Eur. Phys. J. Special Topics 223, 3045–3062 (2014)
https://doi.org/10.1140/epjst/e2014-02318-4

Regular Article

Diffusion coefficients for two-dimensional narrow asymmetric channels embedded on flat and curved surfaces

¹ Physics Department, Universidad Autónoma Metropolitana Iztapalapa, San Rafael Atlixco 186, México D.F. 09340, Mexico
² Applied Mathematics and Systems Department, Universidad Autónoma Metropolitana Cuajimalpa, Vasco de Quiroga 4871, México D.F. 05348, Mexico

^a e-mail: dll@xanum.uam.mx

Received: 17 October 2014
Revised: 5 November 2014
Published online: 15 December 2014

Abstract

This paper focuses on the derivation of a general position-dependent diffusion coefficient to describe the two-dimensional (2D) diffusion in a narrow and smoothly asymmetric channel of varying cross section and non-straight midline embedded in a flat or on a curved surface. We consider the diffusion of non-interacting point-like Brownian particles under no external field. In order to project the 2D diffusion equation into an effective one-dimensional generalized Fick-Jacobs equation in both, flat and curved manifolds using the generalization of the mapping procedure introduced by Kalinay and Percus. The expression obtained is the more general position-dependent diffusion coefficient for 2D narrow channels that lies in a plane, which contains all the well-known previous results both symmetric and asymmetric channels as special cases. In a straightforward manner, previously defining the corresponding Fick-Jacobs equation on a curved surface, this result can be generalized to the case of a narrow 2D channel embedded on a no-flat smooth surface where the full position-dependent diffusion coefficient is modified according to the metric elements that accounts for the curvature of the surface. In addition, the equations for the mean first-passage time are obtained for asymmetrical channels on curved surfaces. As an example we shall solve this equation for the case of an asymmetric channel defined by straight walls embedded on a cylindrical surface having a reflecting wall at the origin and an absorbent one at distance θ_L.