Memory in self-organized criticality

Abstract

Many natural phenomena exhibit power law behaviour in the distribution of event size. This scaling is successfully reproduced by Self Organized Criticality (SOC). On the other hand, temporal occurrence in SOC models has a Poisson-like statistics, i.e. exponential behaviour in the interevent time distribution, in contrast with experimental observations. We present a SOC model with memory: events are nucleated not only as a consequence of the instantaneous value of the local field with respect to the firing threshold, but on the basis of the whole history of the system. The model is able to reproduce the complex behaviour of inter-event time distribution, in excellent agreement with experimental seismic data.After the pioneering work of Bak, Tang and Wiesenfeld [1], Self Organized Criticality (SOC) has been proposed as a successful approach to the understanding of scaling behaviour in many natural phenomena. The term SOC usually refers to a mechanism of slow energy accumulation and fast energy redistribution driving the system toward a critical state. The prototype of SOC systems is the sand-pile model in which particles are randomly added on a two dimensional lattice. When the number of particles σ i in the i-th site exceeds a threshold value σ c , this site is considered unstable and particles are redistributed to nearest neighbor sites. If in any of these sites σ i > σ c , a further redistribution takes place propagating the avalanche. Border sites are dissipatives and discharge particles outside. The system evolves toward a critical state where the distribution of avalanche sizes is a power law obtained without fine tuning: no tunable parameter is present in the model. The simplicity of the mechanism at the basis of SOC has suggested that many physical and biological phenomena characterized by power laws in the size distribution, represent natural realizations of the SOC idea. For instance, SOC has been proposed to model earthquakes [2,3], the evolution of biological systems [4], solar flare occurrence [5], fluctuations in confined plasma [6] snow avalanches [7] and rain fall [8].Moreover, SOC models can be also considered as cellular automata generating stochastic sequences of events. An important quantity showing evidence of time correlations in a sequence is the distribution of time intervals between successive events. Defining ∆t as the time elapsed between the end of an avalanche and the starting of the next one, for the sand-pile model one obtains that ∆t is exponentially distributed [9]. This behaviour reveals the absence of correlations between events typical of a Poissonian process. Conversely the inter-event time distribution N (∆t) of many physical phenomena has a non-exponential shape, as for instance in the case of earthquakes [10], solar flares [9] and confined plasma [11]. The failure in the description of temporal occurrence is generally considered the main restriction for the applicability of SOC ideas to the description of the above phenomena.In this letter we address the problem of introducin...

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