Local density fluctuations, hyperuniformity, and order metrics

Questions concerning the properties and quantification of density fluctuations in point patterns continue to provide many theoretical challenges. The purpose of this paper is to characterize certain fundamental aspects of local density fluctuations associated with general point patterns in any space dimension d. Our specific objectives are to study the variance in the number of points contained within a regularly shaped window Omega of arbitrary size, and to further illuminate our understanding of hyperuniform systems, i.e., point patterns that do not possess infinite-wavelength fluctuations. For large windows, hyperuniform systems are characterized by a local variance that grows only as the surface area (rather than the volume) of the window. We derive two formulations for the number variance: (i) an ensemble-average formulation, which is valid for statistically homogeneous systems, and (ii) a volume-average formulation, applicable to a single realization of a general point pattern in the large-system limit. The ensemble-average formulation (which includes both real-space and Fourier representations) enables us to show that a homogeneous point pattern in a hyperuniform state is at a "critical point" of a type with appropriate scaling laws and critical exponents, but one in which the direct correlation function (rather than the pair correlation function) is long ranged. We also prove that the non-negativity of the local number variance does not add a new realizability condition on the pair correlation. The volume-average formulation is superior for certain computational purposes, including optimization studies in which it is desired to find the particular point pattern with an extremal or targeted value of the variance. We prove that the simple periodic linear array yields the global minimum value of the average variance among all infinite one-dimensional hyperuniform patterns. We also evaluate the variance for common infinite periodic lattices as well as certain nonperiodic point patterns in one, two, and three dimensions for spherical windows, enabling us to rank-order the spatial patterns. Our results suggest that the local variance may serve as a useful order metric for general point patterns. Contrary to the conjecture that the lattices associated with the densest packing of congruent spheres have the smallest variance regardless of the space dimension, we show that for d=3, the body-centered cubic lattice has a smaller variance than the face-centered cubic lattice. Finally, for certain hyperuniform disordered point patterns, we evaluate the direct correlation function, structure factor, and associated critical exponents exactly.