We present a unified dynamical mean-field theory, based on the single site approximation to the master-equation, for stochastic self-organized critical models. In particular, we analyze in detail the properties of sandpile and forest-fire (FF) 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 (absorbing) or supercritical (active) 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 (SOC). 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 (nu=1/2 and gamma=1) in any dimensions. We treat bull; 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 FF) studied in the literature, and determine the full spectrum of critical exponents.


Originally published in Physical Review E, v.57 no.6 (1998), pp.6345-6362. DOI:10.1103/PhysRevE.57.6345. Dr. Vespignani is affiliated with Northeastern University as of the time of deposit.


sandpile models, forest fire models, stationary state, zero-field susceptibility

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Mean field theory, Stochastic models




American Physical Society

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©1998 American Physical Society

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