Necessary Conditions on Realizable Two-Point Correlation Functions of Random Media

TL;DR

This paper reviews known necessary conditions for the realizability of two-point correlation functions in random media and introduces a new positivity condition to test whether proposed functions can correspond to actual two-phase media.

Contribution

It compiles existing necessary conditions and derives a novel simple positivity condition for assessing the realizability of two-point correlation functions.

Findings

Established a comprehensive list of necessary conditions from literature.

Derived a new simple positivity condition for realizability.

Applied conditions to test specific correlation functions.

Abstract

A fascinating inverse problem that has been receiving considerable attention is the construction of realizations of random two-phase heterogeneous media with a target two-point correlation function. However, not every hypothetical two-point correlation function corresponds to a realizable two-phase medium. Here we collect all of the known necessary conditions on the two-point correlation functions scattered throughout a diverse literature and derive a new but simple positivity condition. We apply the necessary conditions to test the realizability of certain classes of proposed correlation functions.