Distribution of maximum velocities in avalanches near the depinning transition

Abstract

We report exact predictions for universal scaling exponents and scaling functions associated with the distribution of the maximum collective avalanche propagation velocities $v_m$ in the mean field theory of the interface depinning transition. We derive the extreme value distribution $P(v_m|T)$ for the maximum velocities in avalanches of fixed duration $T$, and verify the results by numerical simulation near the critical point. We find that the tail of the distribution of maximum velocity for an arbitrary avalanche duration, $v_m$, scales as $P(v_m)\sim v_m^{-2}$ for large $v_m$. These results account for the observed power-law distribution of the maximum amplitudes in acoustic emission experiments of crystal plasticity, and are also broadly applicable to other systems in the mean-field interface depinning universality class, ranging from magnets to earthquakes.