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

Pierre Le Doussal, Kay Jörg Wiese
CNRS-Laboratoire de Physique Théorique de l'Ecole Normale Supérieure, 24 rue Lhomond, 75005 Paris, France

Abstract

We review how the renormalized force correlator Δ(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 Fréchet. For each class we obtain analytically the universal function Δ(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 Δ(u) can be computed for any velocity. For 2-dimensional disorder, we perform large-scale numerical simulations to calculate the renormalized force correlator tensor Δ_ij(u), and to extract the anisotropic scaling exponents ζ_x > ζ_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.

arXiv:0808.3217 [pdf]
Phys. Rev. E 79 (2009) 051105 [pdf]