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

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