Non-steady relaxation and critical exponents at the depinning transition

We study the non-steady relaxation of a driven one-dimensional elastic interface at the depinning transition by extensive numerical simulations concurrently implemented on graphics processing units (GPUs). 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, non-universal microscopic time-regime, we find a non-trivial long crossover towards the non-steady 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 non-steady dynamics of the continuum displacement quenched Edwards Wilkinson equation, getting accurate and consistent depinning exponents for this class: \beta = 0.245 \pm 0.006, z = 1.433 \pm 0.007, \zeta=1.250 \pm 0.005 and \nu=1.333 \pm 0.007. Our study may explain numerical discrepancies (as large as 30% for the velocity exponent \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.