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Functional series with a pointwise random input (the density field of a random set of points ${\bf x}_j $) are considered. The series are rearranged so as the so-called factorial fields of the set ${\bf x}_j $ appear; the obtained series are called factorial. The basic result of the paper states that the factorial series possess virial property. This means that if a random field $u( {\bf x} )$ is expanded as a factorial series, the truncation $u^{( p )} ( {\bf x} )$ of the latter after the p-tuple term coincides, in the statistical sense, with $u( {\bf x} )$ to the order $n^p $, where n is the number density of the set ${\bf x}_j ,\, p = 1,2, \cdots $. The performance of the factorial series in random media problems is illustrated on the example of steady-state diffusion in a random dispersion of spheres whose sink strength differs from that of the matrix. The full statistical solution of this problem, correct to the order $c^2 $, is obtained; in particular, the effective sink-strength of the dispersion i...

The random field created by a system of random points which are centers of perfectly disordered equi-sized spheres of a finite radius is introduced and named the PDS-field. The notion of Perfect Disorder is defined statistically correctly taking the cumulants instead of moments to be $\delta $-functions. The Volterra-Wiener functional expansion with the PDS-field as a basis function is considered and a system of orthogonal Wiener functionals is explicitly constructed. The expansion is employed to the problem of specifying the overall conductivity for a random medium containing an array of perfectly disordered spherical inclusions. An infinite hierarchy of coupled equations for the kernels of the Wiener functionals is derived. It is shown that the Wiener expansion with respect to the PDS-field is also a virial one, i.e. the nth order term contributes quantities of order $c^n $, where c is the volume fraction of the inclusions. This allows the full stochastic solution to the problem of heat conduction throu...

A general method of placing optimal bounds on the overall conductivity of random heterogeneous media is proposed. It utilizes truncated Volterra-Wiener functional series, generated by the random conductivity field K(x), as classes of trial functions for the classical variational principles. The method simplifies, unifies and/or generalizes the earlier proposed variational techniques of Beran (3), Dederichs and Zeller (10), Hori (15), Kr6ner (17) and Prager (25). The general procedure is displayed in detail in the simplest case of interest, the construction of the optimal third-order bounds, that requires knowledge of the two- and three-point correlation functions for K(x). The evaluation of these bounds is reduced to the solution of an integrodifferential equation whose coefficients and kernels are expressed through the said correlation functions. A perturbation solution to this equation is given. For Miller's cell materials the equation is explicitly solved and the obtained optimal bounds are shown to coincide with those of Beran-Miller.