Driven interfaces: from flow to creep through model reduction

TL;DR

This paper introduces a simplified two-degree-of-freedom model to explain the creep law in disordered one-dimensional interfaces, capturing key phenomenology and effects of correlations, and analyzing the transition to linear response.

Contribution

It provides a theoretical justification for the creep law using an effective model, including effects of disorder correlations and finite-size crossover phenomena.

Findings

The model reproduces the creep law velocity-force relationship.

It captures the influence of disorder correlations on low-temperature behavior.

It characterizes the crossover from creep to linear response regimes.

Abstract

The response of spatially extended systems to a force leading their steady state out of equilibrium is strongly affected by the presence of disorder. We focus on the mean velocity induced by a constant force applied on one-dimensional interfaces. In the absence of disorder, the velocity is linear in the force. In the presence of disorder, it is widely admitted, as well as experimentally and numerically verified, that the velocity presents a stretched exponential dependence in the force (the so-called 'creep law'), which is out of reach of linear response, or more generically of direct perturbative expansions at small force. In dimension one, there is no exact analytical derivation of such a law, even from a theoretical physical point of view. We propose an effective model with two degrees of freedom, constructed from the full spatially extended model, that captures many aspects of the creep phenomenology. It provides a justification of the creep law form of the velocity-force characteristics, in a quasistatic approximation. It allows, moreover, to capture the non-trivial effects of short-range correlations in the disorder, which govern the low-temperature asymptotics. It enables us to establish a phase diagram where the creep law manifests itself in the vicinity of the origin in the force--system-size--temperature coordinates. Conjointly, we characterise the crossover between the creep regime and a linear-response regime that arises due to finite system size.