Electromagnetic mixing laws: A supersymmetric approach

In this article we address the old problem of finding the effective dielectric constant of materials described either by a local random dielectric constant, or by a set of non-overlapping spherical inclusions randomly dispersed in a host. We use a unified theoretical framework, such that all the most important Electromagnetic Mixing Laws (EML) can be recovered as the first iterative step of a family of results, thus opening the way to future improvements through the refinements of the approximation schemes. When the material is described by a set of immersed inclusions characterized by their spatial correlation functions, we exhibit an EML which, being featured by a minimal approximation scheme, does not come from the multiple scattering paradigm. It is made of a pure Hori-Yonezawa formula, corrected by a power series of the inclusion density. The coefficients of the latter, which are given as sums of standard diagrams, are recast into electromagnetic quantities which calculation is amenable numerically thanks to codes available on the web. The methods used and developed in this work are generic and can be used in a large variety of areas ranging from mechanics to thermodynamics.