Effective one-dimensional diffusion on curved surfaces: Catenoid and pseudosphere

We present the effective diffusion coefficient for the diffusion in a narrow generally asymmetric channel embedded on a curved surface, in the case of simple diffusion of pointlike particles without interaction and under no external forces. First, we define the diffusion equation for anisotropic diffusion involving a version of the Laplace-Beltrami operator. Then, we choose symmetric surfaces whose metric components only depend on one of the local coordinates and thus, apply the Kalinay-Percus’ projection method. With this method one can project two-dimensional anisotropic diffusion into the corresponding effective one-dimensional generalized Fick-Jacobs equation to the lowest order. The perturbation series to all orders converges and as a general result the effective diffusion coefficient on a curved surface depending on the longitudinal local coordinate was obtained and is presented. It contains metric terms that can be related with the Gaussian curvature of the surface. We illustrate our results by studying asymmetric conical channel configurations on two surfaces, namely, the catenoid that is a minimal surface, and the pseudosphere that is a surface with negative constant curvature.