Fick and Fokker--Planck diffusion law in inhomogeneous media

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.

References in corpus (7)

• Fick's law and Fokker-Planck Equation in inhomogeneous environments
• Microscopic models for uphill diffusion
• Latent heat and the Fourier law
• Non-equilibrium 2D Ising model with stationary uphill diffusion
• Particle models with self sustained current
• Stationary uphill currents in locally perturbed Zero Range Processes
• A lattice model of reduced jamming by barrier

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• Thermodynamics of a Brownian particle in a non-confining potential
• Conditional expectation of the duration of the classical gambler problem with defects
• Transport in quantum multi-barrier systems as random walks on a lattice
• Multilayer heat equations and their solutions via oscillating integral transforms