The diffusion behavior of a Brownian particle in a crystal with randomly distributed topological defects was analyzed by means of the continuum theory of defects, combined with the theory of diffusion on manifolds. A path-integral representation of the diffusion process was used to calculate the cumulants of the particle position, as averaged over a defect ensemble. For a random distribution of disclinations, the problem of Brownian motion reduced to a known random-drift problem. Depending upon the properties of the disclination ensemble, this yielded various types of sub-diffusional behavior. In the case of a random array of parallel screw dislocations, it was found that there was a normal but anisotropic diffusion behavior of the mean-square displacement. The process was non-Gaussian, and exhibited long-time tails in the higher-order cumulants.

R.Bausch, R.Schmitz, L.A.Turski: Zeitschrift für Physik B, 1995, 97[2], 171-7