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Averaging of Dilute Random Media: A Rigorous Proof of the Clausius–Mossotti Formula

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Abstract

We consider a large number of randomly dispersed spherical, identical, inclusions in a bounded domain, with conductivity different than that of the host medium. In the dilute limit, with some mild assumptions on the first few marginal probability densities (no periodicity or stationarity is assumed), we prove convergence in the 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.

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  1. Department of Mathematics, Louisiana State University, Baton Rouge, LA, 70803, USA

    Y. Almog

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  1. Y. Almog
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Correspondence to Y. Almog.

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Communicated by C. Le Bris

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Almog, Y. Averaging of Dilute Random Media: A Rigorous Proof of the Clausius–Mossotti Formula. Arch Rational Mech Anal 207, 785–812 (2013). https://doi.org/10.1007/s00205-012-0581-9

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  • DOI: https://doi.org/10.1007/s00205-012-0581-9

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