Hyperuniformity of Weighted Particle Systems

Abstract: Hyperuniform particle arrangements are characterized by a local number variance that grows more slowly than the volume of the observation window. We generalize this concept to describe particle systems in which particles carry weights: internal degrees of freedom such as scalars, vectors, pseudovectors, directors, tensors, or extrinsic local attributes. Our generalization extends hyperuniformity from fluctuations in particle positions to fluctuations in the spatial distribution of weights. We derive generalized weighted pair correlation, autocovariance, and spectral functions, and show their relation to the local variance in weighted many-particle systems. Applying this formalism to bond-orientational ordered phases, dipolar liquid water, Voronoi-cell volumes, and certain ionic liquids, we demonstrate that hyperuniformity in the particle system does not necessarily translate to hyperuniformity of the weighted system. In fact, cases exist where a hyperuniform particle system becomes antihyperuniform when weighted, and others where nonhyperuniform or antihyperuniform particle systems yield hyperuniform weighted systems. This theoretical framework provides a road map for quantifying large-scale fluctuations in weighted many-particle systems, offering a powerful tool for identifying systems with novel physical properties.

• The paper develops generalized correlation functions, spectral densities, and variance formulas for scalar, vector, complex, and geometric particle weights, extending hyperuniformity beyond particle positions.
• The paper shows that weighting can fundamentally reclassify systems: bond-orientational weights produce antihyperuniformity, while Voronoi volumes and excess side numbers generate class I hyperuniformity through geometric screening.
• The paper demonstrates that uncorrelated weights remain nonhyperuniform even when their mean is zero, while correlated weights can suppress fluctuations, with applications to water dipoles, tessellations, packings, and active or ordered materials.

Overview

This paper by Torquato, Kim, Klatt, Car, and Steinhardt extends the theory of hyperuniformity from fluctuations in particle positions to fluctuations in particle weights—internal degrees of freedom such as scalar charges and masses, vector dipoles and velocities, pseudovector spins, complex bond-orientational order parameters, and extrinsic attributes like Voronoi-cell volumes. The central object is a marked (weighted) point process in $\mathbb{R}^d$, for which the authors derive generalized pair correlation functions, autocovariance functions, spectral densities, and local variances, all reducing to the standard unweighted formalism when every weight is unity. The framework's principal finding is that hyperuniformity classification is not preserved under weighting: a hyperuniform point configuration can become antihyperuniform when weighted, while nonhyperuniform or even antihyperuniform configurations can yield class I hyperuniform weighted systems.

Formalism

The authors treat each particle as carrying a complex $d$-dimensional weight ${\bf f}_i$, forming a generalized $2d$-dimensional coordinate ${\bf Q}_i = ({\bf r}_i, {\bf f}_i)$. From the reduced probability densities $\rho_n({\bf Q}^n)$ they define a generalized pair correlation function $g_{2,f}({\bf r};{\bf f}_1,{\bf f}_2)$ and its weight-averaged counterpart $g_{2,f}({\bf r})$, obtained by contracting with the inner product ${\bf f}_1^* \cdot {\bf f}_2$ over the weight distribution. The weighted Dirac measure $$n_f({\bf x}) = \sum_j {\bf f}_j \delta({\bf x}-{\bf r}_j)$$ leads via Campbell's theorem to a weight-averaged autocovariance function

$d$0

where $d$1 is the weight-averaged total correlation function. Its Fourier transform gives the spectral density $d$2, which coincides (up to forward scattering) with the weight-averaged structure factor. The local variance within a spherical window obeys direct- and Fourier-space formulas analogous to the unweighted case, involving the scaled intersection volume $d$3.

Hyperuniformity of the weighted system is defined by $d$4, equivalent to the sum rule $d$5, which requires negative correlations. The three hyperuniformity classes I–III carry over through the small-$d$6 scaling $d$7; antihyperuniformity corresponds to divergence at the origin. To quantify reclassification upon weighting, the authors introduce an exponent change $d$8 computed from the excess spectral density. Notably, if weights are spatially uncorrelated with zero mean, the weighted system is necessarily nonhyperuniform regardless of the underlying configuration—global charge neutrality alone does not suffice.

Bond-orientational order: hyperuniformity lost under weighting

Applying the formalism to 2D phases characterized by KTHNY theory yields a striking inversion. The isotropic liquid, hexatic, nematic, tetratic, and solid phases all have nonhyperuniform unweighted point configurations ($d$9), but weighting particles by the complex order parameter ${\bf f}_i$0 produces antihyperuniformity wherever quasi-long-range orientational order exists. In the hexatic phase, the algebraic decay ${\bf f}_i$1 with ${\bf f}_i$2 implies ${\bf f}_i$3; analogous results hold for nematic (${\bf f}_i$4) and tetratic (${\bf f}_i$5) order. For the 2D hard-disk solid, a single large configuration (${\bf f}_i$6, ${\bf f}_i$7) from Bernard and Krauth gives ${\bf f}_i$8. Most dramatically, ultradense stealthy hyperuniform packings at packing fraction ${\bf f}_i$9—hyperuniform in their positions—become antihyperuniform under $2d$0 weighting, with $2d$1 and local variance growing as $2d$2, faster than the window volume. This is presented as the first example of such an extreme reclassification.

Dipolar water

Reanalyzing the dipole correlation data of Sharma, Resta, and Car for liquid water, the authors find that equilibrium water is typically nonhyperuniform with respect to its dipole moments: the normalized spectral density approaches a bounded positive constant as $2d$3, and the scaled local variance tends to a positive constant, so dipole variance grows proportionally to window volume. The high dielectric constant thus reflects strong but not fluctuation-suppressing dipolar correlations mediated by the hydrogen-bond network. The paper also connects the integrated autocovariance to the Kirkwood correlation function, interpretable as a cumulative coordination number for dipole orientations.

Voronoi-cell volume weights: hyperuniformity gained

Assigning each particle its Voronoi-cell volume as a scalar weight converts every model studied into a class I hyperuniform system, because the cell volume exactly tiles space and fluctuations are confined to window boundaries. For the 1D Poisson process, exact results are obtained: the weight-averaged total correlation function decays exponentially, the spectral density scales as $2d$4 (so $2d$5), and the local variance is bounded, approaching $2d$6. In 2D, the antihyperuniform hyperplane intersection process (HIP), nonhyperuniform random sequential addition (RSA), and stealthy hyperuniform (SHU) configurations all yield class I weighted systems with exponents converging to $2d$7 (HIP), $2d$8 (RSA), and $2d$9 (SHU). The deviations from 4 are attributed, following Gabrielli, Joyce, and Torquato, to long-range correlations in first moments ${\bf Q}_i = ({\bf r}_i, {\bf f}_i)$0 between particle centers and cell centroids. For 3D maximally random jammed (MRJ) packings—which are class II hyperuniform unweighted—the volume-weighted systems become class I with ${\bf Q}_i = ({\bf r}_i, {\bf f}_i)$1 and variance scaling as ${\bf Q}_i = ({\bf r}_i, {\bf f}_i)$2. The authors also compute a hyperuniformity order metric ${\bf Q}_i = ({\bf r}_i, {\bf f}_i)$3 ranking suppression of large-scale fluctuations; across models the ordering HIP > RSA > SHU persists under both volume and excess-side-number weighting.

Excess side numbers as charges

The paper maps disordered 2D tessellations to nonequilibrium ionic liquids by assigning each particle the integer excess side number ${\bf Q}_i = ({\bf r}_i, {\bf f}_i)$4 of its Voronoi cell, which has zero mean by Euler's formula, ensuring overall charge neutrality. For HIP, RSA, and SHU alike, the resulting charged systems are class I hyperuniform with ${\bf Q}_i = ({\bf r}_i, {\bf f}_i)$5 (measured values ${\bf Q}_i = ({\bf r}_i, {\bf f}_i)$6, ${\bf Q}_i = ({\bf r}_i, {\bf f}_i)$7, ${\bf Q}_i = ({\bf r}_i, {\bf f}_i)$8 at the largest sizes) and variances growing like the window surface area. The mechanism is effective screening imposed by the Voronoi network, producing short-ranged positive and negative correlations. A control experiment demonstrates that neutrality alone is insufficient: randomly shuffling the ${\bf Q}_i = ({\bf r}_i, {\bf f}_i)$9 weights on RSA positions destroys screening, yielding a nonhyperuniform system whose spectral density equals the constant $\rho_n({\bf Q}^n)$0, consistent with the general result for uncorrelated weights.

An appendix extends the analysis to weighted random fields defined by convolution with a localized kernel, showing that a hyperuniform field remains hyperuniform after kernel smoothing, whereas a nonhyperuniform field becomes hyperuniform only if the product $\rho_n({\bf Q}^n)$1 vanishes at the origin.

Limitations and open questions

Several caveats attend these results. The exponent $\rho_n({\bf Q}^n)$2 for the hard-disk solid rests on a single very large configuration, and the asymptotic decay of $\rho_n({\bf Q}^n)$3 in solid phases is not theoretically established. The convergence of measured exponents ($\rho_n({\bf Q}^n)$4, $\rho_n({\bf Q}^n)$5, or $\rho_n({\bf Q}^n)$6; $\rho_n({\bf Q}^n)$7) is inferred from finite-size trends rather than proven. The claim that $\rho_n({\bf Q}^n)$8 universally is supported only for three 2D models; whether it holds in other dimensions, whether class II charged systems exist, and whether an effective screening length can be extracted for these nonequilibrium ionic liquids remain open. The existence of disordered systems that remain hyperuniform under $\rho_n({\bf Q}^n)$9 weighting is likewise unresolved.

Conclusion

The paper supplies a complete statistical-mechanical apparatus—generalized correlation functions, spectral densities, local variances, sum rules, and order metrics—for characterizing large-scale fluctuations in weighted many-particle systems. Its applications demonstrate that positional hyperuniformity and weighted hyperuniformity are logically independent properties: bond-orientational order drives antihyperuniformity even in positionally hyperuniform packings, while geometric constraints from tessellations impose class I hyperuniformity on charge-like weights irrespective of the underlying configuration's classification. The framework is directly applicable to polymer bond vectors, liquid-crystal directors, active-matter velocities, and spin systems, providing a quantitative basis for connecting suppressed or enhanced weight fluctuations to physical properties such as dielectric response and scattering.