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

Salvatore Torquato, Jaeuk Kim, and Michael A. Klatt
Phys. Rev. X 11, 021028 – Published 5 May 2021
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Abstract

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 Rd 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.

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  • Received 10 December 2020
  • Revised 26 February 2021
  • Accepted 3 March 2021

DOI:https://doi.org/10.1103/PhysRevX.11.021028

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Condensed Matter, Materials & Applied PhysicsStatistical Physics & Thermodynamics

Authors & Affiliations

Salvatore Torquato1,2,*, Jaeuk Kim1,†, and Michael A. Klatt1,3,‡

  • 1Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
  • 2Department of Chemistry, Princeton Institute for the Science and Technology of Materials, and Program in Applied and Computational Mathematics, Princeton University, Princeton, New Jersey 08544, USA
  • 3Institut für Theoretische Physik, FAU Erlangen-Nürnberg, Staudtstr. 7, 91058 Erlangen, Germany

  • *torquato@princeton.edu
  • jaeukk@princeton.edu
  • mklatt@princeton.edu

Popular Summary

The emerging field of “hyperuniformity” provides mathematical tools for classifying large ensembles of particles based on the degree to which density fluctuations are suppressed according to their first two moments (the expected value and the variance). Originally formulated to describe how atoms are positioned in crystals, quasicrystals, and amorphous solids, the framework is useful in many studies, from the distribution of prime numbers to photoreceptor cells in avian retina. Here, we stress the importance of characterizing the higher-order moments of any system, hyperuniform or not.

To more completely characterize density fluctuations, we carry out an extensive theoretical and computational study of higher-order moments and the corresponding probability distribution function of a large class of models across the first three space dimensions. These models describe both hyperuniform and nonhyperuniform systems, that is, those in which density fluctuations are greatly suppressed and those in which they are not. Remarkably, we discover that diverse systems—encompassing the majority of our models—can be described by “universal” distribution functions.

Our work elucidates the fundamental importance of higher-order structural information to fully characterize density fluctuations in many-body systems across length scales and dimensions, and thus it has broad implications for condensed-matter physics, engineering, mathematics, and biology.

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Vol. 11, Iss. 2 — April - June 2021

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