Distribution of joint local and total size and of extension for avalanches in the Brownian force model.

Abstract

The Brownian force model is a mean-field model for local velocities during avalanches in elastic interfaces of internal space dimension d, driven in a random medium. It is exactly solvable via a nonlinear differential equation. We study avalanches following a kick, i.e., a step in the driving force. We first recall the calculation of the distributions of the global size (total swept area) and of the local jump size for an arbitrary kick amplitude. We extend this calculation to the joint density of local and global sizes within a single avalanche in the limit of an infinitesimal kick. When the interface is driven by a single point, we find new exponents τ_{0}=5/3 and τ=7/4, depending on whether the force or the displacement is imposed. We show that the extension of a "single avalanche" along one internal direction (i.e., the total length in d=1) is finite, and we calculate its distribution following either a local or a global kick. In all cases, it exhibits a divergence P(ℓ)∼ℓ^{-3} at small ℓ. Most of our results are tested in a numerical simulation in dimension d=1.