Dichotomic ratchet in a two-dimensional corrugated channel

We consider a particle diffusing in a two-dimensional (2D) channel of varying width h(x). It is driven by a force of constant magnitude f but random orientation there or back along the channel. We derive the effective generalized Fick-Jacobs equation for this system, which describes the dynamics of such a particle in the longitudinal coordinate x. Aside from the effective diffusion coefficient D(x), our mapping also generates an additional effective potential -γ(x) added to the entropic potential -log[h(x)]. It acquires an increasing or decreasing component in asymmetric periodic channels, and thus it explains appearance of the ratchet current. We study this effect on a trial example and compare the results of our true 2D theory with a commonly used effective one-dimensional description; the data are verified by the numerical solution of the full 2D problem.