Driven particle in a random landscape: disorder correlator, avalanche distribution and extreme value statistics of records

TL;DR

This paper reviews the measurement and analytical computation of the renormalized force correlator in elastic manifolds with disorder, exploring universality classes, avalanche statistics, and applications to record statistics in random sequences.

Contribution

It provides a comprehensive analysis of the renormalized force correlator Delta(u) across different disorder classes, including new analytical results and numerical validations, and extends to record statistics in random sequences.

Findings

Universal functions for Delta(u) in three extreme value classes

Analytical expressions for critical force corrections and avalanche distributions

Numerical simulations confirming theoretical predictions and exploring anisotropic scaling

Abstract

We review how the renormalized force correlator Delta(u), the function computed in the functional RG field theory, can be measured directly in numerics and experiments on the dynamics of elastic manifolds in presence of pinning disorder. We show how this function can be computed analytically for a particle dragged through a 1-dimensional random-force landscape. The limit of small velocity allows to access the critical behavior at the depinning transition. For uncorrelated forces one finds three universality classes, corresponding to the three extreme value statistics, Gumbel, Weibull, and Frechet. For each class we obtain analytically the universal function Delta(u), the corrections to the critical force, and the joint probability distribution of avalanche sizes s and waiting times w. We find P(s)=P(w) for all three cases. All results are checked numerically. For a Brownian force landscape, known as the ABBM model, avalanche distributions and Delta(u) can be computed for any velocity. For 2-dimensional disorder, we perform large-scale numerical simulations to calculate the renormalized force correlator tensor Delta_{ij}(u), and to extract the anisotropic scaling exponents zeta_x > zeta_y. We also show how the Middleton theorem is violated. Our results are relevant for the record statistics of random sequences with linear trends, as encountered e.g. in some models of global warming. We give the joint distribution of the time s between two successive records and their difference in value w.