Universality in Sandpiles, Interface Depinning, and Earthquake Models

We introduce a new continuous cellular automaton that presents self-organized criticality. It is one-dimensional, totally deterministic, without any embedded randomness, not even in the initial conditions. This system is in the same universality class as the Oslo rice pile, boundary driven interface depinning and the train model for earthquakes. Although the system is chaotic, in the thermodynamic limit chaos occurs only in a microscopic level.In 1987, Bak, Tang and Wiesenfeld showed that fractal behavior, that is, power-law distributions, can be observed in simple dissipative systems with many degrees of freedom without fine tuning of parameters [1]. They called this phenomenon self-organized criticality (SOC). Until then, the studies of fractal structures were basically related to equilibrium systems where fractality appears only at special parameter values where a phase transition takes place.Since the pioneering work of Bak et. al, an enormous amount of numerical, theoretical and experimental studies have been done in systems that present SOC. One of the most interesting experimental studies demonstrating the existence of SOC in Nature was done in a quasi-onedimensional pile of rice by Frette et. al [2]. They found that the occurrence of SOC depends on the shape of the rice. Only with sufficient elongated grains, avalanches with a power-law distribution occurred. If the rice had little asymmetry, a distribution described by a stretched exponential was seen. Christensen et. al [3] introduced a model for the rice pile experiment in which the local critical slope varies randomly between 1 and 2. They found that their model, known as the Oslo rice pile model, reproduced well the experimental results.A good understanding of the Oslo system was achieved by Paczuski and Boettcher [4]. They showed that it could be mapped exactly to a model for interface depinning where the interface is slowly pulled at one end through a medium with quenched random pinning forces. They found that the height of the interface maps to the number of toppling events in the rice pile model. The critical exponents of the two models were identical (within the error bars), showing that they were in the same universality class. Paczuski and Boettcher also conjectured that the train model for earthquakes, which was introduced by Burridge and Knopoff [5], and studied in detail in [6], is also in that same universality class. The train model is the only model that we know (besides the one we introduce here) that presents SOC and has no kind of embedded randomness. However, it is governed by coupled ordinary equations (ODE's), what makes its study very time consuming.A way of making a system governed by ODE's more amenable to computer simulations is to discretize it in time. This was done by Olami, Feder and Christensen (OFC) [7] who introduced a continuous cellular automaton (CCA) to study the two-dimensional version of another Burridge and Knopoff model for earthquakes [5]. [A continuous cellular automaton in SOC is known in chaos theory as coup...

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confidence: 63%

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confidence: 99%

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Simple deterministic self-organized critical system

Vieira

2000

“…The critical state, which can be characterized by various critical exponents and scaling functions, was studied using both theoretical [4,5,6,7,8,9,10,11] and numerical approaches [12,13,14,15,16,17,18,19,20]. These studies stimulated an effort to examine the utility of the SOC framework to the understanding of empirical phenomena such as earthquakes avalanches in granular flow and mass extinctions [21].…”

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confidence: 99%

Sandpile models and random walkers on finite lattices

Shilo

Biham

2003

Abelian sandpile models, both deterministic, such as the Bak, Tang, Wiesenfeld (BTW) model [P. Bak, C. Tang and K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987) L. The first few moments of this distribution are evaluated numerically and their dependence on the system size is examined. The sandpile models are conservative in the sense that grains are conserved in the bulk and can leave the system only through the boundaries. It is shown that the conservation law provides an interesting connection between sandpile models and random walk models. Using this connection, it is shown that the average avalanche sizes, n L , for the BTW and the Manna models are equal to each other, and both are equal to the average path-length of a random walker starting from a random initial site on the same lattice of size L. This is in spite of the fact that sandpile models with deterministic (BTW) and stochastic (Manna) toppling rules exhibit different critical exponents, indicating that they belong to different universality classes.

“…Also, a scaling relation s ∼ L was found numerically in [16], and is a general result for many boundary driven SOC systems [20], giving the result that D(2 − τ ) = 1. Combining the two equations, β = D − 1 = (τ − 1)/(2 − τ ), which agrees very well with numerical results.…”

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confidence: 99%

1/fαnoise from correlations between avalanches in self-organized criticality

Davidsen

Paczuski

2002

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We show that large, slowly driven systems can evolve to a self-organized critical state where long range temporal correlations between bursts or avalanches produce low frequency 1/f α noise. The avalanches can occur instantaneously in the external time scale of the slow drive, and their event statistics are described by power law distributions. A specific example of this behavior is provided by numerical simulations of a deterministic "sandpile" model.