对统计量的几何模糊性。II。非均匀介质
Geometrical Ambiguity of Pair Statistics. II. Heterogeneous Media
摘要：
在先前的论文[Jiao，Stillinger和Torquato，PRE 81，011105（2010）]中，我们考虑了与点配置相关联的对统计的几何模糊性。在这里，我们专注于非均匀介质（材料）的类似问题。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$S_n$。一个复杂的反问题的实际重要性是在何种程度上可以重建介质从两点关联函数的目标介质$S_2$。最近，一般索赔的唯一性重建使用$S_2$已经取得了基于数值研究。在这里，我们提供了一个系统的方法来表征连续两相非均匀介质和它们的数字化表示的$S_2$的几何模糊性在数学上精确的方式。特别是，我们推导出的情况下，两个不同的媒体拥有相同的$S_2$的确切条件。退化条件给出的积分和代数方程的连续介质及其数字化表示，分别。通过检查这些方程并针对具体示例构造其严格解，我们最终表明，一般而言，$S_2$确实不足以唯一地确定介质的结构，这与我们最近关于非均匀介质重建的研究结果一致[Jiao，Stillinger和Torquato，PNAS 106，17634（2009）]。讨论了多相介质重建的唯一性问题和表征非均匀介质所需的附加结构信息。
英文摘要：
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.
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