We reconsider the treatment of Lise and Jensen (Phys. Rev. Lett. 76, 2326Lett. 76, (1996) on the random neighbor Olami-Feder-Christensen stik-slip model, and examine the strong dependence of the results on the approximations used for the distribution of states p(E).PACS number(s): 05.40.+j, 05.70.Jk, 05.70.Ln The work of Olami, Feder and Christensen [1] on a slip-stick earthquake model indicated, some time ago, that selforganized criticality may occur without a local conservation law. Recently, it has been claimed by Lise and Jensen [2] that the random-neighbor version of the OFC model also presents critical behavior above some critical dissipation level α c < α 0 , where α 0 = 1/q is associated with local conservation for q neighbors. These authors based their claims on some theoretical mean field arguments and on numerical simulations with systems with up to N = 400 2 sites. In order to perform the mean-field calculation they had to make many different assumptions about the behavior of the model.More recently, Chabanol and Hakin [3], and Bröker and Grassberger [4] performed a more detailed analysis of the same model, showing that what has been interpreted as a critical behavior in [2] indeed corresponds to a subcritical region with very large (but finite) mean avalanche sizes. Although Bröker and Grassberger [4] gives a comprehensive treatment of the random-neighbor version of the OFC model (which we will designate R-OFC), it may be of interest to detect exactly where the theoretical arguments given in [2] fail, since that point is not transparent in their paper and similar problems may occur or be of interest in the future. This is the aim of our paper. We will show that the problem is not in the method used in [2] (which eventually can give useful informations about the mechanism behind SOC) but in the strong dependence of the output of the calculations on the exact form of the distribution of states p(E) of the system.To reinforce the strong dependence of the results on the specific form of p(E), we revisit the R-OFC model, but this time introducing a simple and small modification on the p(E) distribution, that consists in replacing the interval [0, E c ] where the uniform distribution used by Lise and Jensen was defined by the interval [0, E ⋆ ], with E ⋆ < E c (E c is the threshold value above which the sites become unstable and relax), that iswhere Θ(x) is the Heaviside function (see Figure (1-a)). The random version of the OFC model (R-OFC) consists of N sites initially with an energy E i < E c , for i = 1, ..., N . The sites with energy E bellow E c are stable sites (inative) and will be labelled by the superscript −; the sites with energy E above E c are unstable (active) and will be labelled by the superscript +. The energies of all sites are increased slowly until the instant t when the energy of a certain site i reaches the value E c . This site becomes then unstable and the system relaxes in a very short time scale according to the following rules:where E rn stands for the energy of q othe...
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...
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