Projection of two-dimensional diffusion in a curved midline and narrow varying width channel onto the longitudinal dimension.

This study focuses on the derivation of a general effective diffusion coefficient to describe the two-dimensional (2D) diffusion in a narrow and smoothly asymmetric channel of varying width, in the simple diffusional motion of noninteracting pointlike particles under no external field. We present a generalization to the case of an asymmetric channel using the projection method introduced earlier by Kalinay and Percus [J. Chem. Phys. 122, 204701 (2005); and Phys. Rev. E 74, 041203 (2006)] to project the 2D diffusion equation into an effective one-dimensional generalized Fick-Jacobs equation. The expression for the diffusion coefficient given in Eq. (23) is our main result. This expression is a more general effective diffusion coefficient for narrow channels in 2D, which contains the well-known previous results as special cases, namely, those obtained by Bradley [Phys. Rev. E 80, 061142 (2009)], and more recently by Berezhkovskii and Szabo [J. Chem. Phys. 135, 074108 (2011)]. Finally, we study some specific 2D asymmetric channel configurations to test and show the broader applicability of Eq. (23).