Threshold critical dynamics of driven interfaces in random media

Abstract

The dynamics of driven interfaces in random media are analyzed, focusing on the critical behavior near the depinning threshold. The roughening exponent in the critical region is shown to be independent of the type of disorder, in contrast to the equilibrium static behavior where there are two different universality classes corresponding to random-bond and random-field disorder. This critical dynamic roughening exponent is argued to be equal to its equilibrium static random-field value: ${\mathrm{\ensuremath{\zeta}}}_{\mathit{c}}$=\ensuremath{\epsilon}/3 to all orders in \ensuremath{\epsilon} in 5-\ensuremath{\epsilon} dimensions. All other critical exponents are obtained in terms of z, the dynamic exponent, which is calculated to O(\ensuremath{\epsilon}) to be z=2-2\ensuremath{\epsilon}/9+O(${\mathrm{\ensuremath{\epsilon}}}^{2}$). The results agree fairly well with recent numerical simulations. For random-field disorder, the same results have been obtained earlier by Nattermann et al. [J. Phys. (France) II 2, 1483 (1992)] to O(\ensuremath{\epsilon}). The results above threshold are used, together with scaling laws, to yield conjectures on the critical behavior as threshold is approached from below. In particular, the probability that the diameter of an ``avalanche'' exceeds l decays as ${\mathit{l}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\kappa}}}$ just below threshold with \ensuremath{\kappa}=d-3+\ensuremath{\zeta}.