Langevin equations for competitive growth models

TL;DR

This paper derives Langevin equations for one-dimensional competitive growth models, revealing how surface tension and nonlinear coefficients depend on the probability of correlated deposition, with results matching simulations and scaling theories.

Contribution

It introduces a systematic derivation of Langevin equations for competitive growth models, highlighting the role of step function expansions and their scaling with the probability p.

Findings

Surface tension  p^2 for RD-RDSR models

Nonlinear coefficient  p^{3/2} for RD-BD models

Linear p-dependence of  p for RDSR-BD model

Abstract

Langevin equations for several competitive growth models in one dimension are derived. For models with crossover from random deposition (RD) to some correlated deposition (CD) dynamics, with small probability p of CD, the surface tension \nu and the nonlinear coefficient \lambda of the associated equations have linear dependence on p due solely to this random choice. However, they also depend on the regularized step functions present in the analytical representations of the CD, whose expansion coefficients scale with p according to the divergence of local height differences when p->0. The superposition of those scaling factors gives \nu ~ p^2 for random deposition with surface relaxation (RDSR) as the CD, and \nu ~ p, \lambda ~ p^{3/2} for ballistic deposition (BD) as the CD, in agreement with simulation and other scaling approaches. For bidisperse ballistic deposition (BBD), the same scaling of RD-BD model is found. The Langevin equation for the model with competing RDSR and BD, with probability p for the latter, is also constructed. It shows linear p-dependence of \lambda, while the quadratic dependence observed in previous simulations is explained by an additional crossover before the asymptotic regime. The results highlight the relevance of scaling of the coefficients of step function expansions in systems with steep surfaces, which is responsible for noninteger exponents in some p-dependent stochastic equations, and the importance of the physical correspondence of aggregation rules and equation coefficients.