Macroscopic diffusion on rough surfaces

We consider diffusion on rough and spatially periodic surfaces. The macroscopic diffusion tensor D is determined by averaging the local fluxes over the unit cell. D is proved to be the unit tensor for macroscopically isotropic surfaces. For general surfaces, an asymptotic analysis is applied, when the ratio of the oscillation amplitude to the size of the unit cell is a small parameter epsilon. The microscopic field is determined up to O(epsilon(6)) in analytical form and an algorithm is derived to calculate higher order terms. We also deduce general analytical formulas for D up to O(epsilon(6)) and derive an algorithm to compute D as a series in epsilon(2).