Self-similar grain size distribution in two dimensions: Analytical solution

Consideration of the physics and topology of two-dimensional grain growth suggests that a stochastic treatment is required to determine grain size distribution [Pande CS. Acta Metall 1987;35:2671]. In this paper, a size-based continuum stochastic formulation is presented based on topological considerations. As expected, this analysis leads to a Fokker–Planck equation for the size distribution, which should yield a unique self-similar asymptotic state that could be reached from any arbitrary initial state. The approximate solution of the Fokker–Planck equation presented here is limited to two dimensions and is based on the assumption of quasi-stationary distributions reached in the long time limit. The resulting grain size distribution is shown to be in agreement with that obtained from computer simulations, indicating the validity of the stochastic approach.