Type: Article
Publication Date: 2010-07-06
Citations: 45
DOI: https://doi.org/10.1103/physreve.82.011106
In the first part of this series of two papers [Y. Jiao, F. H. Stillinger, and S. Torquato, Phys. Rev. E 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). Heterogeneous media are ubiquitous in a host of contexts, including composites and granular media, biological tissues, ecological patterns, and astrophysical structures. 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, which implies that ${S}_{2}$ suffices to uniquely determine the structure of a medium within certain numerical accuracy. In this paper, 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 ${S}_{2}$, i.e., they form a degenerate pair. 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 [Y. Jiao, F. H. Stillinger, and S. Torquato, Proc. Natl. Acad. Sci. U.S.A. 106, 17634 (2009)]. The analytical examples include complex patterns composed of building blocks bearing the letter ``T'' and the word ``WATER'' as well as degenerate stacking variants of the densest sphere packing in three dimensions (Barlow films). Several numerical examples of degeneracy (e.g., reconstructions of polycrystal microstructures, laser-speckle patterns and sphere packings) are also given, which are virtually exact solutions of the degeneracy equations. The uniqueness issue of multiphase media reconstructions and additional structural information required to characterize heterogeneous media are discussed, including two-point quantities that contain topological connectedness information about the phases.