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Spherical structure factor and classification of hyperuniformity on the sphere.

Anže Lošdorfer Božič1, Simon Čopar2

  • 1Department of Theoretical Physics, Jožef Stefan Institute, SI-1000 Ljubljana, Slovenia.

Physical Review. E
|April 20, 2019
PubMed
Summary

We introduce a new framework to classify particle arrangements on spheres using hyperuniformity. This method analyzes global order and disorder in spherical distributions, extending Euclidean concepts to curved surfaces.

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Area of Science:

  • Physics
  • Materials Science
  • Computational Mathematics

Background:

  • Particle arrangements on spheres are crucial in diverse systems, from physics to biology.
  • Spherical geometry imposes unique constraints on particle distributions compared to flat spaces.
  • Existing analysis methods for Euclidean space need adaptation for spherical surfaces.

Purpose of the Study:

  • To develop a framework for analyzing and classifying structural order and disorder in particle distributions on a sphere.
  • To extend the concept of hyperuniformity, previously studied in Euclidean space, to spherical surfaces.
  • To provide tools for analyzing spherical computational meshes and biological/synthetic assemblies.

Main Methods:

  • Generalization of the structure factor to spherical surfaces, linked to multipole expansion power spectra.
  • Coupling the spherical structure factor with cap number variance to measure density variations.
  • Deriving analytical forms of variance for different distribution types.
  • Constructing a classification of hyperuniformity for scale-free and other spherical distributions.

Main Results:

  • A novel framework for analyzing particle distributions on spheres is presented.
  • Hyperuniformity on spheres is defined via a vanishing spherical structure factor or cap number variance scaling.
  • The framework extends Euclidean hyperuniformity definitions while highlighting key differences.
  • The study provides a comprehensive tool for detecting long-range order on spheres.

Conclusions:

  • The developed framework offers a robust method for quantifying order in spherical particle systems.
  • This work bridges the gap in analyzing ordered structures on curved surfaces.
  • Applications range from computational science to understanding physical and biological spherical assemblies.