No abstract

Diffusion in a narrow two-dimensional channel of width A(x), depending on the longitudinal coordinate x, is the object of our study. We show how the 2+1 dimensional diffusion equation can be projected onto a 1+1 dimensional one, governing corresponding one-dimensional density, in a steady-state approximation. Then we demonstrate the method on a nontrivial exactly solvable case for A(x)=x and discuss projection of the initial condition.

Diffusion in a quasi-one-dimensional channel, with cross section varying along the longitudinal coordinate, is considered. Using a rigorous mapping of the diffusion equation onto one dimension, eliminating transients in transverse direction(s), we derive an expansion of the effective diffusion coefficient D(x), which represents corrections to the Fick-Jacobs equation.

We derive an extended Fick-Jacobs equation for the diffusion of noninteracting particles in a two- and symmetric three-dimensional channels of varying cross section A(x), using a variational approach. The result is a diffusion differential equation of second order in only one space (longitudinal) coordinate. This equation is tested on the task of calculating the stationary flux through a hyperboloidal tube, and its solution is compared with that of other methods.