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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.

We numerically investigate random walks (RWs) and self-avoiding random walks (SAWs) on critical percolation clusters, basic models for diffusion and flexible polymers in disordered media. While this can be easily done for RWs using a simple enumeration method, it is difficult for long SAWs due to the long-range correlations. We employed a sophisticated algorithm that makes use of the self-similar structure of the critical clusters and allows exact enumeration of several thousand SAW steps. We also investigate a kinetic version of the SAW, the so-called kinetic growth (self-avoiding) walk (KGW), as well static averaging over all RW conformations, which describes the so-called ideal chain. For the KGW, we use a chain-growth Monte Carlo method which is inspired by the pruned-enriched Rosenbluth method. The four walk types are found to be affected in different ways by the fractal, disordered structure of the critical clusters. The simulations were carried out in two and three dimensions.