Avalanche size distributions in mean field plastic yielding models

TL;DR

This paper investigates avalanche size distributions in mean field models of amorphous solids at the yielding transition, revealing protocol-dependent power law exponents and providing a theoretical explanation via a random walk analogy.

Contribution

It demonstrates that the avalanche size distribution exponent varies with the dynamic protocol in mean field models, contrasting with depinning models, and offers a theoretical framework for this behavior.

Findings

Random triggering yields an exponent of 3/2.

External loading results in a smaller exponent close to 1.

Mapping to a random walk explains the protocol dependence.

Abstract

I discuss the size distribution ${\cal N}(S)$ of avalanches occurring at the yielding transition of mean field (i.e., Hebraud-Lequeux) models of amorphous solids. The size distribution follows a power law dependence of the form: ${\cal N}(S)\sim S^{-\tau}$. However (contrary to what is found in its depinning counterpart) the value of $\tau$ depends on details of the dynamic protocol used. For random triggering of avalanches I recover the $\tau=3/2$ exponent typical of mean field models, which in particular is valid for the depinning case. However, for the physically relevant case of external loading through a quasistatic increase of applied strain, a smaller exponent (close to 1) is obtained. This result is rationalized by mapping the problem to an effective random walk in the presence of a moving absorbing boundary.