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

T G Mattos, J G Moreira, A P F Atman
2007 Journal of Physics A: Mathematical and Theoretical  
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)=ρ exp[κΓ_i(t)] for a site i to receive one particle at a time t and all the sites are updated simultaneously. Here ρ and κ are two parameters and Γ_i(t) is a function which depends on height of the site i and its neighbors. Its functional form is
more » ... ied 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 κ is varied.
doi:10.1088/1751-8113/40/44/006 fatcat:tkqt4t56bvcjdkkwgmfhoof2wy