Effective Transport Properties of Lattices

The method of multiple scales is used to calculate the effective macroscopic transport propertiesof a random walk on a periodic lattice with arbitrary, spatially-dependent transitionrates. In contrast to the standard multiple-scales approach for continuous media, the fast scalehere is discrete, and only the slow scale is continuous. The solution is found as the discreteprobability distribution of finding the particle at a particular node in the unit cell, modulatedby a continuous slow function of the position of the unit cell in the macroscopic material. Thislast function represents the macroscopic drift-diffusion of the particle, with drift and diffusioncoefficients calculated in terms of the individual transition probabilities