Non-steady-state diffusion in two-dimensional periodic channels

The dynamics of a freely diffusing particle in a two-dimensional channel with cross-sectional area A(x) can be effectively described by a one-dimensional diffusion equation under the action of a potential of mean force U(x)=-kBTln[A(x)] (where kBT is the thermal energy) in a system with a spatially dependent diffusion coefficient D(x). Several attempts to derive expressions relating D(x) to A(x) and its derivatives have been made, which were based on considering stationary flows in periodic channels. Here we take an alternative approach and consider non-steady-state single-particle diffusion in an open periodic channel. The approach allows us to express D(x) as a series of terms of increasing powers of ϵ-a parameter associated with the aspect ratio of the channel. When the expansion is truncated at the leading term, we recover the expression suggested by Zwanzig [J. Phys. Chem. 96, 3926 (1992)10.1021/j100189a004] for D(x). Furthermore, comparison of the first few terms in our expansion for D(x) with the one proposed by Kalinay and Percus [Phys. Rev. E 74, 041203 (2006)10.1103/PhysRevE.74.041203] shows that they are consistent with each other. In the limit of long-wavelength channels (ϵ?1), the expansion converges rapidly and the leading approximation provides a very accurate description of the two-dimensional dynamics. For short-wavelength channels, the expansion does not converge and the validity of the effective one-dimensional description is questionable.