Local Number Fluctuations in Hyperuniform and Nonhyperuniform Systems: HigherOrder Moments and Distribution Functions
Abstract
The local number variance σ^{2}(R ) associated with a spherical sampling window of radius R enables a classification of manyparticle systems in d dimensional Euclidean space R^{d} according to the degree to which largescale 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 higherorder moments or cumulants, 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 systems with varying degrees of short and longrange order. To carry out this comprehensive program, we derive new theoretical results that apply to general point processes, and we conduct highprecision numerical studies. Specifically, we derive explicit closedform integral expressions for γ_{1}(R ) and γ_{2}(R ) that encode structural information up to threebody and fourbody correlation functions, respectively. We also derive rigorous bounds on γ_{1}(R ), γ_{2}(R ), and P [N (R )] for general point processes and corresponding exact results for general packings of identical spheres. Highquality simulation data for γ_{1}(R ), γ_{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 γ_{1}(R )∼l_{2}(R )∼R^{(d +1 )/2} and γ_{2}(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 onedimensional 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 largeR 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 Gaussianlike behavior, respectively. Our work sheds light on the fundamental importance of higherorder structural information to fully characterize density fluctuations in manybody systems across length scales and dimensions, and thus has broad implications for condensed matter physics, engineering, mathematics, and biology.
 Publication:

Physical Review X
 Pub Date:
 April 2021
 DOI:
 10.1103/PhysRevX.11.021028
 arXiv:
 arXiv:2012.02358
 Bibcode:
 2021PhRvX..11b1028T
 Keywords:

 Condensed Matter  Statistical Mechanics;
 Condensed Matter  Soft Condensed Matter
 EPrint:
 23 pages, 8 figures