The branching rate of the random Olami-Feder-Christensen model with a generic coordination number

Abstract

In this paper we review and discuss some fundamental aspects of the random version of the Olami-FederChristensen model, and its relevance for the understanding of self-organized criticality (SOC). We review the universal character of the exponent τ = 3/2, related to avalanche size distributions in random SOC models, and its connection to branching processes theory. We also generalize previous results, that had been obtained for the random OFC model with four neighbors, to any coordination number. Finally we present some connections between our generalization and recent discussions involving the branching rate approach to this model. I IntroductionThe concept of self-organized criticality was introduced by Bak, Tang and Wiesenfeld [1] in 1987, as a possible explanation of scale invariance in nature. To illustrate their basic ideas, they present a cellular automaton model that became known as the sandpile model, because of an analogy between its dynamical rules and the way sand topples and generates avalanches, in a real sand pile. Since this seminal work, a great number of cellular automata and coupled map models have been investigated, in an attempt to elucidate the essential mechanisms hidden in such a wide class of different non-linear phenomena whose statistics of events (or 'avalanches') are governed by power-laws. However, despite many efforts, up to now, one still lacks from a general theoretical framework for self-organized criticality. Most of the available results are purely numerical. Success in analytical investigations have been achieved mainly in the study of a special class of models that became known as abelian models [2], in mean-field type calculations [3][4][5][6][7][8][9] or through a renormalization group approach [10].In this paper we discuss the random neighbor version of the Olami-Feder-Christensen (R-OFC) model. The original OFC model introduced in 1992 [11] is a two-dimensional coupled map model, defined on a square lattice, whose dynamical rules were inspired in a spring-block model [12] proposed to describe the dynamics of earthquakes, which is related to some empirical power-laws (like the GutenbergRitcher law). With each node of a square lattice we associate a real state variable (or 'energy') z i,j . The model is globally driven, and each time the energy of a given site (i, j) exceeds a threshold value, the system relaxes according to specific rules that will be presented in detail in section III. Within the OFC model there is a dissipation parameter α. If α = 0.25 the dynamical variable z i,j of the model is conserved during the avalanche process, in the bulk of the lattice (there is always dissipation in the boundaries), but if α < 0.25 there is some dissipation also in the bulk of the system. Because of those facts, this model has been widely studied in literature. It is, at the same time, a prototype of self-organization in systems with non-conservative relaxation rules, and also a paradigm of the success of SOC ideas, since it is able to reproduce important aspects of the stati...

Background activity drives criticality of neuronal avalanches

“…In particular, Pinho and Prado [25] have calculated an analytic form of ( 15) for the special case of β = 0. Equation ( 15) can easily be demonstrated, by suitable application of Stirling's relations, to have the asymptotic behaviour P (s) ∼ s −3/2 for s 1 if a = 2b − 1, a condition that can equivalently be expressed as…”

Dissipative self-organized branching in a dynamic population

“…Graphically, Equation (3) can be shown to define a power law with exponent τ s = 3/2 for s >> 1 at the critical condition a = 2b − 1 with b ∈ [0, 1). In particular, for b = 0 the recurrence relation reduces to the analytic prediction in [3,5], which only holds for odd values of s.…”

“…Graphically, Equation (3) can be shown to define a power law with exponent τ s = 3/2 for s >> 1 at the critical condition a = 2b − 1 with b ∈ [0, 1). In particular, for b = 0 the recurrence relation reduces to the analytic prediction in [3,5], which only holds for odd values of s. Dissipation in the model is implemented by allowing q k=0 > 0 or equivalently, α + β < 1. Hence, with probability q 0 = 1 − (α + β), an unstable active site (z = 2) dissipates two grains out of the system.…”

“…In order for the sandpile dynamics to be self-organized critical (SOC), the stationary state of the dynamical system must be a critical state. To show this, we solve for the fixed point solution of Equation (5). Generally, the fixed-point solution of the density of active sites p * does not coincide with its critical value p c .…”