Distribution of velocities in an avalanche, and related quantities: Theory and numerical verification

Abstract

We study several probability distributions relevant to the avalanche dynamics of elastic interfaces driven on a random substrate: The distribution of size, duration, lateral extension or area, as well as velocities. Results from the functional renormalization group and scaling relations involving two independent exponents, roughness ζ, and dynamics z, are confronted to high-precision numerical simulations of an elastic line with short-range elasticity, i.e., of internal dimension d = 1. The latter are based on a novel stochastic algorithm which generates its disorder on the fly. Its precision grows linearly in the time discretization step, and it is parallelizable. Our results show good agreement between theory and numerics, both for the critical exponents as for the scaling functions. In particular, the prediction a = 2-&frac; 2/d+ ζ-z for the velocity exponent is confirmed with good accuracy.