Effective one-dimensional description of confined diffusion biased by a transverse gravitational force

Diffusion of pointlike noninteracting particles in a two-dimensional channel of varying cross section is considered. The particles are biased by a constant force in the transverse direction. A recurrence mapping procedure is applied, which enables the derivation of an effective one-dimensional (1D) evolution equation that governs the 1D density of the particles in the channel. In the limit of stationary flow, an extended Fick-Jacobs equation is reached, which is corrected by an effective diffusion coefficient D(x) that depends on the longitudinal coordinate x. The result is an approximate formula for D(x) that also involves the influence of the transverse force. The calculations are verified by the stationary diffusion in a linear cone, which is exactly solvable.