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

TL;DR

This paper examines how in inhomogeneous environments, the traditional Fick's law may not accurately describe diffusion, emphasizing the importance of the Fokker-Planck Equation and microscopic dynamics in modeling such systems.

Contribution

It clarifies the conditions under which Fick's law applies and highlights the limitations of the Fokker-Planck Equation in sharply inhomogeneous environments.

Findings

Fick's law may not hold in inhomogeneous environments.

The Fokker-Planck Equation's accuracy depends on microscopic dynamics.

Sharp inhomogeneities can invalidate hydrodynamic approximations.

Abstract

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