Kinetic theory of inhomogeneous diffusion

This paper theoretically investigates particle diffusion in a medium where the diffusivity depends on position. We exclusively consider continuous-time, continuous-space transport and our working tool is the linear kinetic theory pertinent to guest particles in a passive host medium (Lorentz’s picture of transport). The host medium may or may not thermalize the guest particles. It may be inhomogeneous in two ways: either particle scattering features depend on position in an explicit way (geometric inhomogeneity), or they depend on position through the medium’s local temperature (thermal inhomogeneity). When the inhomogeneity is geometric, it is found that Fick’s law is valid and the particle-current equation exhibits drift without current. When the inhomogeneity is thermal, current without drift is possible, but there is no generally valid pattern for the current equation. The consistency of our results with non-equilibrium thermodynamics is brought out. The results shed light on thermodiffusion (the Ludwig–Soret effect), which often combines inhomogeneities of both kinds. Finally, a limitation of the Lorentz picture of transport in accounting for thermodiffusion is outlined.