The unreasonable effectiveness of cluster scaling

Equilibrium statistical mechanics requires assumptions such as a Hamiltonian description and ergodicity. Nevertheless, the tools of statistical mechanics have been successfully applied to models in biology and economics. We generalize the Fisher-Stauffer cluster scaling (from percolation theory) to study the avalanches in the nearest-neighbor stochastic Olami-Feder-Christensen (OFC) model, which is believed to be a non-equilibrium system. The OFC model is a two-dimensional lattice of leaky integrate-and-fire(IF) sites. The strength of the noise determines if OFC model is effectively ergodic or non-ergodic. We show that the limit of vanishing dissipation corresponds to a critical point. The Fisher-Stauffer scaling holds when the OFC model is effectively ergodic. We derive universal scaling functions for the avalanche distributions and dynamics. However, the cluster scaling breaks down when the OFC model is non-ergodic, and the system may be characterized by a multi-fractal scaling distribution with infinitely many critical exponents. Our results raise the question whether a Hamiltonian description exists for certain stochastic IF systems. Our work indicates that effective ergodicity may be a sufficient criterion for the validity of cluster scaling methods.