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

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 p i(t) for site i to receive a particle at time t, where p i(t) = rho exp[kGi(t)]. Here r and k are two parameters and gammai(t) is a kernel that depends on height h i(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 gammai(t) = h i+1(t)+h i-1(t)-2h i(t), which follows from the discretization of Ñ2h. By means of simulations and statistical analysis of the height distributions of the profiles, we obtain the roughening exponents, beta, alpha and z, whose values confirm that the processes defined by the method are indeed in the universality class of the original growth equation