Self-similar grain size distribution in three dimensions: A stochastic treatment

In this paper, a stochastic formulation of three-dimensional grain growth is presented. This formulation employs the recent extension of the von Neumann law to three dimensions, and leads to a Fokker-Planck equation for the size distribution. The self-similar solutions of the Fokker-Planck equation presented here are based on the assumption of quasi-stationary distributions reached in the long time limit. The resulting grain size distributions, obtained both numerically and analytically, are shown to be in good agreement with each other and also with those obtained from computer simulations, indicating the validity of the stochastic approach.