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A stochastic algorithm for simulation of fluctuation-induced kinetics ofH$_2$ formation on grain surfaces is suggested as a generalization of thetechnique developed in our recent studies where this method was developed todescribe the annihilation of spatially separate electrons and holes in adisordered semiconductor. The stochastic model is based on the spatiallyinhomogeneous, nonlinear integro-differential Smoluchowski equations withrandom source term. In this paper we derive the general system of Smoluchowskitype equations for the formation of H$_2$ from two hydrogen atoms on thesurface of interstellar dust grains with physisorption and chemisorption sites.We focus in this study on the spatial distribution, and numerically investigatethe segregation in the case of a source with a continuous generation in timeand randomly distributed in space. The stochastic particle method presented isbased on a probabilistic interpretation of the underlying process as astochastic Markov process of interacting particle system in discrete butrandomly progressed time instances. The segregation is analyzed through thecorrelation analysis of the vector random field of concentrations which appearsto be isotropic in space and stationary in time.

A stochastic model and Monte Carlo algorithm for fluctuation-induced H$_2$ formation on the surface of interstellar dust grains