Analyticity of homogenized coefficients under Bernoulli perturbations and the Clausius-Mossotti formulas

TL;DR

This paper proves the analyticity of homogenized coefficients under Bernoulli perturbations in random media, deriving formulas that generalize the Clausius-Mossotti relations for random inclusions.

Contribution

It introduces a new energy estimate approach to establish analyticity and provides the first general proof of Clausius-Mossotti formulas for random inclusions.

Findings

Homogenized coefficients are analytic functions of the Bernoulli parameter.

Derived explicit formulas for derivatives of homogenized coefficients.

First proof of Clausius-Mossotti formulas in the context of random inclusions.

Abstract

This paper is concerned with the behavior of the homogenized coefficients associated with some random stationary ergodic medium under a Bernoulli perturbation. Introducing a new family of energy estimates that combine probability and physical spaces, we prove the analyticity of the perturbed homogenized coefficients with respect to the Bernoulli parameter. Our approach holds under the minimal assumptions of stationarity and ergodicity, both in the scalar and vector cases, and gives analytical formulas for each derivative that essentially coincide with the so-called cluster expansion used by physicists. In particular, the first term yields the celebrated (electric and elastic) Clausius-Mossotti formulas for isotropic spherical random inclusions in an isotropic reference medium. This work constitutes the first general proof of these formulas in the case of random inclusions.