Time distribution and loss of scaling in granular flow

Abstract

Two cellular automata models with directed mass flow and internal time scales are studied by numerical simulations. Relaxation rules are a combination of probabilistic critical height (probability of toppling $p$) and deterministic critical slope processes with internal correlation time $t_c$ equal to the avalanche lifetime, in Model A, and $t_c\equiv 1$, in Model B. In both cases nonuniversal scaling properties of avalanche distributions are found for $p\ge p^{\star }$, where $p^{\star }$ is related to directed percolation threshold in $d=3$. Distributions of avalanche durations for $p\ge p^{\star }$ are studied in detail, exhibiting multifractal scaling behavior in model A, and finite size scaling behavior in model B, and scaling exponents are determined as a function of $p$. At $p=p^{\star }$ a phase transition to noncritical steady state occurs. Due to difference in the relaxation mechanisms, avalanche statistics at $p^{\star }$ approaches the parity conserving universality class in Model A, and the mean-field universality class in Model B. We also estimate roughness exponent at the transition.