Local Number Fluctuations in Hyperuniform and Nonhyperuniform Systems: Higher-Order Moments and Distribution Functions

The local number variance σ²(R ) associated with a spherical sampling window of radius R enables a classification of many-particle systems in d -dimensional Euclidean space 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 γ₁(R ), excess kurtosis γ₂(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 γ₁(R ) and γ₂(R ) that encode structural information up to three-body and four-body correlation functions, respectively. We also derive rigorous bounds on γ₁(R ), γ₂(R ), and P [N (R )] for general point processes and corresponding exact results for general packings of identical spheres. High-quality simulation data for γ₁(R ), γ₂(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 l₂(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 γ₁(R )∼l₂(R )∼R^{-(d +1 )/2} and γ₂(R )∼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 γ₁(R ), γ₂(R ), and l₂(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.