Time distribution and loss of scaling in granular flow

TL;DR

This study investigates how different relaxation mechanisms in cellular automata models affect avalanche distributions and scaling properties in granular flow, revealing phase transitions and universality classes at critical points.

Contribution

It introduces two cellular automata models with distinct internal time scales and analyzes their avalanche scaling behaviors and phase transitions.

Findings

Avalanche distributions exhibit nonuniversal scaling for p ≥ p*

Multifractal and finite size scaling behaviors are observed in models A and B

At p* a phase transition occurs to a noncritical steady state

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.