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Hyperuniform many-particle systems are characterized by a structure factor S(k) that is precisely zero as |k|→0; and stealthy hyperuniform systems have S(k)=0 for the finite range 0

The local number variance σ^{2}(R) associated with a spherical sampling window of radius R enables a classification of many-particle systems in d-dimensional Euclidean space R^{d} according to the degree to which large-scale density fluctuations are suppressed, resulting in a demarcation between hyperuniform and nonhyperuniform phyla. To more completely characterize density fluctuations, we carry out an extensive study of higher-order moments or cumulants, including the skewness γ_{1}(R), excess kurtosis γ_{2}(R), and the corresponding probability distribution function P[N(R)] of a large family of models across the first three space dimensions, including both hyperuniform and nonhyperuniform systems with varying degrees of short- and long-range order. To carry out this comprehensive program, we derive new theoretical results that apply to general point processes, and we conduct high-precision numerical studies. Specifically, we derive explicit closed-form integral expressions for γ_{1}(R) and γ_{2}(R) that encode structural information up to three-body and four-body correlation functions, respectively. We also derive rigorous bounds on γ_{1}(R), γ_{2}(R), and P[N(R)] for general point processes and corresponding exact results for general packings of identical spheres. High-quality simulation data for γ_{1}(R), γ_{2}(R), and P[N(R)] are generated for each model. We also ascertain the proximity of P[N(R)] to the normal distribution via a novel Gaussian “distance” metric l_{2}(R). Among all models, the convergence to a central limit theorem (CLT) is generally fastest for the disordered hyperuniform processes in two or higher dimensions such that γ_{1}(R)∼l_{2}(R)∼R^{-(d+1)/2} and γ_{2}(R)∼R^{-(d+1)} for large R. The convergence to a CLT is slower for standard nonhyperuniform models and slowest for the “antihyperuniform” model studied here. We prove that one-dimensional hyperuniform systems of class I or any d-dimensional lattice cannot obey a CLT. Remarkably, we discover a type of universality in that, for all of our models that obey a CLT, the gamma distribution provides a good approximation to P[N(R)] across all dimensions for intermediate to large values of R, enabling us to estimate the large-R scalings of γ_{1}(R), γ_{2}(R), and l_{2}(R). For any d-dimensional model that “decorrelates” or “correlates” with d, we elucidate why P[N(R)] increasingly moves toward or away from Gaussian-like behavior, respectively. Our work sheds light on the fundamental importance of higher-order structural information to fully characterize density fluctuations in many-body systems across length scales and dimensions, and thus has broad implications for condensed matter physics, engineering, mathematics, and biology.

We derive exact nonlocal homogenized constitutive relations for the effective electromagnetic wave properties of disordered two-phase composites and metamaterials from first principles. This exact formalism enables us to extend the long-wavelength limitations of conventional homogenization estimates of the effective dynamic dielectric constant tensor ϵ_{e}(k_{I},ω) for arbitrary microstructures so that it can capture spatial dispersion well beyond the quasistatic regime (where ω and k_{I} are the frequency and wave vector of the incident radiation). We accomplish this task by deriving nonlocal strong-contrast expansions that exactly account for complete microstructural information (infinite set of n-point correlation functions) and hence multiple scattering to all orders for the range of wave numbers for which our extended homogenization theory applies, i.e., 0≤|k_{I}|ℓ≲1 (where ℓ is a characteristic heterogeneity length scale). Because of the fast-convergence properties of such expansions, their lower-order truncations yield accurate closed-form approximate formulas for ϵ_{e}(k_{I},ω) that apply for a wide class of microstructures. These nonlocal formulas are resummed representations of the strong-contrast expansions that still accurately capture multiple scattering to all orders via the microstructural information embodied in the spectral density, which is easy to compute for any composite. The accuracy of these microstructure-dependent approximations is validated by comparison to full-waveform simulation computations for both 2D and 3D ordered and disordered models of composite media. Thus, our closed-form formulas enable one to predict accurately and efficiently the effective wave characteristics well beyond the quasistatic regime for a wide class of composite microstructures without having to perform full-blown simulations. We find that disordered hyperuniform media are generally less lossy than their nonhyperuniform counterparts. We also show that certain disordered hyperuniform particulate composites exhibit novel wave characteristics, including the capacity to act as low-pass filters that transmit waves “isotropically” up to a selected wave number and refractive indices that abruptly change over a narrow range of wave numbers. Our results demonstrate that one can design the effective wave characteristics of a disordered composite by engineering the microstructure to possess tailored spatial correlations at prescribed length scales. Thus, our findings can accelerate the discovery of novel electromagnetic composites.

Disordered hyperuniform packings (or dispersions) are unusual amorphous two-phase materials that are endowed with exotic physical properties. Such hyperuniform systems are characterized by an anomalous suppression of volume-fraction fluctuations at infinitely long-wavelengths, compared to ordinary disordered materials. While there has been growing interest in such singular states of amorphous matter, a major obstacle has been an inability to produce large samples that are perfectly hyperuniform due to practical limitations of conventional numerical and experimental methods. To overcome these limitations, we introduce a general theoretical methodology to construct perfectly hyperuniform packings in d-dimensional Euclidean space Rd. Specifically, beginning with an initial general tessellation of space by disjoint cells that meets a "bounded-cell" condition, hard particles of general shape are placed inside each cell such that the local-cell particle packing fractions are identical to the global packing fraction. We prove that the constructed packings with a polydispersity in size are perfectly hyperuniform in the infinite-sample-size limit, regardless of particle shapes, positions, and numbers per cell. We use this theoretical formulation to devise an efficient and tunable algorithm to generate extremely large realizations of such packings. We employ two distinct initial tessellations: Voronoi as well as sphere tessellations. Beginning with Voronoi tessellations, we show that our algorithm can remarkably convert extremely large nonhyperuniform packings into hyperuniform ones in R² and R³. Implementing our theoretical methodology on sphere tessellations, we establish the hyperuniformity of the classical Hashin-Shtrikman multiscale coated-spheres structures, which are known to be two-phase media microstructures that possess optimal effective transport and elastic properties. A consequence of our work is a rigorous demonstration that packings that have identical tessellations can either be nonhyperuniform or hyperuniform by simply tuning local characteristics. It is noteworthy that our computationally designed hyperuniform two-phase systems can easily be fabricated via state-of-the-art methods, such as 2D photolithographic and 3D printing technologies. In addition, the tunability of our methodology offers a route for the discovery of novel disordered hyperuniform two-phase materials. [ABSTRACT FROM AUTHOR]