Ballistic and sub-ballistic motion of interfaces in a field of random obstacles

TL;DR

This paper analyzes a discretized model of interface propagation through a random obstacle field, proving conditions for ballistic motion and demonstrating non-ballistic propagation in certain dependent obstacle scenarios.

Contribution

It introduces a nonlinear lattice-based model for interface motion in random fields and establishes conditions for ballistic and non-ballistic propagation.

Findings

Ballistic propagation occurs with sufficient driving force.

No stationary solutions exist for certain dependent obstacle configurations.

Propagation can be non-ballistic despite the absence of stationary states.

Abstract

We consider a discretized version of the quenched Edwards-Wilkinson model for the propagation of a driven interface through a random field of obstacles. Our model consists of a system of ordinary differential equations on a $d$-dimensional lattice coupled by the discrete Laplacian. At each lattice point, the system is subject to a constant driving force and a random obstacle force impeding free propagation. The obstacle force depends on the current state of the solution and thus renders the problem non-linear. For independent and identically distributed obstacle strengths with exponential moment we prove ballistic propagation (i.e., propagation with a positive velocity) of the interface if the driving force is large enough. For a specific case of dependent obstacles, we show that no stationary solution exists, but still the propagation of the front is not ballistic.