Mean-field theory of avalanches in self-organized critical states

Katori, Makoto; Kobayashi, Hirotsugu

doi:10.1016/0378-4371(96)00003-9

Cited by 12 publications

(10 citation statements)

References 32 publications

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“…We also verify that at ζ = ζ c , each active site has, on average, one critical nearest neighbor. The overall density of critical sites is ρ c = 0.434, again in agreement with driven sandpile simulations [37] (At the critical point, about 10% of critical sites have heights in excess of 4. )…”

Section: Simulation Resultssupporting

confidence: 79%

Self-organized criticality as an absorbing-state phase transition

1998

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We explore the connection between self-organized criticality and phase transitions in models with absorbing states. Sandpile models are found to exhibit criticality only when a pair of relevant parameters -dissipation ⑀ and driving field h -are set to their critical values. The critical values of ⑀ and h are both equal to zero. The first result is due to the absence of saturation ͑no bound on energy͒ in the sandpile model, while the second result is common to other absorbing-state transitions. The original definition of the sandpile model places it at the point (⑀ϭ0,hϭ0 ϩ ): it is critical by definition. We argue power-law avalanche distributions are a general feature of models with infinitely many absorbing configurations, when they are subject to slow driving at the critical point. Our assertions are supported by simulations of the sandpile at ⑀ϭhϭ0 and fixed energy density ͑no drive, periodic boundaries͒, and of the slowly driven pair contact process. We formulate a field theory for the sandpile model, in which the order parameter is coupled to a conserved energy density, which plays the role of an effective creation rate. ͓S1063-651X͑98͒08805-9͔

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Section: Simulation Resultssupporting

confidence: 79%

Self-organized criticality as an absorbing-state phase transition

1998

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“…For the original BTW model (g = 2d), extensive numerical simulations on the density of energy levels can be found in Ref [34]. As expected the agreement with the MF result increases for high dimensional systems and we recover the exact result in the limit d → ∞.…”

Section: Mean-field Analysis Of Sandpile Modelssupporting

confidence: 80%

How self-organized criticality works: A unified mean-field picture

1998

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We present a unified dynamical mean-field theory, based on the single site approximation to the masterequation, for stochastic self-organized critical models. In particular, we analyze in detail the properties of sandpile and forest-fire ͑FF͒ models. In analogy with other nonequilibrium critical phenomena, we identify an order parameter with the density of ''active'' sites, and control parameters with the driving rates. Depending on the values of the control parameters, the system is shown to reach a subcritical ͑absorbing͒ or supercritical ͑active͒ stationary state. Criticality is analyzed in terms of the singularities of the zero-field susceptibility. In the limit of vanishing control parameters, the stationary state displays scaling characteristics of self-organized criticality ͑SOC͒. We show that this limit corresponds to the breakdown of space-time locality in the dynamical rules of the models. We define a complete set of critical exponents, describing the scaling of order parameter, response functions, susceptibility and correlation length in the subcritical and supercritical states. In the subcritical state, the response of the system to small perturbations takes place in avalanches. We analyze their scaling behavior in relation with branching processes. In sandpile models, because of conservation laws, a critical exponents subset displays mean-field values ( ϭ 1 2 and ␥ϭ1) in any dimensions. We treat bulk and boundary dissipation and introduce a critical exponent relating dissipation and finite size effects. We present numerical simulations that confirm our results. In the case of the forest-fire model, our approach can distinguish between different regimes ͑SOC-FF and deterministic FF͒ studied in the literature, and determine the full spectrum of critical exponents. ͓S1063-651X͑98͒09805-5͔

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“…A realspace renormalization scheme predicted that both models belong to the same universality class [22,23]. Here, the authors used a mean-field-type approximation [24] in order to perform a block transformation. Therefore it is not clear if this renormalization ansatz is an appropriate tool to take the additional fluctuations of the energy distribution of the Manna model into account.…”

Section: Models and Simulationsmentioning

confidence: 99%