Geometrical Ambiguity of Pair Statistics. II. Heterogeneous Media

TL;DR

This paper investigates whether the two-point correlation function $S_2$ uniquely determines the structure of heterogeneous media, providing mathematical conditions for when different media can share identical $S_2$, thus highlighting limitations in reconstructing media from pair statistics.

Contribution

The paper offers a systematic, mathematically rigorous characterization of the geometrical ambiguity of $S_2$ in heterogeneous media, including exact conditions for non-uniqueness.

Findings

$S_2$ does not generally uniquely determine media structure.

Derived integral and algebraic conditions for media sharing identical $S_2$.

Constructed explicit examples demonstrating non-uniqueness.

Abstract

In a previous paper [Jiao, Stillinger and Torquato, PRE 81, 011105 (2010)], we considered the geometrical ambiguity of pair statistics associated with point configurations. Here we focus on the analogous problem for heterogeneous media (materials). The complex structures of heterogeneous media are usually characterized via statistical descriptors, such as the $n$-point correlation function $S_n$. An intricate inverse problem of practical importance is to what extent a medium can be reconstructed from the two-point correlation function $S_2$ of a target medium. Recently, general claims of the uniqueness of reconstructions using $S_2$ have been made based on numerical studies. Here, we provide a systematic approach to characterize the geometrical ambiguity of $S_2$ for both continuous two-phase heterogeneous media and their digitized representations in a mathematically precise way. In particular, we derive the exact conditions for the case where two distinct media possess identical $S_2$. The degeneracy conditions are given in terms of integral and algebraic equations for continuous media and their digitized representations, respectively. By examining these equations and constructing their rigorous solutions for specific examples, we conclusively show that in general $S_2$ is indeed not sufficient information to uniquely determine the structure of the medium, which is consistent with the results of our recent study on heterogeneous media reconstruction [Jiao, Stillinger and Torquato, PNAS 106, 17634 (2009)]. The uniqueness issue of multiphase media reconstructions and additional structural information required to characterize heterogeneous media are discussed.