A One Dimensional Stochastic Model of Coarsening

Abstract

A one dimensional model of the coarsening of intervals on a line is considered in which the boundary points between adjacent intervals execute independent random walk with a common diffusion coefficient D/2; when two boundary points meet, they coalesce into a single point that continues to execute random walk. We calculate the following quantities in the asymptotic limit of long times: 1) The average interval length (i.e., $\;\left\langle l \right\rangle = \sqrt {\pi Dt}$ , 2) the time-independent probability density for the reduced length $\sigma = 1/\left\langle l \right\rangle$ , and 3) the expected value of dl/dt for a given l, which is positive for and negative for $l > {l_c} = \sqrt {2/\pi } \left\langle l \right\rangle$ and negative for l < l _c . The model is similar to one proposed by Louat for grain growth. Although it is not a good representation of the details of most physical processes of coarsening, it is of theoretical interest since it is one of the few cases for which analytic results can be obtained.