Size distributions of shocks and static avalanches from the Functional Renormalization Group

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

Abstract

Interfaces pinned by quenched disorder are often used to model jerky self-organized critical motion. We study static avalanches, or shocks, defined here as jumps between distinct global minima upon changing an external field. We show how the full statistics of these jumps is encoded in the functional-renormalization-group fixed-point functions. This allows us to obtain the size distribution P(S) of static avalanches in an expansion in the internal dimension d of the interface. Near and above d = 4 this yields the mean-field distribution P(S) ∼ S^-3/2exp(-S/4S_m) where S_m is a large-scale cutoff, in some cases calculable. Resumming all 1-loop contributions, we find P(S) ∼ S^-τ exp(C(S/S_m)^1/2 -B (S/S_m)^δ), where B, C, δ and τ are obtained to first order in ε = 4 - d. Our result is consistent to O(ε) with the relation τ = τ_ζ := 2 - 2/(d + ζ), where ζ is the static roughness exponent, often conjectured to hold at depinning. Our calculation applies to all static universality classes, including random-bond, random-field and random-periodic disorder. Extended to long-range elastic systems, it yields a different size distribution for the case of contact-line elasticity, with an exponent compatible with τ = 2 - 1/(d + ζ) to O(ε = 2 - d). We discuss consequences for avalanches at depinning and for sandpile models, relations to Burgers turbulence and the possibility that the relation τ = τ_ζ be violated to higher loop order. Finally, we show that the avalanche-size distribution on a hyper-plane of co-dimension one is in mean-field (valid close to and above d = 4) given by P(S) ∼ K_1/3(S)/S, where K is the Bessel-K function, thus τ_{hyper plane} = 4/3.

arXiv:0812.1893 [pdf]
Phys. Rev. E 79, 051106 (2009) [pdf] [MR report]