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
Local Number Fluctuations in Hyperuniform and Nonhyperuniform Systems: Higher-Order Moments and Distribution Functions
摘要：
与半径为$R$的球形采样窗口相关联的局部数方差使得能够根据大规模密度波动被抑制的程度对$d$维欧几里得空间中的多粒子系统进行分类，从而导致超均匀和非超均匀门之间的划分。为了更好地表征密度波动，我们进行了广泛的研究高阶矩，包括偏度$γ_1(R)$，过剩峰度$γ_2(R)$和相应的概率分布函数$P[N(R)]$的一个大家庭的模型在前三个空间维度，包括超均匀和nonhyperuniform模型。具体来说，我们推导出明确的积分表达式$γ_1(R)$和$γ_2(R)$涉及到三个和四体关联函数，分别。我们还得到了$γ_1(R)$，$γ_2(R)$和$P[N(R)]$的严格界.为每个模型生成这些量的高质量模拟数据。我们还确定通过一种新的高斯距离度量$l_2(R)$的接近$P[N(R)]$的正态分布。在所有的模型中，无序超一致过程的中心极限定理（CLT）的收敛速度通常是最快的。对于标准的非超均匀模型，收敛到CLT的速度较慢，而对于这里研究的反超均匀模型，收敛速度最慢。我们证明了一维超一致系统的第一类或任何$d$维格不能服从CLT。值得注意的是，我们发现伽玛分布为所有服从CLT的模型提供了一个很好的近似$P[N(R)]$，使我们能够估计$γ_1(R)$，$γ_2(R)$和$l_2(R)$的大- $R$标度。对于任何$d$维模型，“去相关”或“相关”与$d$，我们阐明了为什么$P[N(R)]$越来越多地走向或远离高斯样的行为，分别。
英文摘要：
The local number variance associated with a spherical sampling window of radius $R$ enables a classification of many-particle systems in $d$-dimensional Euclidean space according to the degree to which large-scale density fluctuations are suppressed, resulting in a demarcation between hyperuniform and nonhyperuniform phyla. To better characterize density fluctuations, we carry out an extensive study of higher-order moments, including the skewness $γ_1(R)$, excess kurtosis $γ_2(R)$ 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 models. Specifically, we derive explicit integral expressions for $γ_1(R)$ and $γ_2(R)$ involving up to three- and four-body correlation functions, respectively. We also derive rigorous bounds on $γ_1(R)$, $γ_2(R)$ and $P[N(R)]$. High-quality simulation data for these quantities 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. 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 discovered that the gamma distribution provides a good approximation to $P[N(R)]$ for all models that obey a CLT, enabling us to estimate the large-$R$ scalings of $γ_1(R)$, $γ_2(R)$ 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.
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