Approximations of the generalized Fick-Jacobs equation

We analyze the generalized Fick-Jacobs equation, obtained by a rigorous mapping of the diffusion equation in a quasi-one-dimensional (quasi-1D) (narrow 2D or 3D) channel with varying cross section A (x) onto the longitudinal coordinate x. We show that for constructing approximations and understanding their applicability in practice, it is crucial to study the 2D (3D) density inside the channel in the regime of stationary flow. We present algorithms enabling us to derive approximate formulas for the effective diffusion coefficient involving derivatives of A (x) higher than A′ (x) and give examples for 2D channels. Effects of the boundary conditions at the ends of a finite channel and the case of nonsmooth A (x) are also discussed.