New method to study stochastic growth equations: a cellular automata perspective

TL;DR

This paper presents a cellular automata-based method to simulate and analyze stochastic growth equations, successfully reproducing known universality classes and observing crossover behaviors.

Contribution

The authors introduce a novel cellular automata approach to study stochastic growth equations, capturing universality classes and crossover phenomena.

Findings

Recovered roughening exponents consistent with original equations

Observed crossover from random deposition to correlated regimes

Validated method for linear and non-linear growth equations

Abstract

We introduce a new method based on cellular automata dynamics to study stochastic growth equations. The method defines an interface growth process which depends on height differences between neighbors. The growth rule assigns a probability $p_{i}(t)=\rho$ exp$[\kappa \Gamma_{i}(t)]$ for a site $i$ to receive one particle at a time $t$ and all the sites are updated simultaneously. Here $\rho$ and $\kappa$ are two parameters and $\Gamma_{i}(t)$ is a function which depends on height of the site $i$ and its neighbors. Its functional form is specified through discretization of the deterministic part of the growth equation associated to a given deposition process. In particular, we apply this method to study two linear equations - the Edwards-Wilkinson (EW) equation and the Mullins-Herring (MH) equation - and a non-linear one - the Kardar-Parisi-Zhang (KPZ) equation. Through simulations and statistical analysis of the height distributions of the profiles, we recover the values for roughening exponents, which confirm that the processes generated by the method are indeed in the universality classes of the original growth equations. In addition, a crossover from Random Deposition to the associated correlated regime is observed when the parameter $\kappa$ is varied.