Nonsteady relaxation and critical exponents at the depinning transition

Abstract

We study the nonsteady relaxation of a driven one-dimensional elastic interface at the depinning transition by extensive numerical simulations concurrently implemented on graphics processing units. We compute the time-dependent velocity and roughness as the interface relaxes from a flat initial configuration at the thermodynamic random-manifold critical force. Above a first, nonuniversal microscopic time regime, we find a nontrivial long crossover towards the nonsteady macroscopic critical regime. This “mesoscopic” time regime is robust under changes of the microscopic disorder, including its random-bond or random-field character, and can be fairly described as power-law corrections to the asymptotic scaling forms, yielding the true critical exponents. In order to avoid fitting effective exponents with a systematic bias we implement a practical criterion of consistency and perform large-scale ( L ≃ 2 25 ) simulations for the nonsteady dynamics of the continuum displacement quenched Edwards-Wilkinson equation, getting accurate and consistent depinning exponents for this class: β = 0.245 ± 0.006 , z = 1.433 ± 0.007 , ζ = 1.250 ± 0.005 , and ν = 1.333 ± 0.007 . Our study may explain numerical discrepancies (as large as 30 % for the velocity exponent β ) found in the literature. It might also be relevant for the analysis of experimental protocols with driven interfaces keeping a long-term memory of the initial condition.