The non-linear diffusion equation was solved as a function of temperature, assuming a time-dependent diffusion coefficient. The resultant analytical solutions described the diffusion profiles for increasing temperatures, and for the case of excited systems; where vacancies and impurity atoms were not in thermal equilibrium with the lattice. It was noted that the relationship between the temperature function and the population of excited states of atoms surrounding vacancies allowed for the possibility of super-diffusion.

The Nonlinear Diffusion in the Nonisothermal Case. A.J.Janavicius: Acta Physica Polonica A, 1998, 93[3], 505-12