Elliptic equation for random walks. Application to transport in microporous media

We consider a process of random walks with arbitrary residence time distribution. We show that in many cases this process may not be described by the classical (Fick) parabolic diffusion equation, but an elliptic equation. An additional term proportional to the second time derivative takes into account the distribution of the residence times of molecules in pores. The new elliptic diffusion equation is strictly derived by the operator approach. A criterion showing where the new equation should be applied instead of the standard diffusion equation is obtained. Boundary conditions are studied and a principle for selection of a unique bounded solution is formulated. Fundamental solutions are obtained and compared with the results of direct simulation of the random walks.