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

TL;DR

This paper analyzes the distribution and extension of avalanches in the Brownian force model, providing exact solutions and new exponents, with results supported by numerical simulations in one dimension.

Contribution

It extends the calculation of avalanche size distributions to joint local and global sizes and introduces new exponents for driven interfaces.

Findings

Distribution of joint local and global avalanche sizes derived.

New exponents τ₀=5/3 and τ=7/4 identified for specific driving conditions.

Avalanche extension distribution exhibits a divergence with exponent -3 at small lengths.

Abstract

The Brownian force model (BFM) is a mean-field model for the local velocities during avalanches in elastic interfaces of internal space dimension $d$, driven in a random medium. It is exactly solvable via a non-linear 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 $\tau_0=5/3$ and $\tau=7/4$, 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 $P(\ell) \sim \ell^{-3}$ at small $\ell$. Most of our results are tested in a numerical simulation in dimension $d=1$.