Effective permittivity of log-normal isotropic random media

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Abstract
A method of deriving the effective permittivity epsilon eff of log-normal d-dimensional isotropic media is developed. The approach is based on applying the regularized Green function technique for the Laplace operator. The feasibility of exact determining of the effective permittivity is analysed by deriving epsilon eff at third order in the log-permittivity variance. It is shown that the 3D effective permittivity is a functional of the spectrum of inhomogeneities and therefore cannot be expressed by a closed formula. In particular, the formula for the effective permittivity discussed previously by Landau and Lifshitz(1960) and Matheron(1967) deviates from the exact one in the third-order term.