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Disordered hyperuniform many-particle systems are recently discovered exotic states of matter, characterized by a complete suppression of normalized infinite-wavelength density fluctuations and lack of conventional long-range order. Here, we begin a program to quantify the structural properties of nonhyperuniform and hyperuniform networks. In particular, large two-dimensional (2D) Voronoi networks (graphs) containing approximately 10,000 nodes are created from a variety of different point configurations, including the antihyperuniform HIP, nonhyperuniform Poisson process, nonhyperuniform RSA saturated packing, and both non-stealthy and stealthy hyperuniform point processes. We carry out an extensive study of the Voronoi-cell area distribution of each of the networks through determining multiple metrics that characterize the distribution, including their higher-cumulants. We show that the HIP distribution is far from Gaussian; the Poisson and non-stealthy hyperuniform distributions are Gaussian-like distributions, the RSA and the highest stealthy hyperuniform distributions are also non-Gaussian, with diametrically opposite non-Gaussian behavior of the HIP. Moreover, we compute the Voronoi-area correlation functions $C_{00}(r)$ for the networks [M. A. Klatt and S. Torquato, Phys. Rev. E {\bf 90}, 052120 (2014)]. We show that the correlation functions $C_{00}(r)$ qualitatively distinguish the antihyperuniform, nonhyperuniform and hyperuniform Voronoi networks. We find strong anticorrelations in $C_{00}(r)$ (i.e., negative values) for the hyperuniform networks., Comment: 4 figures, 11 pages

Disordered hyperuniform many-body systems are exotic states of matter with novel optical, transport, and mechanical properties. These systems are characterized by an anomalous suppression of large-scale density fluctuations compared to typical liquids, i.e., the structure factor obeys the scaling relation $S(k)\sim \mathcal{B}k^\alpha$ with $\mathcal{B}, \alpha>0$ in the limit $k$\,$\rightarrow$\,$ 0$. Ground-state $d$-dimensional free fermionic gases, which are fundamental models for many metals and semiconductors, are key examples of \textit{quantum} disordered hyperuniform states with important connections to random matrix theory. However, the effects of electron-electron interactions as well as the polarization of the electron liquid on hyperuniformity have not been explored thus far. In this work, we systematically address these questions by deriving the analytical small-$k$ behaviors (and associatedly, $\alpha$ and $\mathcal{B}$) of the total and spin-resolved structure factors of quasi-1D, 2D, and 3D electron liquids for varying polarizations and interaction parameters. We validate that these equilibrium disordered ground states are hyperuniform, as dictated by the fluctuation-compressibility relation. Interestingly, free fermions, partially polarized interacting fermions, and fully polarized interacting fermions are characterized by different values of the small-$k$ scaling exponent $\alpha$ and coefficient $\mathcal{B}$. In particular, partially polarized fermionic liquids exhibit a unique form of \textit{multihyperuniformity}, in which the net configuration exhibits a stronger form of hyperuniformity (i.e., larger $\alpha$) than each individual spin component., Comment: Accepted at PRB

Formulating order metrics that sensitively quantify the degree of order/disorder in many-particle systems in $d$-dimensional Euclidean space $\mathbb{R}^d$ across length scales is an outstanding challenge in physics, chemistry, and materials science. Since an infinite set of $n$-particle correlation functions is required to fully characterize a system, one must settle for a reduced set of structural information, in practice. We initiate a program to use the number variance $\sigma_N^2(R)$ associated with a spherical sampling window of radius $R$ (which encodes pair correlations) and an integral measure derived from it $\Sigma_N(R_i,R_j)$ that depends on two specified radial distances $R_i$ and $R_j$. Across the first three space dimensions ($d = 1,2,3$), we find these metrics can sensitively describe and categorize the degree of order/disorder of 41 different models of antihyperuniform, nonhyperuniform, disordered hyperuniform, and ordered hyperuniform many-particle systems at a specified length scale $R$. Using our local variance metrics, we demonstrate the importance of assessing order/disorder with respect to a specific value of $R$. These order metrics could also aid in the inverse design of structures with prescribed scale-specific degrees of order/disorder that yield desired physical properties. In future work, it would be fruitful to explore the use of higher-order moments [S. Torquato et al., Phys. Rev. X, 11, 021028 (2021)] to devise even more sensitive order metrics., Comment: 19 pages, 8 figures

The probability of finding a spherical "hole" of a given radius $r$ contains crucial structural information about many-body systems. Such hole statistics, including the void conditional nearest-neighbor probability functions $G_V(r)$, have been well studied for hard-sphere fluids in $d$-dimensional Euclidean space $\mathbb{R}^d$. However, little is known about these functions for hard-sphere crystals for values of $r$ beyond the hard-sphere diameter, as large holes are extremely rare in crystal phases. To overcome these computational challenges, we introduce a biased-sampling scheme that accurately determines hole statistics for equilibrium hard spheres on ranges of $r$ that far extend those that could be previously explored. We discover that $G_V(r)$ in crystal and hexatic states exhibits oscillations whose amplitudes increase rapidly with the packing fraction, which stands in contrast to $G_V(r)$ for fluid states that is monotonic in $r$. The oscillations in $G_V(r)$ for 2D crystals are strongly correlated with the local orientational order metric in the vicinity of the holes, and variations in $G_V(r)$ for 3D states indicate a transition between tetrahedral and octahedral holes, demonstrating the power of $G_V(r)$ as a probe of local coordination geometry. To further study the statistics of interparticle spacing in hard-sphere systems, we compute the local packing fraction distribution $f(\phi_l)$ of Delaunay cells, and find that for $d\leq 3$, the excess kurtosis of $f(\phi_l)$ switches sign at a certain transitional global packing fraction. Our investigation facilitates the study of structural and bulk properties of materials that involve the creation of rare large holes, such as the solubility of alloys.

In this article, we employ multiscale physics-informed neural networks (MscalePINNs) for the inverse design of finite-size photonic materials with stealthy hyperuniform (SHU) disordered geometries. Specifically, we show that MscalePINNs can capture the fast spatial variations of complex fields scattered by arrays of dielectric nanocylinders arranged according to isotropic SHU point patterns, thus enabling a systematic methodology to inversely retrieve their effective dielectric profiles. Our approach extends the recently developed high-frequency homogenization theory of hyperuniform media and retrieves more general permittivity profiles for applications-relevant finite-size SHU systems, unveiling unique features related to their isotropic nature. In particular, we numerically corroborate the existence of a transparency region beyond the long-wavelength approximation, enabling effective and isotropic homogenization even without disorder-averaging, in contrast to the case of uncorrelated Poisson random patterns. The flexible multiscale network approach introduced here enables the efficient inverse design of more general effective media and finite-size optical metamaterials with isotropic electromagnetic responses beyond the limitations of traditional homogenization theories.

Hyperuniform point patterns can be classified by the hyperuniformity scaling exponent $\alpha > 0$, that characterizes the power-law scaling behavior of the structure factor $S(\mathbf{k})$ as a function of wavenumber $k\equiv|\mathbf{k}|$ in the vicinity of the origin, e.g., $S(\mathbf{k})\sim|\mathbf{k}|^{\alpha}$ in cases where $S(\mathbf{k})$ varies continuously with $k$ as $k\rightarrow0$. In this paper, we show that the spreadability is an effective method for determining $\alpha$ for quasiperiodic systems where $S(\mathbf{k})$ is discontinuous and consists of a dense set of Bragg peaks. We first transform quasiperiodic and limit-periodic point patterns into two-phase media by mapping them onto packings of identical nonoverlapping disks, where space interior to the disks represents one phase and the space in exterior to them represents the second phase. We then compute the spectral density of the packings, and finally compute and fit the long-time behavior of their excess spreadabilities. Specifically, we show that the excess spreadability can be used to accurately extract $\alpha$ for the 1D limit-periodic period doubling chain and the 1D quasicrystalline Fibonacci chain to within $0.02\%$ of the analytically known exact results. Moreover, we obtain a value of $\alpha = 5.97\pm0.06$ for the 2D Penrose tiling, which had not been computed previously. We also show that one can truncate the small-$k$ region of the scattering information used to compute the spreadability and still obtain an accurate value of $\alpha$. The methods described here offer a simple way to characterize the large-scale translational order present in quasicrystalline and limit-periodic media in any space dimension that are self-similar. Moreover, the scattering information extracted from these two-phase media encoded in the spectral density can be used to estimate their physical properties. (abridged), Comment: 15 pages, 11 figures

The isothermal compressibility (i.e., the asymptotic number variance) of equilibrium liquid water as a function of temperature is minimal near ambient conditions. This anomalous non-monotonic temperature dependence is due to a balance between thermal fluctuations and the formation of tetrahedral hydrogen-bond networks. Since tetrahedrality is a many-body property, it will also influence the higher-order moments of density fluctuations, including the skewness and kurtosis. To gain a more complete picture, we examine these higher-order moments that encapsulate many-body correlations using a recently developed, advanced platform for local density fluctuations. We study an extensive set of simulated phases of water across a range of temperatures (80 K to 1600 K) with various degrees of tetrahedrality, including ice phases, equilibrium liquid water, supercritical water, and disordered nonequilibrium quenches. We find clear signatures of tetrahedrality in the higher-order moments, including the skewness and excess kurtosis, that scale for all cases with the degree of tetrahedrality. More importantly, this scaling behavior leads to non-monotonic temperature dependencies in the higher-order moments for both equilibrium and non-equilibrium phases. Specifically, at near-ambient conditions, the higher-order moments vanish most rapidly for large length scales, and the distribution quickly converges to a Gaussian in our metric. However, at non-ambient conditions, higher-order moments vanish more slowly and hence become more relevant especially for improving information-theoretic approximations of hydrophobic solubility. The temperature non-monotonicity that we observe in the full distribution across length-scales could shed light on water's nested anomalies, i.e., reveal new links between structural, dynamic, and thermodynamic anomalies., Comment: 14 pages, 8 figures

3D bicontinuous two-phase materials are increasingly gaining interest because of their unique multifunctional characteristics and advancements in techniques to fabricate them. Due to their complex topological and structural properties, it still has been nontrivial to develop explicit microstructure-dependent formulas to predict accurately their physical properties. A primary goal of the present paper is to ascertain various microstructural and transport characteristics of five different models of triply periodic bicontinuous porous materials at a porosity $\phi_1=1/2$: those in which the two-phase interfaces are the Schwarz P, Schwarz D and Schoen G minimal surfaces as well as two different pore-channel structures. We ascertain their spectral densities, pore-size distribution functions, local volume-fraction variances, and hyperuniformity order metrics and then use this information to estimate certain effective transport properties via closed-form microstructure-property formulas. Specifically, we estimate the recently introduced time-dependent diffusion spreadability exactly from the spectral density. Moreover, we accurately estimate the fluid permeability of such porous materials from the second moment of the pore-size function and the formation factor, a measure of the tortuosity of the pore space. We also rigorously bound the permeability from above using the spectral density. For the five models with identical cubic unit cells, we find that the permeability, inverse of the specific surface, hyperuniformity order metric, pore-size second moment and long-time spreadability behavior are all positively correlated and rank order the structures in exactly the same way. We also conjecture what structures maximize the fluid permeability for arbitrary porosities and show that this conjecture must be true in the extreme porosity limits by identifying the corresponding optimal structures., Comment: 16 pages, 6 figures, and 5 tables

In previous work [Phys. Rev. X 5, 021020 (2015)], it was shown that stealthy hyperuniform systems can be regarded as hard spheres in Fourier-space in the sense that the the structure factor is exactly zero in a spherical region around the origin in analogy with the pair-correlation function of real-space hard spheres. In this work, we exploit this correspondence to confirm that the densest Fourier-space hard-sphere system is that of a Bravais lattice. This is in contrast to real-space hard-spheres, whose densest configuration is conjectured to be disordered. We also extend the virial series previously suggested for disordered stealthy hyperuniform systems to higher dimensions in order to predict spatial decorrelation as function of dimension. This prediction is then borne out by numerical simulations of disordered stealthy hyperuniform ground states in dimensions $d=2$-$8$., Comment: 9 pages; 6 figures

In photonic crystals the propagation of light is governed by their photonic band structure, an ensemble of propagating states grouped into bands, separated by photonic band gaps. Due to discrete symmetries in spatially strictly periodic dielectric structures their photonic band structure is intrinsically anisotropic. However, for many applications, such as manufacturing artificial structural color materials or developing photonic computing devices, but also for the fundamental understanding of light-matter interactions, it is of major interest to seek materials with long range non-periodic dielectric structures which allow the formation of {\it isotropic} photonic band gaps. Here, we report the first ever 3D isotropic photonic band gap for an optimized disordered stealthy hyperuniform structure for microwaves. The transmission spectra are directly compared to a diamond pattern and an amorphous structure with similar node density. The band structure is measured experimentally for all three microwave structures, manufactured by 3D-Laser-printing for meta-materials with refractive index up to $n=2.1$. Results agree well with finite-difference-time-domain numerical investigations and a priori calculations of the band-gap for the hyperuniform structure: the diamond structure shows gaps but being anisotropic as expected, the stealthy hyperuniform pattern shows an isotropic gap of very similar magnitude, while the amorphous structure does not show a gap at all. The centimeter scaled microwave structures may serve as prototypes for micrometer scaled structures with bandgaps in the technologically very interesting region of infrared (IR)., Comment: 9 pages, 6 figures

Disordered stealthy hyperuniform (SHU) two-phase media are a special subset of hyperuniform structures with novel physical properties due to their hybrid crystal-liquid nature. We have previously shown that the strong-contrast expansion of a linear fractional form of the effective dynamic dielectric constant leads to accurate approximations for disordered two-phase composites when truncated at the two-point level for distinctly different microstructural symmetries in three dimensions. Here, we further elucidate the extraordinary optical and transport properties of disordered SHU media. Among other results, we prove in detail that SHU layered and transversely isotropic media are perfectly transparent (i.e., no Anderson localization, in principle) within finite wavenumber intervals through the third-order terms. Remarkably, the results for these SHU media imply that there can be no Anderson localization within the predicted perfect transparency interval in practice because the localization length is much larger than any practically large sample size. We further contrast and compare the extraordinary physical properties of SHU layered, transversely isotropic, and fully isotropic media to other model nonstealthy microstructures, including their attenuation characteristics, as measured by the imaginary part of effective dielectric constant, and transport properties, as measured by the time-dependent diffusion spreadability. We demonstrate cross-property relations between them: they are positively correlated as the structures span from nonhyperuniform, nonstealthy hyperuniform, and SHU media. Establishing cross-property relations for SHU media for other wave phenomena (e.g., elastodynamics) and transport properties will also be useful. Cross-property relations are generally useful because they enable one to estimate one property, given a measurement of another., Comment: 38 pages, 12 figures, 2 table

Knowledge of exact analytical functional forms for the pair correlation function $g_2(r)$ and its corresponding structure factor $S(k)$ of disordered many-particle systems is limited. For fundamental and practical reasons, it is highly desirable to add to the existing data base of analytical functional forms for such pair statistics. Here, we design a plethora of such pair functions in direct and Fourier spaces across the first three Euclidean space dimensions that are realizable by diverse many-particle systems with varying degrees of correlated disorder across length scales, spanning a wide spectrum of hyperuniform, typical nonhyperuniform and antihyperuniform ones. This is accomplished by utilizing an efficient inverse algorithm that determines equilibrium states with up to pair interactions at positive temperature that precisely match targeted forms for both $g_2(r)$ and $S(k)$. Among other results, we realize an example with the strongest hyperuniform property among known positive-temperature equilibrium states, critical-point systems (implying unusual 1D systems with phase transitions) that are not in the Ising universality class, systems that attain self-similar pair statistics under Fourier transformation, and an experimentally feasible polymer model. We show that our pair functions enable one to achieve systems with a wide range of translational order and self-diffusion coefficients $\cal D$, which are inversely related to one another. One can design other realizable pair statistics via linear combinations of our functions or by applying our inverse procedure to other desirable functional forms. Our approach facilitates the inverse design of materials with desirable physical and chemical properties by tuning their pair statistics.

Torquato and Kim [Phys. Rev. X 11, 296 021002 (2021)] derived exact nonlocal strong-contrast expansions of the effective dynamic dielectric constant tensor that treat general three-dimensional (3D) two-phase composites, which are valid well beyond the long-wavelength regime. Here, we demonstrate that truncating this general rapidly converging series at the two- and three-point levels is a powerful theoretical tool for extracting accurate approximations suited for various microstructural symmetries. We derive such closed-form formulas applicable to transverse polarization in layered media and transverse magnetic polarization in transversely isotropic media, respectively. We use these formulas to estimate effective dielectric constant for models of 3D disordered hyperuniform layered and transversely isotropic media: nonstealthy hyperuniform and stealthy hyperuniform (SHU) media. In particular, we show that SHU media are perfectly transparent (trivially implying no Anderson localization, in principle) within finite wave number intervals through the third-order terms. For these two models, we validate that the second-order formulas, which depend on the spectral density, are already very accurate well beyond the long-wavelength regime by showing very good agreement with the finite-difference time-domain simulations. The high predictive power of the second-order formulas implies that higher-order contributions are negligibly small, and thus, it very accurately approximates multiple scattering effects. Therefore, there can be no Anderson localization in practice within the predicted perfect transparency interval in SHU media because the localization length should be very large compared to any practically large sample size. Our predictive theory provides a foundation for the inverse design of novel effective wave characteristics of disordered and statistically anisotropic structures., Comment: 16 pages, 7 figures

The maximally random jammed (MRJ) state is the most random configuration of strictly jammed (mechanically rigid) nonoverlapping objects. MRJ packings are hyperuniform, meaning their long-wavelength density fluctuations are anomalously suppressed compared to typical disordered systems, i.e., their structure factors $S(\mathbf{k})$ tend to zero as the wavenumber $|\mathbf{k}|$ tends to zero. Here, we show that generating high-quality strictly jammed states for space dimensions $d = 3,4,$ and $5$ is of paramount importance in ensuring hyperuniformity and extracting precise values of the hyperuniformity exponent $\alpha > 0$ for MRJ states, defined by the power-law behavior of $S(\mathbf{k})\sim|\mathbf{k}|^{\alpha}$ in the limit $|\mathbf{k}|\rightarrow0$. Moreover, we show that for fixed $d$ it is more difficult to ensure jamming as the particle number $N$ increases, which results in packings that are nonhyperuniform. Free-volume theory arguments suggest that the ideal MRJ state does not contain rattlers, which act as defects in numerically generated packings. As $d$ increases, we find that the fraction of rattlers decreases substantially. Our analysis of the largest truly jammed packings suggests that the ideal MRJ packings for all dimensions $d\geq3$ are hyperuniform with $\alpha = d - 2$, implying the packings become more hyperuniform as $d$ increases. The differences in $\alpha$ between MRJ packings and recently proposed Manna-class random close packed (RCP) states, which were reported to have $\alpha = 0.25$ in $d=3$ and be nonhyperuniform ($\alpha = 0$) for $d = 4$ and $d = 5$, demonstrate the vivid distinctions between the large-scale structure of RCP and MRJ states in these dimensions. Our work clarifies the importance of the link between true jamming and hyperuniformity and motivates the development of an algorithm to produce rattler-free three-dimensional MRJ packings., Comment: 15 pages, 8 figures

Disordered hyperuniform materials are an emerging class of exotic amorphous states of matter that endow them with singular physical properties. Here, we generalize the Fourier-space based numerical construction procedure for designing {\it isotropic} disordered hyperuniform two-phase heterogeneous materials (i.e., composites) developed by Chen and Torquato [Acta Mater. {\bf 142}, 152 (2018)] to {\it anisotropic} microstructures by explicitly incorporating the {\it vector-dependent} spectral density function ${\tilde \chi}_{_V}({\bf k})$ of {\it arbitrary form} that is realizable. We demonstrate the utility of the procedure by generating a wide spectrum of {\it anisotropic} stealthy hyperuniform (SHU) microstructures with ${\tilde \chi}_{_V}({\bf k}) = 0$ for ${\bf k} \in \Omega$. We show how different exclusion-region shapes with various discrete symmetries and varying size affect the resulting statistically anisotropic microstructures as a function of the and phase volume fraction. We find that, among other properties, the directional hyperuniform behaviors imposed by the shape asymmetry (or anisotropy) of certain exclusion regions give rise to distinct anisotropic structures and degree of uniformity in the distribution of the phases on intermediate and large length scales along different directions. Moreover, while the anisotropic exclusion regions impose strong constraints on the {\it global} symmetry of the resulting media, they can still possess almost isotropic {\it local} structures. Our construction algorithm enables one to control the statistical anisotropy of composite microstructures which is crucial to engineering directional optical, transport and mechanical properties of two-phase composite media., Comment: 22 figures, 24 pages

Disordered stealthy hyperuniform dielectric composites exhibit novel electromagnetic wave transport properties in two and three dimensions. Here, we carry out the first study of the electromagnetic properties of one-dimensional (1D) disordered stealthy hyperuniform layered media. From an exact nonlocal theory, we derive an approximation formula for the effective dynamic dielectric constant tensor ${\boldsymbol \varepsilon}_e({\bf k}_q,\omega)$ of general 1D media that is valid well beyond the quasistatic regime and apply it to 1D stealthy hyperuniform systems. We consider incident waves of transverse polarization, frequency $\omega$, and wavenumber $k_q$. Our formula for ${\boldsymbol \varepsilon}_e({k}_q,\omega)$, which is given in terms of the spectral density, leads to a closed-form relation for the transmittance $T$. Our theoretical predictions are in excellent agreement with finite-difference time-domain (FDTD) simulations. Stealthy hyperuniform layered media have perfect transparency intervals up to a finite wavenumber, implying no Anderson localization, but non-stealthy hyperuniform media are not perfectly transparent. Our predictive theory provides a new path for the inverse design of the wave characteristics of disordered layered media, which are readily fabricated, by engineering their spectral densities., Comment: 8 pages, 6 figures

Hyperuniform many-particle systems are characterized by a structure factor $S({\mathbf{k}})$ that is precisely zero as $|\mathbf{k}|\rightarrow0$; and stealthy hyperuniform systems have $S({\mathbf{k}})=0$ for the finite range $0 < |{\mathbf{k}}| \le K$, called the "exclusion region." Through a process of collective-coordinate optimization, energy-minimizing disordered stealthy hyperuniform systems of moderate size have been made to high accuracy, and their novel physical properties have shown great promise. However, minimizing $S(\mathbf{k})$ in the exclusion region is computationally intensive as the system size becomes large. In this Letter, we present an improved methodology to generate such states using double-double precision calculations on GPUs that reduces the deviations from zero within the exclusion region by a factor of approximately $10^{30}$ for systems sizes more than an order of magnitude larger. We further show that this ultra-high accuracy is required to draw conclusions about their corresponding characteristics, such as the nature of the associated energy landscape and the presence or absence of Anderson localization, which might be masked, even when deviations are relatively small., Comment: 7 pages, 3 figures

An outstanding problem in statistical mechanics is the determination of whether prescribed functional forms of the pair correlation function $g_2(r)$ [or equivalently, structure factor $S(k)$] at some number density $\rho$ can be achieved by $d$-dimensional many-body systems. The Zhang-Torquato conjecture states that any realizable set of pair statistics, whether from a nonequilibrium or equilibrium system, can be achieved by equilibrium systems involving up to two-body interactions. In light of this conjecture, we study the realizability problem of the nonequilibrium iso-$g_2$ process, i.e., the determination of density-dependent effective potentials that yield equilibrium states in which $g_2$ remains invariant for a positive range of densities. Using a precise inverse methodology that determines effective potentials that match hypothesized functional forms of $g_2(r)$ for all $r$ and $S(k)$ for all $k$, we show that the unit-step function $g_2$, which is the zero-density limit of the hard-sphere potential, is remarkably numerically realizable up to the packing fraction $\phi=0.49$ for $d=1$. For $d=2$ and 3, it is realizable up to the maximum ``terminal'' packing fraction $\phi_c=1/2^d$, at which the systems are hyperuniform, implying that the explicitly known necessary conditions for realizability are sufficient up through $\phi_c$. For $\phi$ near but below $\phi_c$, the large-$r$ behaviors of the effective potentials are given exactly by the functional forms $\exp[-\kappa(\phi) r]$ for $d=1$, $r^{-1/2}\exp[-\kappa(\phi) r]$ for $d=2$, and $r^{-1}\exp[-\kappa(\phi) r]$ (Yukawa form) for $d=3$, where $\kappa^{-1}(\phi)$ is a screening length, and for $\phi=\phi_c$, the potentials at large $r$ are given by the pure Coulomb forms in the respective dimensions, as predicted by Torquato and Stillinger [\textit{Phys. Rev. E}, 68, 041113 1-25 (2003)].

The Zhang-Torquato conjecture [Phys. Rev. E 101, 032124 (2020)] states that any realizable pair correlation function $g_2({\bf r})$ or structure factor $S({\bf k})$ of a translationally invariant nonequilibrium system can be attained by an equilibrium ensemble involving only (up to) effective two-body interactions. To test this conjecture, we consider two singular nonequilibrium models of recent interest that also have the exotic hyperuniformity property: a 2D "perfect glass" and a 3D critical absorbing-state model. We find that each nonequilibrium target can be achieved accurately by equilibrium states with effective one- and two-body potentials, lending further support to the conjecture. To characterize the structural degeneracy of such nonequilibrium-equilibrium correspondence, we compute higher-order statistics for both models, as well as those for a hyperuniform 3D uniformly randomized lattice (URL), whose higher-order statistics can be very precisely ascertained. Interestingly, we find that the differences in the higher-order statistics between nonequilibrium and equilibrium systems with matching pair statistics, as measured by the "hole" probability distribution, provides measures of the degree to which a system is out of equilibrium. We show that all three systems studied possess the \textit{bounded-hole} property, and that holes near the maximum hole size in the URL are much rarer than those in the underlying simple cubic lattice. Remarkably, upon quenching, the effective potentials for all three systems possess local energy minima with stronger forms of hyperuniformity compared to their target counterparts. Our work is expected to facilitate the self-assembly of tunable hyperuniform soft-matter systems.

The determination of the pair potential $v({\bf r})$ that accurately yields an equilibrium state at positive temperature $T$ with a prescribed pair correlation function $g_2({\bf r})$ or corresponding structure factor $S({\bf k})$ in $d$-dimensional Euclidean space $\mathbb{R}^d$ is an outstanding inverse statistical mechanics problem with far-reaching implications. Recently, Zhang and Torquato conjectured that any realizable $g_2({\bf r})$ or $S({\bf k})$ corresponding to a translationally invariant nonequilibrium system can be attained by a classical equilibrium ensemble involving only (up to) effective pair interactions. Testing this conjecture for nonequilibrium systems as well as for nontrivial equilibrium states requires improved inverse methodologies. We have devised a novel optimization algorithm to find effective pair potentials that correspond to pair statistics of general translationally invariant disordered many-body equilibrium or nonequilibrium systems at positive temperatures. This methodology utilizes a parameterized family of pointwise basis functions for the potential function whose initial form is informed by small- and large-distance behaviors dictated by statistical-mechanical theory. Subsequently, a nonlinear optimization technique is utilized to minimize an objective function that incorporates both the target pair correlation function $g_2({\bf r})$ and structure factor $S({\bf k})$ so that the small- and large-distance correlations are very accurately captured. To illustrate the versatility and power of our methodology, we accurately determine the effective pair interactions of the following four diverse target systems. We found that the optimized pair potentials generate corresponding pair statistics that accurately match their corresponding targets with total $L_2$-norm errors that are an order of magnitude smaller than that of previous methods.

Studies of disordered heterogeneous media and galaxy cosmology share a common goal: analyzing the distribution of particles at `microscales' to predict physical properties at `macroscales', whether for a liquid, composite material, or entire Universe. The former theory provides an array of techniques to characterize a wide class of microstructures; in this work, we apply them to the distributions of galaxies. We focus on the lower-order correlation functions, `void' and `particle' nearest-neighbor functions, pair-connectedness functions, percolation properties, and a scalar order metric. Compared to homogeneous Poisson and typical disordered systems, the cosmological simulations exhibit enhanced large-scale clustering and longer tails in the nearest-neighbor functions, due to the presence of quasi-long-range correlations. On large scales, the system appears `hyperuniform', due to primordial density fluctuations, whilst on the smallest scales, the system becomes almost `antihyperuniform', and, via the order metric, is shown to be a highly correlated disordered system. Via a finite scaling analysis, we show that the percolation threshold of the galaxy catalogs is significantly lower than for Poisson realizations; this is consistent with the observation that the galaxy distribution contains larger voids. However, the two sets of simulations share a fractal dimension, implying that they lie in the same universality class. Finally, we consider the ability of large-scale clustering statistics to constrain cosmological parameters using simulation-based inference. Both the nearest-neighbor distribution and pair-connectedness function considerably tighten bounds on the amplitude of cosmological fluctuations at a level equivalent to observing twenty-five times more galaxies. These are a useful alternative to the three-particle correlation, and are computable in much reduced time. (Abridged), Comment: 29 pages, accepted by Phys. Rev. X

The capacity to devise order metrics for microstructures of multiphase heterogeneous media is a highly challenging task, given the richness of the possible geometries and topologies of the phases that can arise. This investigation initiates a program to formulate order metrics to characterize the degree of order/disorder of the microstructures of two-phase media in $d$-dimensional Euclidean space $\mathbb{R}^d$ across length scales. In particular, we propose the use of the local volume-fraction variance $\sigma^2_{_V}(R)$ associated with a spherical window of radius $R$ as an order metric. We determine $\sigma^2_{_V}(R)$ as a function of $R$ for 22 different models across the first three space dimensions, including both hyperuniform and nonhyperuniform systems with varying degrees of short- and long-range order. We find that the local volume-fraction variance as well as asymptotic coefficients and integral measures derived from it provide reasonably robust and sensitive order metrics to categorize disordered and ordered two-phase media across all length scales.

A variety of performance demands are being placed on material systems, including desirable mechanical, thermal, electrical, optical, acoustic and flow properties. The purpose of the present article is to review the emerging field of disordered hyperuniform composites and their novel multifunctional characteristics. Disordered hyperuniform media are exotic amorphous states of matter that are characterized by an anomalous suppression of large-scale volume-fraction fluctuations compared to those in "garden-variety" disordered materials. Such unusual composites can have advantages over their periodic counterparts, such as unique or nearly optimal, direction-independent physical properties and robustness against defects. It will be shown that disordered hyperuniform composites and porous media can be endowed with a broad spectrum of extraordinary physical properties, including photonic, phononic, transport, chemical and mechanical characteristics that are only beginning to be discov, Comment: 12 figures

Using the concepts of mixed volumes and quermassintegrals of convex geometry, we derive an exact formula for the exclusion volume for a general convex body that applies in any space dimension, including both the rotationally-averaged exclusion volume and with the same orientation. We show that the sphere minimizes the dimensionless exclusion volume $v_{ex}(K)/v(K)$ among all convex bodies, whether randomly oriented or uniformly oriented, for any $d$, where $v(K)$ is the volume of $K$. When the bodies have the same orientation, the simplex maximizes the dimensionless exclusion volume for any $d$ with a large-$d$ asymptotic scaling behavior of $2^{2d}/d^{3/2}$, which is to be contrasted with the scaling of $2^d$ for the sphere. We present explicit formulas for quermassintegrals for many nonspherical convex bodies as well as as well as lower-dimensional bodies. These results are utilized to determine the rotationally-averaged exclusion volume for these shapes for dimensions 2 through 12. While the sphere is the shape possessing the minimal dimensionless exclusion volume, among the convex bodies considered that are sufficiently compact, the simplex possesses the maximal dimensionless exclusion volume with a scaling behavior of $2^{1.6618\ldots d}$. We also determine the corresponding second virial coefficient $B_2(K)$ of the aforementioned hard hyperparticles and compute estimates of the continuum percolation threshold $\eta_c$ derived previously by the authors. We conjecture that overlapping spheres possess the maximal value of $\eta_c$ among all identical nonzero-volume convex overlapping bodies for $d \ge 2$, randomly or uniformly oriented, and that, among all identical, oriented nonzero-volume convex bodies, overlapping simplices have the minimal value of $\eta_c$ for $d\ge 2$.