1. Introduction. The paper is devoted to the problem of estimating the effective scalar transport properties (in the context of thermal conductivity) for random dispersions of spheres, i.e. arrays of equisized (with radius a) nonoverlapping spheres haphazardly distributed throughout a matrix. Such dispersions, as is well known, have been considered by many authors starting perhaps with J. Maxwell [1] who gave a heuristic formula for their effective conductivity k_*. After Maxwell many authors studied the effective properties of the random dispersions struggling with the profound problem how to account properly for the multi-particle interactions among the spheres, see, e.g., [2,3]. The difficulties in calculating k have made many authors consider the simplified problem in which, say, the effective conductivity k_* , is to be evaluated to the order c only, because this is the simplest situation in which the particle interaction shows up in a nontrivial way, see [4-6] et al.; hereafter c denotes the volume fraction of the spheres. A problem, closely related to the evaluation of the effective properties, consists in bounding of their values making use of the limited statistical information concerning the random constitution. This problem appears in a natural way in practice since very little is usually known about the internal structure of a real heterogeneous material of technological interest, and of a random dispersion in particular. That is why the bounding problem was considered by many authors, most of them making use of variational principles, see the survey [7] for details and references. It appeared that the existing variational procedures could be conveniently unifiedImaking use of truncated functional series as classes of trial functions [8].
The outline of the paper is as follows. We first recall in &2 the bounding procedure of [8] and its realization(&3) that leads to so-called Beran's bounds [9], and then we pose, after [10], the problem of optimality of the latter. Optimality is understood here in the sense that they should be the most restrictive ones under a given amount of statistical information; for Beran's bounds this information comprises the prescribed two- and three-point correlation functions for the medium. In this sense the said bounds are third-order. Having briefly recalled the methods of a statistical description for random dispersions, we modify in &4 the variational procedure and introduce the so-called cluster bounds of Torquato [11]; they appeared toe coincide with Beran's as a simple consequence of the assumption of nonoverlapping. In &4 it is shown that Beran's, and therefore Torquato's bounds as well, are optimal to the order c; however, they are not optimal to the higher orders of c, and these are the central results of the paper.