Disordered systems are ubiquitous in physical, biological, and material sciences. Examples include liquid and glassy states of condensed matter, colloids, granular materials, porous media, composites, alloys, packings of cells in avian retina, and tumor spheroids, to name but a few. A comprehensive understanding of such disordered systems requires, as the first step, systematic quantification, modeling, and representation of the underlying complex configurations and microstructure, which is generally very challenging to achieve. Recently, we introduced a set of hierarchical statistical microstructural descriptors, i.e., the “$n$-point polytope functions” $P_{\mathrm{n}}$, which are derived from the standard $n$-point correlation functions $S_{\mathrm{n}}$, and successively included higher-order $n$-point statistics of the morphological features of interest in a concise, explainable, and expressive manner. Here we investigate the information content of the $P_{\mathrm{n}}$ functions via optimization-based realization rendering. This is achieved by successively incorporating higher-order $P_{\mathrm{n}}$ functions up to $n=8$ and quantitatively assessing the accuracy of the reconstructed systems via unconstrained statistical morphological descriptors (e.g., the lineal-path function). We examine a wide spectrum of representative random systems with distinct geometrical and topological features. We find that, generally, successively incorporating higher-order $P_{\mathrm{n}}$ functions and, thus, the higher-order morphological information encoded in these descriptors leads to superior accuracy of the reconstructions. However, incorporating more $P_{\mathrm{n}}$ functions into the reconstruction also significantly increases the complexity and roughness of the associated energy landscape for the underlying stochastic optimization, making it difficult to convergence numerically.

中文翻译：

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探究分层n点多态性功能的信息内容，以量化和重建无序系统。

在物理，生物和材料科学中，无序的系统无处不在。例子包括凝聚态的液态和玻璃态，胶体，颗粒状材料，多孔介质，复合材料，合金，禽视网膜中细胞堆积和肿瘤球体，仅举几例。对此类无序系统的全面了解需要第一步，即系统地量化，建模和表示基本的复杂配置和微结构，这通常很难实现。最近，我们引入了一组分层的统计微观结构描述符，即“$ñ$点多面体功能” $P_{\mathrm{ñ}}$，这是从标准派生的 $ñ$点相关函数 ${\mathrm{小号}}_{\mathrm{ñ}}$，并依次包含高阶 $ñ$以简洁，可解释和富有表现力的方式对感兴趣的形态特征进行定点统计。在这里，我们调查了$P_{\mathrm{ñ}}$通过基于优化的实现呈现功能。这是通过相继合并高阶来实现的$P_{\mathrm{ñ}}$ 功能可达 $ñ=8$并通过无约束的统计形态描述符（例如，线性路径函数）定量评估重建系统的准确性。我们研究了具有代表性的具有随机几何和拓扑特征的随机系统。我们发现，通常，先后合并高阶$P_{\mathrm{ñ}}$因此，这些描述符中编码的高阶形态学信息会产生更高的准确性。但是，合并更多$P_{\mathrm{ñ}}$ 重构中的函数还显着增加了相关能量分布的复杂性和粗糙度，以进行潜在的随机优化，使其难以在数值上收敛。