Statistical theory of effective electrical, thermal, and magnetic properties of random heterogeneous materials. III. Perturbation treatment of the effective permittivity in completely random heterogeneous materials

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Abstract
Perturbation expansion series are derived for the effective permittivity of completely random heterogeneous materials. The formulation is performed by regarding a completely random material as a limiting case of an isotropic cell material. It is emphasized that, in order to obtain a physically reasonable and mathematically correct result, the ``exclusion effect'' must be taken into account in the averaging procedure. Prescription for evaluating the perturbation coefficient of an arbitrary order is given and explicit forms of leading terms are presented. The results bear a wide variety of applications in calculating effective physical constants such as dielectric constant, magnetic permeability, electrical and thermal conductivity, and diffusion constant. It is mentioned that the idea and formulation in this article are important for more general approximations (to be studied in the succeeding Paper IV) especially in connection with the problem of electron localization in some disordered systems.