Avalanche shape and exponents beyond mean-field theory

CNRS-Laboratoire de Physique Théorique de l'Ecole Normale Supérieure - 24 rue Lhomond, 75005 Paris, France

Received: 11 August 2014
Accepted: 1 December 2014

Abstract

Elastic systems, such as magnetic domain walls, density waves, contact lines, and cracks, are pinned by substrate disorder. When driven, they move via avalanches, with power law distributions of size, duration and velocity. Their exponents, and the shape of an avalanche, defined as its mean velocity as a function of time, were studied. They are known approximatively from experiments and simulations, and were predicted from mean-field models, such as the Brownian force model (BFM), where each point of the elastic interface sees a force field which itself is a random walk. As we showed in EPL, 97 (2012) 46004, the BFM is the starting point for an expansion around the upper critical dimension, with for short-ranged elasticity, and for long-ranged elasticity. Here we calculate analytically the , i.e. 1-loop, correction to the avalanche shape at fixed duration T, for both types of elasticity. The exact expression, though different from the phenomenological form presented by Laurson et al. in Nat. Commun., 4 (2013) 2927, is well approximated by , 0 < x < 1. The asymmetry is negative for d close to , skewing the avalanche towards its end, as observed in numerical simulations in d = 2 and 3. The exponent is given by the two independent exponents at depinning, the roughness ζ and the dynamical exponent z. We propose a general procedure to predict other avalanche exponents in terms of ζ and z. We finally introduce and calculate the shape at fixed avalanche size, not yet measured in experiments or simulations.