Ricerc@Sapienza

We discuss particle diffusion in a spatially inhomogeneous medium. From the micro-
scopic 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 equa-
tion 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 inho-
mogeneous 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.