N-State Exclusive Diffusion Models for Avalanche Processes Showing Self-Organized Criticality

Abstract

We introduce new avalanche models on the d -dimensional hyper-cubic lattices which show the self-organized criticality (SOC). They are generalization of the two-state exclusive diffusion model introduced in our previous paper and called the n -state exclusive diffusion models (1≤ n ≤2 d ). If n =2 d , this model is the sandpile model of Bak, Tang and Wiesenfeld. By Monte Carlo simulations, we evaluated the height distributions of particles and the avalanche exponents for d =2,3, and 4. Numerical results imply that the n -state exclusive diffusion models belong to the same universality class for each d , if 2≤ n ≤2 d . We apply the mean-field approximation and evaluate the height distributions of particles in the SOC states. It is shown that the mean-field theory by Tang and Bak is the d →∞ limit of our mean-field approximation.