Microscopic equation for growing interfaces in quenched disordered media

TL;DR

This paper derives a microscopic equation for the growth of interfaces in quenched disordered media, specifically for the Tang and Leschhorn model, highlighting the interplay between diffusion and substrate effects.

Contribution

It introduces a simple derivation of the microscopic evolution equations for interface height, mean height, and roughness, including an equation for interface activity density.

Findings

Equations reveal the interplay between diffusion and substratum effects.

Provides a new microscopic framework for interface growth in quenched media.

Simplifies the derivation of key interface evolution equations.

Abstract

We present the microscopic equation of growing interface with quenched noise for the Tang and Leschhorn model [L. H. Tang and H. Leschhorn, Phys. Rev. A {\bf 45}, R8309 (1992)]. The evolution equation for the height, the mean height, and the roughness are reached in a simple way. An equation for the interface activity density (or free sites density) as function of time is obtained. The microscopic equation allows us to express these equations into two contributions: the diffusion and the substratum contributions. All these equations shows the strong interplay between the diffusion and the substratum contribution in the dynamics.