The Clausius-Mossotti formula for dilute random media of perfectly conducting inclusions

TL;DR

This paper proves that in dilute random media with perfectly conducting inclusions, the expected heat distribution converges to an effective medium described by the Clausius-Mossotti formula, with explicit error bounds.

Contribution

It establishes a rigorous convergence result for the expected solution of the heat equation in dilute media, deriving the Clausius-Mossotti formula as the effective conductivity.

Findings

Convergence of expected solutions in H^1 norm

Explicit error estimates for the approximation

Validation of the Clausius-Mossotti formula in this setting

Abstract

We consider a large number of randomly dispersed spherical, identical, perfectly conducting inclusions (of infinite conductivity) in a bounded domain. The host medium's conductivity is finite and can be inhomogeneous. In the dilute limit, with some boundedness assumption on a large number (proportional to the global volume fraction raised to the power of -1/2) of marginal probability densities, we prove convergence in H^1 norm of the expectation of the solution of the steady state heat equation, to the solution of an effective medium problem, where the conductivity is given by the Clausius-Mossotti formula. Error estimates are provided as well.