avalanches of size 1, with the weight of larger avalanches converging towards zero as the system size increases.

Finite-size effects of avalanche dynamics

We study the avalanche dynamics of a system of globally coupled threshold elements receiving random input. The model belongs to the same universality class as the random-neighbor version of the Olami-Feder-Christensen stick-slip model. A closed expression for avalanche size distributions is derived for arbitrary system sizes N using geometrical arguments in the system's configuration space. For finite systems, approximate power-law behavior is obtained in the nonconservative regime, whereas for N--> infinity, critical behavior with an exponent of -3/2 is found in the conservative case only. We compare these results to the avalanche properties found in networks of integrate-and-fire neurons, and relate the different dynamical regimes to the emergence of synchronization with and without oscillatory components.

Nonconservative earthquake model of self-organized criticality on a random graph

We numerically investigate the Olami-Feder-Christensen model on a quenched random graph. Contrary to the case of annealed random neighbors, we find that the quenched model exhibits self-organized criticality deep within the nonconservative regime. The probability distribution for avalanche size obeys finite size scaling, with universal critical exponents. In addition, a power law relation between the size and the duration of an avalanche exists. We propose that this may represent the correct mean-field limit of the model rather than the annealed random neighbor version.

Type of self-organized criticality model based on neural networks

Based on the standard self-organizing map neural network model, we introduce a kind of coupled map lattice system to investigate self-organized criticality (SOC) in the activity of model neural populations. Our system is simulated by a more detailed integrate-and-fire mechanism and a kind of local perturbation driving rule; it can display SOC behavior in a certain range of system parameters, even with period boundary condition. More importantly, when the influence of synaptic plasticity is adequately considered, we can find that our system's learning process plays a promotive role in the emergence of SOC behavior.

Defensive alliances in spatial models of cyclical population interactions

As a generalization of the three-strategy Rock-Scissors-Paper game dynamics in space, cyclical interaction models of six mutating species are studied on a square lattice, in which each species is supposed to have two dominant, two subordinated, and a neutral interacting partner. Depending on their interaction topologies, all imaginable systems can be classified into four (isomorphic) groups exhibiting significantly different behaviors as a function of mutation rate. In three out of four cases three (or four) species form defensive alliances that maintain themselves in a self-organizing polydomain structure via cyclic invasions. Varying the mutation rate, this mechanism results in an ordering phenomenon analogous to that of magnetic Ising systems. The model explains a very basic mechanism of community organization, which might gain important applications in biology, economics, and sociology.

Scaling in a nonconservative earthquake model of self-organized criticality

We numerically investigate the Olami-Feder-Christensen model for earthquakes in order to characterize its scaling behavior. We show that ordinary finite size scaling in the model is violated due to global, system wide events. Nevertheless we find that subsystems of linear dimension small compared to the overall system size obey finite (subsystem) size scaling, with universal critical coefficients, for the earthquake events localized within the subsystem. We provide evidence, moreover, that large earthquakes responsible for breaking finite-size scaling are initiated predominantly near the boundary.

Self-organized criticality and universality in a nonconservative earthquake model

We make an extensive numerical study of a two-dimensional nonconservative model proposed by Olami, Feder, and Christensen to describe earthquake behavior [Phys. Rev. Lett. 68, 1244 (1992)]. By analyzing the distribution of earthquake sizes using a multiscaling method, we find evidence that the model is critical, with no characteristic length scale other than the system size, in agreement with previous results. However, in contrast to previous claims, we find a convergence to universal behavior as the system size increases, over a range of values of the dissipation parameter alpha. We also find that both "free" and "open" boundary conditions tend to the same result. Our analysis indicates that, as L increases, the behavior slowly converges toward a power law distribution of earthquake sizes P(s) approximately equal s(-tau) with an exponent tau approximately equal 1.8. The universal value of tau we find numerically agrees quantitatively with the empirical value (tau=B+1) associated with the Gutenberg-Richter law.

Dissipative Abelian sandpiles and random walks

We show that the dissipative Abelian sandpile on a graph L can be related to a random walk on a graph that consists of L extended with a trapping site. From this relation it can be shown, using exact results and a scaling assumption, that the correlation length exponent nu of the dissipative sandpiles always equals 1/d(w), where d(w) is the fractal dimension of the random walker. This leads to a new understanding of the known result that nu=1/2 on any Euclidean lattice. Our result is, however, more general, and as an example we also present exact data for finite Sierpinski gaskets, which fully confirm our predictions.

Forest-fire models as a bridge between different paradigms in self-organized criticality

We turn the stochastic critical forest-fire model introduced by Drossel and Schwabl [Phys. Rev. Lett. 69, 1629 (1992)] into a completely deterministic threshold model. This model has many features in common with sandpile and earthquake models of self-organized criticality. Our deterministic forest-fire model exhibits in detail the same macroscopic statistical properties as the original Drossel-Schwabl model. We use the deterministic model to elaborate on the relation between forest-fire, sandpile, and earthquake models.

Spatial evolutionary prisoner's dilemma game with three strategies and external constraints

The emergency of mutual cooperation is studied in a spatially extended evolutionary prisoner's dilemma game in which the players are located on the sites of cubic lattices for dimensions d=1, 2, and 3. Each player can choose one of the three following strategies: cooperation (C), defection (D) or "tit for tat" (T). During the evolutionary process the randomly chosen players adopt one of their neighboring strategies if the chosen neighbor has a higher payoff. Moreover, an external constraint imposes that the players always cooperate with probability p. The stationary state phase diagram is computed by both using generalized mean-field approximations and Monte Carlo simulations. Nonequilibrium second-order phase transitions associated with the extinction of one of the possible strategies are found and the corresponding critical exponents belong to the directed percolation universality class. It is shown that externally forcing the collaboration does not always produce the desired result.

Self-organized criticality in the olami-feder-christensen model

A system is in a self-organized critical state if the distribution of some measured events obeys a power law. The finite-size scaling of this distribution with the lattice size is usually enough to assume that the system displays self-organized criticality. This approach, however, can be misleading. In this paper we analyze the behavior of the branching rate sigma of the events to establish whether a system is in a critical state. We apply this method to the Olami-Feder-Christensen model to obtain evidence that, in contrast to previous results, the model is critical in the conservative regime only.

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.

Rapid local synchronization of action potentials: toward computation with coupled integrate-and-fire neurons

The collective behavior of interconnected spiking nerve cells is investigated. It is shown that a variety of model systems exhibit the same short-time behavior and rapidly converge to (approximately) periodic firing patterns with locally synchronized action potentials. The dynamics of one model can be described by a downhill motion on an abstract energy landscape. Since an energy landscape makes it possible to understand and program computation done by an attractor network, the results will extend our understanding of collective computation from models based on a firing-rate description to biologically more realistic systems with integrate-and-fire neurons.

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.