Classical diffusion in channels with a spatially varying cross-section

We study the diffusion of classical particles in channels with varying boundaries. The problem is characterized by the Neumann boundary condition (zero normal current) in contrast to the Dirichlet boundary condition (zero function) for "quantum confinement" problems. Eliminating transverse modes, we derive an effective diffusion equation that describes particle propagation in the space of reduced dimension in the presence of a frozen drift field. The latter stems from boundary variations of the original boundary problem. Boundary variations may thus result in an appreciable change of the particle transport and, in particular, in a nonlinear response to an external field. We show also that there is a difference between the nonlinear responses of open and closed channels.