The local number variance ${\sigma }^2(R)$ associated with a spherical sampling window of radius $R$ enables a classification of many-particle systems in $d$-dimensional Euclidean space ${\mathbb{R}}^d$ according to the degree to which large-scale density fluctuations are suppressed, resulting in a demarcation between hyperuniform and nonhyperuniform phyla. To more completely characterize density fluctuations, we carry out an extensive study of higher-order moments or cumulants, including the skewness ${\gamma }_1(R)$, excess kurtosis ${\gamma }_2(R)$, and the corresponding probability distribution function $P[N(R)]$ of a large family of models across the first three space dimensions, including both hyperuniform and nonhyperuniform systems with varying degrees of short- and long-range order. To carry out this comprehensive program, we derive new theoretical results that apply to general point processes, and we conduct high-precision numerical studies. Specifically, we derive explicit closed-form integral expressions for ${\gamma }_1(R)$ and ${\gamma }_2(R)$ that encode structural information up to three-body and four-body correlation functions, respectively. We also derive rigorous bounds on ${\gamma }_1(R)$, ${\gamma }_2(R)$, and $P[N(R)]$ for general point processes and corresponding exact results for general packings of identical spheres. High-quality simulation data for ${\gamma }_1(R)$, ${\gamma }_2(R)$, and $P[N(R)]$ are generated for each model. We also ascertain the proximity of $P[N(R)]$ to the normal distribution via a novel Gaussian “distance” metric $l_2(R)$. Among all models, the convergence to a central limit theorem (CLT) is generally fastest for the disordered hyperuniform processes in two or higher dimensions such that ${\gamma }_1(R)\sim l_2(R)\sim R^{-(d+1)/2}$ and ${\gamma }_2(R)\sim R^{-(d+1)}$ for large $R$. The convergence to a CLT is slower for standard nonhyperuniform models and slowest for the “antihyperuniform” model studied here. We prove that one-dimensional hyperuniform systems of class I or any $d$-dimensional lattice cannot obey a CLT. Remarkably, we discover a type of universality in that, for all of our models that obey a CLT, the gamma distribution provides a good approximation to $P[N(R)]$ across all dimensions for intermediate to large values of $R$, enabling us to estimate the large-$R$ scalings of ${\gamma }_1(R)$, ${\gamma }_2(R)$, and $l_2(R)$. For any $d$-dimensional model that “decorrelates” or “correlates” with $d$, we elucidate why $P[N(R)]$ increasingly moves toward or away from Gaussian-like behavior, respectively. Our work sheds light on the fundamental importance of higher-order structural information to fully characterize density fluctuations in many-body systems across length scales and dimensions, and thus has broad implications for condensed matter physics, engineering, mathematics, and biology.

中文翻译：

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超均匀和非超均匀系统中的局部数涨落：高阶矩和分布函数

本地数方差 ${\sigma }^{\mathrm{2个}}（\mathrm{[R}）$ 与半径为球面的采样窗口相关联 $\mathrm{[R}$ 可以对中的多粒子系统进行分类 $d$维欧氏空间 ${\mathbb{[R}}^d$根据抑制大规模密度波动的程度，导致超均匀门和非超均匀门的分界。为了更完整地描述密度波动，我们对高阶矩或累积量（包括偏度）进行了广泛的研究。${\gamma }_{\mathrm{1个}}（\mathrm{[R}）$，过量峰度 ${\gamma }_{\mathrm{2个}}（\mathrm{[R}）$，以及相应的概率分布函数 $P[ñ（\mathrm{[R}）]$在前三个空间维度上的大型模型集合，包括超均匀和非超均匀系统，它们具有不同程度的短程和远程顺序。为了执行此综合程序，我们得出了适用于一般点过程的新理论结果，并进行了高精度数值研究。具体来说，我们推导了以下形式的显式封闭形式积分表达式${\gamma }_{\mathrm{1个}}（\mathrm{[R}）$ 和 ${\gamma }_{\mathrm{2个}}（\mathrm{[R}）$分别将结构信息编码为三体和四体相关函数。我们还得出了严格的界限${\gamma }_{\mathrm{1个}}（\mathrm{[R}）$， ${\gamma }_{\mathrm{2个}}（\mathrm{[R}）$， 和 $P[ñ（\mathrm{[R}）]$对于一般的点过程，以及对于相同球体的一般填充物的相应精确结果。的高质量模拟数据${\gamma }_{\mathrm{1个}}（\mathrm{[R}）$， ${\gamma }_{\mathrm{2个}}（\mathrm{[R}）$， 和 $P[ñ（\mathrm{[R}）]$为每个模型生成。我们还要确定$P[ñ（\mathrm{[R}）]$ 通过一种新颖的高斯“距离”度量来回归正态分布 ${升}_{\mathrm{2个}}（\mathrm{[R}）$。在所有模型中，对于二维或更高维度的无序超均匀过程，收敛到中心极限定理（CLT）通常最快。${\gamma }_{\mathrm{1个}}（\mathrm{[R}）〜{升}_{\mathrm{2个}}（\mathrm{[R}）〜{\mathrm{[R}}^{-（d+\mathrm{1个}）/\mathrm{2个}}$ 和 ${\gamma }_{\mathrm{2个}}（\mathrm{[R}）〜{\mathrm{[R}}^{-（d+\mathrm{1个}）}$ 大 $\mathrm{[R}$。对于标准非超均匀模型，对CLT的收敛较慢，而对于本文研究的“反超均匀”模型，对CLT的收敛最慢。我们证明了I类或任何一维的一维超均匀系统$d$维晶格不能遵守CLT。值得注意的是，我们发现了一种通用性，因为对于我们所有符合CLT的模型，伽玛分布都可以很好地近似于$P[ñ（\mathrm{[R}）]$ 涵盖所有维度，从中等到较大的值 $\mathrm{[R}$，使我们能够估算出$\mathrm{[R}$ 标度 ${\gamma }_{\mathrm{1个}}（\mathrm{[R}）$， ${\gamma }_{\mathrm{2个}}（\mathrm{[R}）$， 和 ${升}_{\mathrm{2个}}（\mathrm{[R}）$。对于任何$d$与“去相关”或“相关”的三维模型 $d$，我们说明原因 $P[ñ（\mathrm{[R}）]$分别逐渐趋向或远离高斯式行为。我们的工作揭示了高阶结构信息对于充分表征长度和尺寸范围内的多体系统中密度波动的根本重要性，因此对凝聚态物理，工程，数学和生物学具有广泛的意义。