Diffusion with drift on a finite line

Abstract

The authors consider diffusion in the presence of drift (caused, e.g. by a constant external field) on a finite continuous line terminated by reflecting boundaries. Exact analytical expressions are obtained for the positional probability density and the generalised diffusion coefficient (or the dynamic mobility), and thence the frequency-dependent dielectric response. For a given ratio of the diffusion and drift time scales, the generalized diffusivity is shown to be a universal function of omega tau L where tau L is the diffusion time on a strand of length L. They also present an exact solution for the strand-length-averaged diffusion coefficient in the absence of drift, which is the continuum analogue of the corresponding solution in the bond-percolation model. Finally, they apply their results to spectral diffusion and recover the nonDebye pulse relaxation known to occur in certain time regimes owing to the interplay of diffusion and drift.