Revisiting random deposition with surface relaxation: approaches from growth rules to Edwards-Wilkinson equation

TL;DR

This paper explores multiple methods to derive the Edwards-Wilkinson equation from the RDSR model, clarifying the roles of deposition, diffusion, and volume conservation in surface growth dynamics.

Contribution

It introduces a novel procedure to decompose transition moments and derives the Langevin equation coefficients using three distinct approaches, enhancing understanding of surface relaxation models.

Findings

The diffusion process relates to antisymmetric contributions.

Volume conservation renormalizes to zero in the coarse-grained limit.

The methods are applicable to other discrete models with relaxation.

Abstract

We present several approaches for deriving the coarse-grained continuous Langevin equation (or Edwards-Wilkinson equation) from a random deposition with surface relaxation (RDSR) model. First we introduce a novel procedure to divide the first transition moment into the three fundamental processes involved: deposition, diffusion and volume conservation. We show how the diffusion process is related to antisymmetric contribution and the volume conservation process is related to symmetric contribution, which renormalizes to zero in the coarse-grained limit. In another approach, we find the coefficients of the continuous Langevin equation, by regularizing the discrete Langevin equation. Finally, in a third approach, we derive these coefficients from the set of test functions supported by the stationary probability density function (SPDF) of the discrete model. The applicability of the used approaches to other discrete random deposition models with instantaneous relaxation to a neighboring site is discussed.