Curvature Dependent Diffusion Flow on Surface with Thickness

Particle diffusion in a two dimensional curved surface embedded in $R_3$ is considered. In addition to the usual diffusion flow, we find a new flow with an explicit curvature dependence. New diffusion equation is obtained in $\epsilon$ (thickness of surface) expansion. As an example, the surface of elliptic cylinder is considered, and curvature dependent diffusion coefficient is calculated.