Generalizing Alessandro-Beatrice-Bertotti-Montorsi (AMMB) Models to the Case of Velocity-dependent Dissipation

TL;DR

This paper extends the AMMB models to include velocity-dependent dissipation, deriving the Fokker-Planck equation and analyzing avalanche size and duration distributions under various driving conditions.

Contribution

It introduces a generalized AMMB model with velocity-dependent dissipation and analyzes its stationary and non-stationary behavior, including avalanche distributions.

Findings

Avalanche size distribution remains unchanged with velocity-dependent dissipation.

Duration distribution exhibits exponential decay for large durations.

Under certain conditions, duration distribution shows power-law behavior for small durations.

Abstract

We study a more general class of the Alessandro-Beatrice-Bertotti-Montorsi (AMMB) models with velocity-dependent dissipation. We obtain the Fokker-Planck equation describing the evolution of an arbitrary initial probability distribution, and from there the stationary distribution under constant driving. For this class of models, we show that the distribution of the size of an avalanche is the same as when the dissipation is velocity-independent. As for durations, we show that, under non-stationary driving known as "kicks", although no closed-form solution seems to be available for an arbitrary velocity-dependent dissipation, for large durations the distribution seems to demonstrate an exponential fall-off, while for small durations (under some extra conditions to be made clear in the paper) the distribution seems to show a characteristic power-law behaviour.