Distribution of velocities and acceleration for a particle in Brownian correlated disorder: inertial case

TL;DR

This paper investigates the velocity and acceleration distributions of a particle in a disordered environment with inertia, extending the ABBM model to include mass effects and analyzing their impact through analytical and numerical methods.

Contribution

It introduces an inertial extension of the ABBM model, providing analytical and numerical insights into velocity and acceleration distributions in disordered systems with inertia.

Findings

Large-deviation functions for positive velocities are identical across models at high driving velocities.

Analytical solutions for small and large mass cases are obtained.

Inertial effects influence the distribution of velocities and accelerations, especially at small scales.

Abstract

We study the motion of an elastic object driven in a disordered environment in presence of both dissipation and inertia. We consider random forces with the statistics of random walks and reduce the problem to a single degree of freedom. It is the extension of the mean field ABBM model in presence of an inertial mass m. While the ABBM model can be solved exactly, its extension to inertia exhibits complicated history dependence due to oscillations and backward motion. The characteristic scales for avalanche motion are studied from numerics and qualitative arguments. To make analytical progress we consider two variants which coincide with the original model whenever the particle moves only forward. Using a combination of analytical and numerical methods together with simulations, we characterize the distributions of instantaneous acceleration and velocity, and compare them in these three models. We show that for large driving velocity, all three models share the same large-deviation function for positive velocities, which is obtained analytically for small and large m, as well as for m =6/25. The effect of small additional thermal and quantum fluctuations can be treated within an approximate method.