Local Number Fluctuations in Hyperuniform and Nonhyperuniform Systems: Higher-Order Moments and Distribution Functions
Salvatore Torquato, Jaeuk Kim, Michael A. Klatt
Abstract
The local number variance 2 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 1 R, excess kurtosis 2 R, and the corresponding probability distribution function PNR 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 1 R and 2 R that encode structural information up to three-body and four-body correlation functions, respectively. We also derive rigorous bounds on 1 R, 2 R, and PNR for general point processes and corresponding exact results for general packings of identical spheres. High-quality simulation data for 1 R, 2 R, and PNR are generated for each model. We also ascertain the proximity of PNR 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 1 R l 2 R R -d1=2 and 2 R R -d1 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 PNR across all dimensions for intermediate to large values of R, enabling us to estimate the large-R scalings of 1 R, 2 R, and l 2 R. For any d-dimensional model that "decorrelates" or "correlates" with d, we elucidate why PNR 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.