The effective one-dimensional description of diffusion in two-dimensional channels of varying cross section is revisited. The effective diffusion coefficient $D\left(x\right)$, extending Fick-Jacobs equation, depending on the longitudinal coordinate $x$, is derived here without use of scaling of the transverse coordinates. The result of the presented method is an integral formula for $D\left(x\right)$, calculating its value at $x$ as an integral of contributions from the neighboring positions $x^{\prime }$ depending on $h\left(x^{\prime }\right)$, a function shaping the channel. Unlike the standard formulas based on the scaling, the new proposed formula also describes $D\left(x\right)$ correctly near the cusps, or in wider channels.

• Received 22 April 2016

DOI:https://doi.org/10.1103/PhysRevE.94.012102