A new method to study stochastic growth equations: Application to the Edwards-Wilkinson equation

Abstract

In this work we introduce a method to study stochastic growth equations, which follows a dynamics based on cellular automata modeling. The method defines an interface growth process that depends on height differences between neighbors. The growth rules assign a probability pi(t) for site i to receive a particle at time t, where pi(t) = ρ exp[κΓl(t)]. Here ρ and κ are two parameters and Γl(t) is a kernel that depends on height hl(t) of site i and on heights of its neighbors, at time t. We specify the functional form of this kernel by the discretization of the deterministic part of the equation that describes some growth process. In particular, we study the Edwards-Wilkinson (EW) equation which describes growth processes where surface relaxation plays a major role. In this case we have a Laplacian term dominating in the growth equation and Γi(t) = hi+1 (t) + h i-1 (t) - 2hi(t), which follows from the discretization of ∇2h. By means of simulations and statistical analysis of the height distributions of the profiles, we obtain the roughening exponents, β, α and z, whose values confirm that the processes defined by the method are indeed in the universality class of the original growth equation.