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

Diffusion of point-like non interacting particles in a two-dimensional (2D) channel of varying cross section is considered. The particles are biased by a constant force in the transverse direction. We apply our recurrence mapping procedure, which enables us to derive an effective one-dimensional (1D) evolution equation, governing the 1D density of the particles in the channel. In the limit of stationary flow, we arrive at an extended Fick-Jacobs equation, corrected by an effective diffusion coefficient D(x), depending on the longitudinal coordinate x. Our result is an approximate formula for D(x), involving also influence of the transverse force. Our calculations are verified on the stationary diffusion in a linear cone, which is exactly solvable.