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 $\zeta$, 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 ${\sf a} = 2 - \frac{2}{d+ \zeta - z}$ for the velocity exponent is confirmed with good accuracy.