Fick and Fokker--Planck diffusion law in inhomogeneous media

TL;DR

This paper explores the microscopic and macroscopic aspects of diffusion in inhomogeneous media, deriving the Fokker-Planck equation and Fick's law from geometric and probabilistic principles, and discussing related properties like Einstein relation and uphill diffusion.

Contribution

It provides a geometric interpretation of reversibility and derives macroscopic diffusion laws from microscopic particle dynamics in inhomogeneous media.

Findings

Reversibility corresponds to a geometric condition involving weights on edges and vertices.

The Fokker-Planck equation is obtained as the macroscopic limit of microscopic particle dynamics.

Fick's law with inhomogeneous diffusion matrix is derived under specific conditions.

Abstract

We discuss diffusion of particles in a spatially inhomogeneous medium. From the microscopic viewpoint we consider independent particles randomly evolving on a lattice. We show that the reversibility condition has a discrete geometric interpretation in terms of weights associated to un--oriented edges and vertices. We consider the hydrodynamic diffusive scaling that gives, as a macroscopic evolution equation, the Fokker--Planck equation corresponding to the evolution of the probability distribution of a reversible spatially inhomogeneous diffusion process. The geometric macroscopic counterpart of reversibility is encoded into a tensor metrics and a positive function. The Fick's law with inhomogeneous diffusion matrix is obtained in the case when the spatial inhomogeneity is associated exclusively with the edge weights. We discuss also some related properties of the systems like a non-homogeneous Einstein relation and the possibility of uphill diffusion.