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In inhomogeneous environments, the correct expression of the diffusive fluxis often not given by the Fick's law $\Gamma = - D \nabla n $. The most generalhydrodynamic equation modelling diffusion is indeed the Fokker-Planck Equation(FPE). The microscopic dynamics of each specific system may affect the form ofthe FPE, either establishing connections between the diffusion and theconvection term, as well as providing supplementary terms. In particular, theFick's form for the Diffusion Equation may arise only in consequence of aspecific kind of microscopic dynamics. It is also shown how, in the presence ofsharp inhomogeneities, even the hydrodynamic FPE limit may becomes inaccurateand mask some features of the true solution, as computed from the MasterEquation.

Fick's law and Fokker-Planck Equation in inhomogeneous environments