Diffusion in stochastic sandpiles

TL;DR

This paper investigates particle diffusion in one-dimensional stochastic sandpiles, demonstrating that the diffusion constant scales with activity density and exploring critical behavior and theoretical predictions through simulations and series expansion.

Contribution

It introduces the relation between diffusion constant and activity density as an order parameter and develops a series expansion for the diffusion coefficient in a specific sandpile model.

Findings

Diffusion constant scales with activity density near criticality.

Critical behavior is similar in restricted and unrestricted sandpiles.

Series expansion matches simulation results for the diffusion coefficient.

Abstract

We study diffusion of particles in large-scale simulations of one-dimensional stochastic sandpiles, in both the restricted and unrestricted versions. The results indicate that the diffusion constant scales in the same manner as the activity density, so that it represents an alternative definition of an order parameter. The critical behavior of the unrestricted sandpile is very similar to that of its restricted counterpart, including the fact that a data collapse of the order parameter as a function of the particle density is only possible over a very narrow interval near the critical point. We also develop a series expansion, in inverse powers of the density. for the collective diffusion coefficient in a variant of the stochastic sandpile in which the toppling rate at a site with $n$ particles is $n(n-1)$, and compare the theoretical prediction with simulation results.