Diffusion in inhomogeneous and anisotropic media

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Abstract
Despite common practice, inhomogeneous and/or anisotropic diffusion cannot be considered without regarding the microscopic details breaking the translational and/or angular symmetry. The macroscopic diffusion equation and the stationary solution are determined by the microscopic model and depend in general on all the microscopic parameters and not simply on the combination in the diffusion tensor. The traditional diffusion equation is only valid under special conditions and it cannot, in general, be used for anisotropic diffusion. An alternative form of the diffusion equation has a wider range of applicability. It is shown that for isotropic diffusion all variants of the diffusion equation are mathematically (but not physically) equivalent and can be transformed into each other by introduction of effective potentials. This is not the case for anisotropic diffusion where the traditional diffusion equation in most cases will give incorrect results. Two examples illustrate the differences between the two dynamic equations with respect to stationary solutions and detailed balance.

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