Uniqueness of the thermodynamic limit for driven disordered elastic interfaces

Abstract

We study the finite-size fluctuations at the depinning transition for a one-dimensional elastic interface of size L displacing in a disordered medium of transverse size M = kLζ with periodic boundary conditions, where ζ is the depinning roughness exponent and k is a finite aspect-ratio parameter. We focus on the crossover from the infinitely narrow (k → 0) to the infinitely wide (k → ∞) medium. We find that at the thermodynamic limit both the value of the critical force and the precise behaviour of the velocity?force characteristics are unique and k-independent. We also show that the finite-size fluctuations of the critical force (bias and variance) as well as the global width of the interface cross over from a power-law to a logarithm as a function of k. Our results are relevant for understanding anisotropic size effects in force-driven and velocity-driven interfaces.