Dynamic exponent in Extremal models of Pinning

TL;DR

This paper investigates the dynamic exponent in extremal models of pinning, revealing a size-dependent initial roughness growth and proposing an analytic expression for the dynamic exponent, confirmed across various models.

Contribution

It introduces a new understanding of the initial roughness growth in extremal pinning models and provides an analytic formula for the dynamic exponent.

Findings

Initial roughness growth depends on system size.

Analytic expression for the dynamic exponent is proposed.

Confirmed across multiple extremal models.

Abstract

The depinning transition of a front moving in a time-independent random potential is studied. The temporal development of the overall roughness w(L,t) of an initially flat front, $w(t)\propto t^\beta$, is the classical means to have access to the dynamic exponent. However, in the case of front propagation in quenched disorder via extremal dynamics, we show that the initial increase in front roughness implies an extra dependence over the system size which comes from the fact that the activity is essentially localized in a narrow region of space. We propose an analytic expression for the $\beta$ exponent and confirm this for different models (crack front propagation, Edwards-Wilkinson model in a quenched noise, ...).