Projection of two-dimensional diffusion in a curved midline and narrow varying width channel embedded on a curved surface

This study focuses on the derivation of a general effective diffusion coefficient to describe the twodimensional (2D) diffusion in a narrow and smoothly asymmetric channel of varying width that lies on a curved surface, in the simple diffusional motion of noninteracting point-like particles under no external field. To this end we extend the generalization of the Kalinay-Percus’ projection method [J. Chem. Phys. 122, 204701 (2005); Phys. Rev. E 74, 041203 (2006)] for the asymmetric channels introduced in [J. Chem. Phys. 137, 024107 (2012)], to project the anisotropic 2D diffusion equation on a smooth curved manifold into an effective one-dimensional generalized Fick-Jacobs equation which is modified due to the curvature of the surface. The lowest order in the perturbation parameter, corresponding to the Fick-Jacobs equation, contains an extra term that accounts for the curvature of the surface. We found explicitly the first order correction for the invariant effective concentration, which is defined as the correct marginal concentration in one variable, and we obtain the first approximation to the effective diffusion coefficient analogous to Bradley’s coefficient [Phys. Rev. E 80, 061142 (2009)] as a function of metric elements of the surface. Straightforwardly we study the perturbation series up to the n-th order, and we derive the full effective diffusion coefficient for 2D diffusion in a narrow asymmetric channel, which have modifications due to the curved metric. Finally, as an example we show how to use our formula to calculate the effective diffusion coefficient considering the case of an asymmetric conical channel embedded on a torus.