Distribution Functions for the Number of Distinct Sites Visited in a Random Walk on Cubic Lattices: Relation to Defect Annealing

Distribution functions for the number of distinct sites $S\left(n\right)\mathrm{}$ visited by a point defect executing a symmetric random walk of $n$ jumps on two- and three-dimensional lattices were computed using the Monte Carlo method. The square planar, simple cubic, bcc, and fcc lattices were treated. In three dimensions, the normal distribution appears to describe $S\left(n\right)\mathrm{}$ for $n>\mathrm{}{10}^4$ jumps and at ${10}^4$ jumps the derivative $\frac{d\overline{S}\mathrm{}\left(n\right)\mathrm{}}{\mathrm{dt}}$ of the average number, $\overline{S}\mathrm{}\left(n\right)\mathrm{}$, of distinct sites is within 0.5% of the value given by Vineyard's exact asymptotic solution. The defect annealing rate was computed using the $S\left(n\right)\mathrm{}$ distribution in a simple example and this result compared with an analog Monte Carlo solution. The comparison indicated that fluctuations in the initial defect concentration must be considered in computing the initial annealing rate and the mobile defect concentration as a function of time. After 500 jumps the annealing rate, but not the concentration, can be closely approximated without accounting for fluctuations in the initial concentration.