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

The local number variance σ^{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 hyp...

Descripción completa

Guardado en:

Detalles Bibliográficos
Autores principales:	Salvatore Torquato, Jaeuk Kim, Michael A. Klatt
Formato:	article
Lenguaje:	EN
Publicado:	American Physical Society 2021
Materias:	Physics QC1-999
Acceso en línea:	https://doaj.org/article/81498106a5d9484e82f0ecf812a05038
Etiquetas:	Agregar Etiqueta Sin Etiquetas, Sea el primero en etiquetar este registro!

id	oai:doaj.org-article:81498106a5d9484e82f0ecf812a05038
record_format	dspace
spelling	oai:doaj.org-article:81498106a5d9484e82f0ecf812a050382021-12-02T17:25:07ZLocal Number Fluctuations in Hyperuniform and Nonhyperuniform Systems: Higher-Order Moments and Distribution Functions10.1103/PhysRevX.11.0210282160-3308https://doaj.org/article/81498106a5d9484e82f0ecf812a050382021-05-01T00:00:00Zhttp://doi.org/10.1103/PhysRevX.11.021028http://doi.org/10.1103/PhysRevX.11.021028https://doaj.org/toc/2160-3308The local number variance σ^{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 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 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 γ_{1}(R) and γ_{2}(R) that encode structural information up to three-body and four-body 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. High-quality 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 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 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. 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.Salvatore TorquatoJaeuk KimMichael A. KlattAmerican Physical SocietyarticlePhysicsQC1-999ENPhysical Review X, Vol 11, Iss 2, p 021028 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	Physics QC1-999
spellingShingle	Physics QC1-999 Salvatore Torquato Jaeuk Kim Michael A. Klatt Local Number Fluctuations in Hyperuniform and Nonhyperuniform Systems: Higher-Order Moments and Distribution Functions
description	The local number variance σ^{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 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 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 γ_{1}(R) and γ_{2}(R) that encode structural information up to three-body and four-body 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. High-quality 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 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 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. 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.
format	article
author	Salvatore Torquato Jaeuk Kim Michael A. Klatt
author_facet	Salvatore Torquato Jaeuk Kim Michael A. Klatt
author_sort	Salvatore Torquato
title	Local Number Fluctuations in Hyperuniform and Nonhyperuniform Systems: Higher-Order Moments and Distribution Functions
title_short	Local Number Fluctuations in Hyperuniform and Nonhyperuniform Systems: Higher-Order Moments and Distribution Functions
title_full	Local Number Fluctuations in Hyperuniform and Nonhyperuniform Systems: Higher-Order Moments and Distribution Functions
title_fullStr	Local Number Fluctuations in Hyperuniform and Nonhyperuniform Systems: Higher-Order Moments and Distribution Functions
title_full_unstemmed	Local Number Fluctuations in Hyperuniform and Nonhyperuniform Systems: Higher-Order Moments and Distribution Functions
title_sort	local number fluctuations in hyperuniform and nonhyperuniform systems: higher-order moments and distribution functions
publisher	American Physical Society
publishDate	2021
url	https://doaj.org/article/81498106a5d9484e82f0ecf812a05038
work_keys_str_mv	AT salvatoretorquato localnumberfluctuationsinhyperuniformandnonhyperuniformsystemshigherordermomentsanddistributionfunctions AT jaeukkim localnumberfluctuationsinhyperuniformandnonhyperuniformsystemshigherordermomentsanddistributionfunctions AT michaelaklatt localnumberfluctuationsinhyperuniformandnonhyperuniformsystemshigherordermomentsanddistributionfunctions
_version_	1718380920755453952

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

Ejemplares similares