Local Number Fluctuations in Hyperuniform and Nonhyperuniform Systems: Higher-Order Moments and Distribution Functions
Abstract
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 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 models. Specifically, we derive explicit integral expressions for γ1(R) and γ2(R) involving up to three- and four-body correlation functions, respectively. We also derive rigorous bounds on γ1(R), γ2(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 l2(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 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.