The simulation of spinodal decomposition in two dimensions: A comparison of Monte Carlo and Langevin dynamics

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Abstract
Langevin dynamics and Monte Carlo methods have recently been developed for the simulation of spinodal decomposition. In this paper we show, using the central limit theorem, that the Monte Carlo method (in the limit of small step sizes) is equivalent to Langevin dynamics (in the high friction limit). In this limit an exact relationship is established between the number of trials in a Monte Carlo simulation and Langevin dynamics time. The relationship between the Monte Carlo and Langevin dynamics methods is demonstrated for practical simulations and the effects of deviations from the small step size limit in the Monte Carlo simulation is a linear distortion of the effective time scale, provided the step size is not too large. The equivalence of the Monte Carlo and Langevin dynamics methods (in the appropriate limits) are demonstrated for the case of spinodal decomposition in two dimensions. However, our theoretical analysis is quite general and is not restricted to spinodal decomposition or two-dimensional systems.