Homogenization, Symmetry, and Periodization in Diffusive Random Media

We present a systematic study of homogenization of diffusion in random media with emphasis on tile-based random microstructures. We give detailed examples of several such media starting from their physical descriptions, then construct the associated probability spaces and verify their ergodicity. After a discussion of material symmetries of random media, we derive criteria for the isotropy of the homogenized limits in tile-based structures. Furthermore, we study the periodization algorithm for the numerical approximation of the homogenized diffusion tensor and study the algorithm's rate of convergence. For one dimensional tile-based media, we prove a central limit result, giving a concrete rate of convergence for periodization. We also provide numerical evidence for a similar central limit behavior in the case of two dimensional tile-based structures.