Hyperuniformity in point patterns and two-phase random heterogeneous media

By Chase E. Zachary, Salvatore Torquato. 2009

149 citations recorded by the source corpus.

Hyperuniform point patterns are characterized by vanishing infinite-wavelength density fluctuations and encompass all crystal structures, certain quasiperiodic systems, and special disordered point patterns (Torquato and Stillinger 2003 Phys. Rev. E 68 041113). This paper generalizes the notion of hyperuniformity to include also two-phase random heterogeneous media. Hyperuniform random media do not possess infinite-wavelength volume fraction fluctuations, implying that the variance in the local volume fraction in an observation window decays faster than the reciprocal window volume as the window size increases. For microstructures of impenetrable and penetrable spheres, we derive an upper bound on the asymptotic coefficient governing local volume fraction fluctuations in terms of the corresponding quantity describing the variance in the local number density (i.e., number variance). Extensive calculations of the asymptotic number variance coefficients are performed for a number of disordered (e.g., quasiperiodic tilings, classical ‘stealth’ disordered ground states, and certain determinantal point processes), quasicrystal, and ordered (e.g., Bravais and non-Bravais periodic systems) hyperuniform structures in various Euclidean space dimensions, and our results are consistent with a quantitative order metric characterizing the degree of hyperuniformity. We also present corresponding estimates for the asymptotic local volume fraction coefficients for several lattice families. Our results have interesting implications for a certain problem in number theory.

Subfields: Applied Mathematics, Computational Mechanics, Materials Chemistry

View Paper: https://www.semanticscholar.org/paper/5df84c67c4f4491b578b64f864dc1bf1fc668b80

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In Applied Mathematics

In Computational Mechanics

In Materials Chemistry

Citations in the corpus

Jammed hard-particle packings: From Kepler to Bernal and beyond (2010, background): “…calculation of the local number variance for a variety of crystal, quasicrystal and “hyperuniform” disordered point patterns reveals that it provides a useful rank-order of these hyperuniform spatial patterns at large length scales (Torquato and Stillinger, 2003; Zachary and Torquato, 2009).”