Mapping of forced diffusion in quasi-one-dimensional systems

Diffusion in an external potential in a two-dimensional channel of varying cross section is considered. We show that a rigorous mapping procedure applied on the corresponding Smoluchowski equation yields a one-dimensional evolution equation of the Fick-Jacobs type corrected by an effective coefficient D(x). The procedure enables us to derive this function within a recurrence scheme. We test this result on a model of stationary diffusion in a linear cone in a homogeneous potential, which is exactly solvable. Extension of the approximate formulas for D(x) derived for the diffusion alone is discussed.