Stochastic Functional Expansion for Random Media with Perfectly Disordered Constitution

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 through the random medium in question to be broken into consecutive steps. The first-order approximation yields a formula for the overall conductivity which coincides with the linear part of the knownt Maxwell relation. A heuristic approach in which only the first term in the Wiener expansion is retained is shown to yield, within the frame of the so-called singular approximation, the full Maxwell relation for the overall conductivity. The equations for the second-order approximation are investigated and the kernels of the respective Wiener functionals are related to the temperature fields in an unbounded material, containing one or two spherical inclusions.