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 ($L\ensuremath{\simeq}{2}^{25}$) simulations for the nonsteady dynamics of the continuum displacement quenched Edwards-Wilkinson equation, getting accurate and consistent depinning exponents for this class: $\ensuremath{\beta}=0.245\ifmmode\pm\else\textpm\fi{}0.006$, $z=1.433\ifmmode\pm\else\textpm\fi{}0.007$, $\ensuremath{\zeta}=1.250\ifmmode\pm\else\textpm\fi{}0.005$, and $\ensuremath{\nu}=1.333\ifmmode\pm\else\textpm\fi{}0.007$. Our study may explain numerical discrepancies (as large as $30%$ for the velocity exponent $\ensuremath{\beta}$) 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.