Necessary and sufficient conditions for the two-point phase probability function of two-phase random media

Abstract

Constructing realizations of random media with a specified two-point phase probability function S 2 has attracted considerable attention in the recent literature. However, little is known about conditions under which a prescribed S 2 is realizable. The only known necessary and sufficient condition, due to McMillan, involves a class of square matrices, called corner-positive matrices, about which almost nothing is known except their definition. As a result, McMillan's theorem has gone mostly unused in the literature for over 50 years. In this paper, we present a general decomposition formula for cornerpositive matrices, which allows McMillan's theorem to be written in a significantly more tractable and testable form. We also connect McMillan's theorem with many known but heretofore unrelated necessary conditions on S 2 , extending many of these conditions.

Hyperuniformity, quasi-long-range correlations, and void-space constraints in maximally random jammed particle packings. I. Polydisperse spheres

“…We remark that it is an open problem to identify additional necessary and sufficient conditions that the autocovariance function must satisfy in order to correspond to a binary stochastic process [11,[35][36][37][38].…”

Hyperuniformity, quasi-long-range correlations, and void-space constraints in maximally random jammed particle packings. II. Anisotropy in particle shape

“…It follows from these considerations that if a packing has quasi-long-range correlations (as at the MRJ state), the contribution to S (p) 2 from the pair correlation function must be responsible for this behavior. The corresponding two-point autocovariance function χ(r) [21,22,26] is obtained by subtracting the long-range behavior φ …”

Realisability conditions for second-order marginals of biphased media

“…The "unit field formulation" is preferred in [15,18,19]. The natural symmetry of the set {−1, 1} confers to this version a more handy combinatorial structure, and is exploited in the companion paper [9] to give theoretical realisability results, transferred in this paper to other marginals.…”

“…The positivity problem, central in this paper, is about the compatibility of a candidate p with the algebraic properties of a set covariance, and is of combinatorial nature.This problem has been posed by McMillan [13] in the field of telecommunications. It is more or less implicit in many articles, and has been to the author's knowledge first addressed directly by Shepp [17] and more recently by Quintanilla [15]. A series of works by Torquato and his coauthors (see [5] and [18, Sec.…”

“…It is more or less implicit in many articles, and has been to the author's knowledge first addressed directly by Shepp [19], and more recently by Quintanilla [18]. A series of works by Torquato and his coauthors (see [8] and [20,Sec.…”