Uniqueness of the thermodynamic limit for driven disordered elastic interfaces

TL;DR

This paper investigates the finite size effects at the depinning transition of a one-dimensional elastic interface in a disordered medium, demonstrating the uniqueness of the thermodynamic limit and analyzing the crossover behavior related to the aspect ratio.

Contribution

It establishes the uniqueness of the critical force and velocity-force behavior in the thermodynamic limit for driven disordered elastic interfaces, and characterizes the crossover of finite size fluctuations.

Findings

Critical force and velocity-force behavior are unique and independent of aspect ratio in the thermodynamic limit.

Finite size fluctuations transition from power-law to logarithmic dependence on aspect ratio.

Results inform understanding of anisotropic size-effects in driven 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 $M=k L^\zeta$ with periodic boundary conditions, where $\zeta$ is the depinning roughness exponent and $k$ is a finite aspect ratio parameter. We focus on the crossover from the infinitely narrow ($k\to 0$) to the infinitely wide ($k\to \infty$) medium. We find that at the thermodynamic limit both the value of the critical force and the precise behavior of the velocity-force characteristics are {\it 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.