Universal Statistics of the Critical Depinning Force of Elastic Systems in Random Media

Abstract

We study the rescaled probability distribution of the critical depinning force of an elastic system in a random medium. We put in evidence the underlying connection between the critical properties of the depinning transition and the extreme value statistics of correlated variables. The distribution is Gaussian for all periodic systems, while in the case of random manifolds there exists a family of universal functions ranging from the Gaussian to the Gumbel distribution. Both of these scenarios are a priori experimentally accessible in finite, macroscopic, disordered elastic systems.

Universal distribution of threshold forces at the depinning transition

“…Following Ref. [28] let us introduce for every configuration α of the interface in the sample of width M = kL ζ the depinning force f d (α) and then associate the threshold force of the whole sample with the following maximal value f r c = max α {f d (α)}. In each sample there are only ≈ M/w independent pinned configuration, so that the distribution of the maximum of the corresponding critical forces can be written as…”

“…Each configuration has a slightly different critical force which is distributed according to our FRG result. If one defines the total critical force as a maximum of all the critical forces of these subsamples, it becomes M -dependent and its shifted distribution tends to the distribution of the extreme value statistics [28]. Following Ref.…”

“…In numerical studies of depinning of elastic interfaces, either via an exact determination of the critical state, or via Langevin dynamics [20,27,28,29,30] a cylindrical system which is periodic in both directions was studied: longitudinal with period L and transverse with period M . Hence this is equivalent to a periodic disorder with period M .…”

“…The finite-size sample-tosample distribution of f c (L, M ) was computed numerically [28] and found to depend on the aspect ratio k = M/L ζ of the cylinder. This should be expected since for large k one recovers a zero-dimensional problem and the interface will be blocked by rare disorder configurations, hence dominated by extremal statistics.…”

Random Walks and Polymers in the Presence of Quenched Disorder

“…Exactly at criticality, the typical and averaged correlations are also very different, since the typical correlation decays as C typ (r) ∼ e −w √ r , where w is a random variable of order 1, whereas the averaged correlation is dominated by rare events and decays algebraically C(r) ∼ 1/r (3− √ 5)/2 . Other critical points with two different correlation length exponents are discussed in [16,51,101,102,105]. …”