Mean field theory for skewed height profiles in KPZ growth processes

TL;DR

This paper develops a mean field theory for 1+1 dimensional KPZ interface growth, capturing key features like skewed profiles and crossover behavior, while simplifying spatial correlations.

Contribution

It introduces a mean field approach that accounts for nonlinear effects and surface tension, accurately reproducing qualitative features of KPZ growth.

Findings

Correctly reproduces Edwards-Wilkinson features

Shows crossover to KPZ behavior with z=3/2

Captures skewed interface profiles and wall effects

Abstract

We propose a mean field theory for interfaces growing according to the Kardar-Parisi-Zhang (KPZ) equation in 1+1 dimensions. The mean field equations are formulated in terms of densities at different heights, taking surface tension and the influence of the nonlinear term in the KPZ equation into account. Although spatial correlations are neglected, the mean field equations still reflect the spatial dimensionality of the system. In the special case of Edwards-Wilkinson growth,our mean field theory correctly reproduces all features. In the presence of a nonlinear term one observes a crossover to a KPZ-like behavior with the correct dynamical exponent $z=3/2$. In particular we compute the skewed interface profile during roughening, and we study the influence of a co-moving reflecting wall, which has been discussed recently in the context of nonequilibrium wetting and synchronization transitions. Also here the mean field approximation reproduces all qualitative features of the full KPZ equation, although with different values of the surface exponents.