A discrete method to study stochastic growth equations: a cellular automata perspective

Using cellular automata dynamics, a discrete technique to study stochastic growth equations (SGE) is presented. By analogy to deposition models in which the growth rule depends on height differences between neighbours, we introduce an interface growth process with synchronous updating in which the transition probability for a given site i to receive a particle at a time t is defined as pi(t) = ρexp [κΓi(t)]. ρ and κ are the model parameters and Γi(t) is a function which depends on the height of the site i and its neighbours, and its functional form is specified through discretization of the deterministic part of the growth equation associated with a given deposition process. To validate the method, we study its application to two linear SGE—the Edwards–Wilkinson equation and the Mullins–Herring equation, and a nonlinear one—the Kardar–Parisi–Zhang equation. The statistical analysis of the height distributions in simulations recovered the correct values for roughening exponents, confirming that the processes generated are indeed in the universality classes of the original growth equations. We also observed a crossover from random deposition to the correlated regime when the parameter κ is varied in each case studied.