Jiao Yang, Stillinger Frank H, Torquato Salvatore
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, New Jersey 08544, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2010 Jul;82(1 Pt 1):011106. doi: 10.1103/PhysRevE.82.011106. Epub 2010 Jul 6.
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 Sn. An intricate inverse problem of practical importance is to what extent a medium can be reconstructed from the two-point correlation function S2 of a target medium. Recently, general claims of the uniqueness of reconstructions using S2 have been made based on numerical studies, which implies that S2 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 S2 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 S2 , 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 S2 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.
在这两篇论文系列的第一篇 [焦阳、F. H. 斯蒂林格和 S. 托尔夸托，《物理评论E》81, 011105 (2010)] 中，我们考虑了与点构型相关的对关联统计的几何模糊性。这里我们关注非均匀介质（材料）的类似问题。非均匀介质在许多情况下普遍存在，包括复合材料和颗粒介质、生物组织、生态模式以及天体物理结构。非均匀介质的复杂结构通常通过统计描述符来表征，例如n点关联函数Sn。一个具有实际重要性的复杂反问题是，在多大程度上可以从目标介质的两点关联函数S2重建介质。最近，基于数值研究对使用S2进行重建的唯一性提出了一般性断言，这意味着S2足以在一定数值精度内唯一确定介质的结构。在本文中，我们提供了一种系统方法，以数学上精确的方式表征连续两相非均匀介质及其数字化表示的S2的几何模糊性。特别是，我们推导了两种不同介质具有相同S2的情况（即它们形成退化对）的精确条件。退化条件分别以连续介质及其数字化表示的积分方程和代数方程给出。通过研究这些方程并为具体示例构建其严格解，我们最终表明，一般来说S2确实不足以唯一确定介质的结构，这与我们最近关于非均匀介质重建的研究结果 [焦阳、F. H. 斯蒂林格和 S. 托尔夸托，《美国国家科学院院刊》106, 17634 (2009)] 一致。解析示例包括由带有字母“T”和单词“WATER”的积木组成的复杂图案，以及三维中最密球体堆积的退化堆积变体（巴洛薄膜）。还给出了几个退化的数值示例（例如多晶微观结构、激光散斑图案和球体堆积的重建），它们实际上是退化方程的精确解。讨论了多相介质重建的唯一性问题以及表征非均匀介质所需的额外结构信息，包括包含相拓扑连通性信息的两点量。
