Hyperuniformity in point patterns and two-phase random heterogeneous media

Abstract

Hyperuniform point patterns are characterized by vanishing infinite-wavelength densityfluctuations and encompass all crystal structures, certain quasiperiodic systems, andspecial disordered point patterns (Torquato and Stillinger 2003 Phys. Rev. E 68041113). This paper generalizes the notion of hyperuniformity to include alsotwo-phase random heterogeneous media. Hyperuniform random media do not possessinfinite-wavelength volume fraction fluctuations, implying that the variance in thelocal volume fraction in an observation window decays faster than the reciprocalwindow volume as the window size increases. For microstructures of impenetrableand penetrable spheres, we derive an upper bound on the asymptotic coefficientgoverning local volume fraction fluctuations in terms of the corresponding quantitydescribing the variance in the local number density (i.e., number variance). Extensivecalculations of the asymptotic number variance coefficients are performed for anumber of disordered (e.g., quasiperiodic tilings, classical ‘stealth’ disordered groundstates, and certain determinantal point processes), quasicrystal, and ordered (e.g.,Bravais and non-Bravais periodic systems) hyperuniform structures in variousEuclidean space dimensions, and our results are consistent with a quantitative ordermetric characterizing the degree of hyperuniformity. We also present correspondingestimates for the asymptotic local volume fraction coefficients for several latticefamilies. Our results have interesting implications for a certain problem in numbertheory.