Dynamic Scaling of Width Distribution in Edwards--Wilkinson Type Models of Interface Dynamics

TL;DR

This paper derives exact and approximate scaling forms for the distribution of interface width in Edwards--Wilkinson models, revealing how initial conditions affect the dynamics and confirming results with Monte Carlo simulations.

Contribution

It provides an exact calculation of the width distribution in 1+1D Edwards--Wilkinson models and explores the effects of initial conditions on scaling behavior.

Findings

Exact scaling form for width distribution with flat initial condition

Scaling depends on initial amplitude for complex initial states

Distribution approximates a log-normal form at short times

Abstract

Edwards--Wilkinson type models are studied in 1+1 dimensions and the time-dependent distribution, P_L(w^2,t), of the square of the width of an interface, w^2, is calculated for systems of size L. We find that, using a flat interface as an initial condition, P_L(w^2,t) can be calculated exactly and it obeys scaling in the form <w^2>_\infty P_L(w^2,t) = Phi(w^2 / <w^2>_\infty, t/L^2) where <w^2>_\infty is the stationary value of w^2. For more complicated initial states, scaling is observed only in the large- time limit and the scaling function depends on the initial amplitude of the longest wavelength mode. The short-time limit is also interesting since P_L(w^2,t) is found to closely approximate the log-normal distribution. These results are confirmed by Monte Carlo simulations on a `roof-top' model of surface evolution.