Renormalization approach to the self-organized critical behavior of sandpile models

Abstract

We introduce a renormalization scheme of a type that is able to describe the self-organized critical state (SOC) of sandpile models. We have defined a characterization of the phase space that allows us to study the evolution of the dynamics under change of scale. In addition, a stationarity condition provides a feedback mechanism that drives the system to its critical state. We obtain an attractive fixed point in the phase space of the parameters that clarifies the self-organized critical nature of these systems. The universality class of several models is identified by studying the properties of the basin of attraction of this fixed point. We compute analytically the avalanche exponent τ and the dynamical exponent z for sandpile models in d=2. The values obtained are in very good agreement with computer simulations. The renormalization scheme can also be applied to study nonconservative sandpile models. The result is that the introduction of a dissipation parameter destroys the critical properties as suggested from simulations. The present theoretical framework seems particularly suitable for all SOC problems and can be naturally extended to other systems showing a critical nonequilibrium stationary state.