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Local density fluctuations, hyperuniformity, and order metrics.

Salvatore Torquato1, Frank H Stillinger

  • 1Department of Chemistry and Princeton Materials Institute, Princeton University, Princeton, New Jersey 08544, USA.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|December 20, 2003
PubMed
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This study quantifies density fluctuations in point patterns, revealing hyperuniform systems exhibit minimal variance. The research introduces new formulations for variance calculation and identifies optimal patterns, offering a new metric for spatial order.

Area of Science:

  • Statistical Physics
  • Materials Science
  • Computational Geometry

Background:

  • Understanding density fluctuations in point patterns is crucial for various scientific disciplines.
  • Hyperuniform systems, characterized by suppressed large-scale density fluctuations, are of particular interest.
  • Existing methods for quantifying these fluctuations present theoretical and computational challenges.

Purpose of the Study:

  • To characterize fundamental aspects of local density fluctuations in general point patterns across any dimension.
  • To study the variance in point counts within windows of varying sizes.
  • To further elucidate the properties of hyperuniform systems and their unique scaling behaviors.

Main Methods:

  • Derivation of two formulations for number variance: ensemble-average (for homogeneous systems) and volume-average (for single realizations).

Related Experiment Videos

  • Analysis using real-space and Fourier representations.
  • Evaluation of variance for periodic and nonperiodic patterns in 1, 2, and 3 dimensions using spherical windows.

    • Main Results:

    • Hyperuniform systems exhibit local variance scaling with surface area, not volume, for large windows.
    • Homogeneous hyperuniform patterns are shown to be at a critical point with long-range direct correlation functions.
    • The simple periodic linear array minimizes average variance in 1D hyperuniform systems; BCC lattice shows lower variance than FCC in 3D.

    Conclusions:

    • The local variance serves as a valuable order metric for general point patterns.
    • The study provides exact evaluations for correlation functions and critical exponents in certain hyperuniform disordered systems.
    • Dispels the conjecture that densest packing lattices universally minimize variance, demonstrating dimension-specific behaviors.