Diffusion of point-like particles driven by a constant longitudinal force in two-dimensional channels of periodically varying width is studied. The dynamics of such systems can be effectively described by the one-dimensional Smoluchowski(-Fick-Jacobs) equation in the longitudinal coordinate x, extended by a space dependent effective diffusion coefficient D(x). Our paper is focused on calculation of this function for an arbitrary channel shaping function h(x). Unlike the previous algorithms based on scaling of the transverse lengths, the method presented here uses periodicity of the channel. Instead of complicated expansion containing higher order derivatives of h(x), the proposed algorithm results in an integral formula for D(x), enabling us to study the system for wide range of the driving force and various (periodic) shaping functions h(x).
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