HOW SELF-ORGANIZED CRITICALITY WORKS - A UNIFIED MEAN-FIELD PICTURE

We present a unified dynamical mean-field theory, based on the single site approximation to the master-equation, for stochastic self-organiz ed critical models. In particular, we analyze in detail the properties of sandpile and forest-fire (FF) models. In analogy with other nonequ ilibrium critical phenomena, we identify an order parameter with the d ensity of ''active'' sites, and control parameters with the driving ra tes. Depending on the values of the control parameters, the system is shown to reach a subcritical (absorbing) or supercritical (active) sta tionary state. Criticality is analyzed in terms of the singularities o f the zero-field susceptibility. In the limit of vanishing control par ameters, the stationary state displays scaling characteristics of self -organized criticality (SOC). We show that this limit corresponds to t he breakdown of space-time locality in the dynamical rules of the mode ls. We define a complete set of critical exponents, describing the sca ling of order parameter, response functions, susceptibility and correl ation length in the subcritical and supercritical states. In the subcr itical state, the response of the system to small perturbations takes place in avalanches. We analyze their scaling behavior in relation wit h branching processes. In sandpile models, because of conservation law s, a critical exponents subset displays mean-field values (nu=1/2 and gamma=1) in any dimensions. We treat bull; and boundary dissipation an d introduce a critical exponent relating dissipation and finite size e ffects. We present numerical simulations that confirm our results. In the case of the forest-fire model, our approach can distinguish betwee n different regimes (SOC-FF and deterministic FF) studied in the liter ature, and determine the full spectrum of critical exponents.