On the covariant description of diffusion in two-dimensional confined environments

A covariant description of diffusion of point-size Brownian particles in bounded geometries is presented. To this end, we provide a formal theoretical framework using differential geometry. We propose a coordinate transformation to map the boundaries of a general two-dimensional channel into a straight channel in a non-Cartesian geometry. The new shape of the boundaries naturally suggests a reduction to one dimension. As a consequence of this coordinate transformation, the Fick equation with boundary conditions transforms as a generalized Fick-Jacobs-like equation, in which the leading-order term is equivalent to the Fick-Jacobs approximation. The expression for the effective diffusion coefficient derived here depends on the position and the derivatives of the channel's width and centerline. Finally, we validate our analytic predictions for the effective diffusion coefficients for two periodic channels.