An analytical approximation to the diffusion coefficient in overdamped multidimensional systems

An analytical approximation for the mobility of an overdamped particle in a periodic multi-dimensional system is presented. Attention is focused on two dimensions (quasi-2D approximation) in the most generic case of a 2D-coupled periodic potential in a rectangular lattice and of a position-dependent friction. The approximation is derived in the framework of the Linear Response Theory by fixing the value of one coordinate and solving the problem of diffusion along the other coordinate as strictly 1D. This is expected to be essentially correct if all the most relevant diffusion paths are straight lines. Two different specific applications have been considered: diffusion in a square egg-carton potential and diffusion in absence of potential in a 2D channel with unsurmountable periodic walls. Exact results are available in literature in the latter case and are obtained in the first case by solving the Smoluchowski equation (matrix continued fraction method). Comparisons with the quasi-2D approximation show that the agreement is excellent for the egg-carton potential but far less satisfying for migration in the 2D periodically shaped channel, characterized by important diffusion paths not being straight lines.