对统计量的几何模糊性。I.点配置
Geometrical Ambiguity of Pair Statistics. I. Point Configurations
摘要：
点组态作为模型系统在凝聚态物理、材料科学和生物学中有着广泛的应用。统计描述符，如$n$ -身体分布函数$g_n$通常被用来描述点的配置，其中最广泛使用的是对分布函数$g_2$。一个有趣的反问题的实际重要性，一直受到相当大的关注是在何种程度上可以重建一个点的配置从对分布函数的目标配置。尽管已知$g_2$中包含的对距离信息通常不足以唯一地确定点配置，但这一概念似乎并未得到广泛认可，并且已经基于数值研究做出了使用对信息重建的唯一性的一般声明。在本文中，我们介绍的距离空间，称为$\mathbb{D}$空间的想法。然后，特定点配置的对距离由$\mathbb{D}$空间中的单个点表示。我们导出了与点组态相关联的对距离的条件，这些条件等价于对分布函数$g_2$的可实现性条件。此外，我们推导出的条件对距离，可以组装成不同的配置。这些条件定义了$\mathbb{D}$空间中的有界区域。通过明确地使用$\mathbb{D}$空间构建各种退化点配置，我们表明，对信息确实是不足以唯一地确定一般的配置。我们还讨论了基于$\mathbb{D}$空间的统计物理中的几个重要问题。
英文摘要：
Point configurations have been widely used as model systems in condensed matter physics, materials science and biology. Statistical descriptors such as the $n$-body distribution function $g_n$ is usually employed to characterize the point configurations, among which the most extensively used is the pair distribution function $g_2$. An intriguing inverse problem of practical importance that has been receiving considerable attention is the degree to which a point configuration can be reconstructed from the pair distribution function of a target configuration. Although it is known that the pair-distance information contained in $g_2$ is in general insufficient to uniquely determine a point configuration, this concept does not seem to be widely appreciated and general claims of uniqueness of the reconstructions using pair information have been made based on numerical studies. In this paper, we introduce the idea of the distance space, called the $\mathbb{D}$ space. The pair distances of a specific point configuration are then represented by a single point in the $\mathbb{D}$ space. We derive the conditions on the pair distances that can be associated with a point configuration, which are equivalent to the realizability conditions of the pair distribution function $g_2$. Moreover, we derive the conditions on the pair distances that can be assembled into distinct configurations. These conditions define a bounded region in the $\mathbb{D}$ space. By explicitly constructing a variety of degenerate point configurations using the $\mathbb{D}$ space, we show that pair information is indeed insufficient to uniquely determine the configuration in general. We also discuss several important problems in statistical physics based on the $\mathbb{D}$ space.
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