Diffusion tensor for atomic migration in crystals

The diffusion of holes or interstitials in a crystal is treated by several types of asymptotic analyses, of which the starting point is the Fokker-Planck equation. First, we show that the standard Smoluchowski equation holds, for large as well as intermediate or small values of the interatomic potential $\Phi$, as an approximation of the Fokker-Planck equation. Next, we derive from the Smoluchowski equation a "homogenized" Smoluchowski equation, which contains only constant coefficients. Finally, we expand these coefficients for large $\Phi$ and obtain explicit formulas for the diffusion coefficients in terms of $\Phi$ and other physical quantities.