Unified continuum description for sub-diffusion random walks on multi-dimensional comb model

The anisotropic sub-diffusion random walks on multi-dimensional comb structure model have been studied in the continuum approximation. The problem is that in the considered model mean square displacements on different directions have different power temporal dependencies. So this case essentially differs from usual anisotropic diffusion case and it is not obviously how to describe such a diffusion by a unified way. Nevertheless the unified continuum description for such random walks has been developed. As result the new generalization of Fick’s law for diffusion current has been obtained for anisotropic anomalous sub-diffusion random walks in multi-dimensional case. Namely, it has been shown that the effective diffusion coefficient in the generalized Fick’s law has new operator form: components of diffusion tensor instead of constant values have the fractional order temporal derivatives.