Avalanche-size distribution at the depinning transition: A numerical test of the theory
Alberto Rosso1, Pierre Le Doussal2, Kay Jörg Wiese2
1
LPTMS; CNRS and Université
Paris-Sud, UMR 8626, ORSAY CEDEX 91405, France.
2CNRS-Laboratoire de Physique Théorique de
l'Ecole Normale Supérieure,
24 rue Lhomond, 75005 Paris, France.
Abstract
We calculate numerically the sizes S of jumps (avalanches) between successively pinned configurations of
an elastic line (d=1) or interface (d=2), pulled by a spring of (small) strength m2 in a random-field landscape.
We obtain strong evidence that the size distribution, away from the small-scale cutoff, takes the form P(S) = ⟨S⟩/Sm2 p(S/Sm) where Sm:= ⟨S2⟩/(2⟨S⟩) ∼ m-d-ζ is the scale of avalanches, and ζ the roughness exponent at the depinning transition. Measurement of the scaling function f(s) := sτ p(s) is compared with the predictions from a recent Functional RG (FRG) calculation, both at mean-field and one-loop level. The avalanche-size exponent τ is found in good agreement with the conjecture τ = 2-2/(d+ζ), recently confirmed to one loop via the FRG. The function f(s) exhibits a shoulder and a stretched exponential decay at large s, ln f(s) ∼ - sδ, with δ ≈ 7/6 in d=1. The function f(s), universal ratios of moments, and the generating function ⟨e λs⟩ are found in excellent agreement with the one-loop FRG predictions. The distribution of local avalanche sizes Sφ, i.e. of the jumps of a subspace of the manifold of dimension dφ, is also computed and compared to our FRG predictions, and to the conjecture τφ = 2-2/(dφ+ζ).
arXiv:0904.1123 [pdf]
Phys. Rev. B 80, 144204 (2009) [pdf]
Copyright (C) by Kay Wiese. Last edited April 9, 2009.