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.

Figures (12)

• Figure 1: Schematic of a point configuration with (complex-valued) vector weights of different magnitudes and directions.
• Figure 2: Point configurations derived from 2D ultradense hyperuniform packings with packing fraction $\phi=0.86$To25aKi25a weighted with the sixfold bond-orientational order parameter $\psi_6$. (a) Representative image of a 2D weighted point configuration. The colors depict the argument of the complex-valued parameter $\psi_6$. The high degree of polycrystallinity is clearly picked up by the 6-fold bond-orientational order. (b) Log-log plot of the dimensionless spectral densities $\tilde{\chi}(k)/\rho$ and $\tilde{\chi}_{\psi_6}(k)/\rho$ for 2D unweighted and weighted point configurations. The weighted one is antihyperuniform with $\alpha_{\psi_6}\approx -0.90\pm 0.17$. (c) Log-log plot of the dimensionless local variances $\sigma_N^2(R)$ and $\sigma_{\psi_6}^2(R)$ for the unweighted and weighted ones, respectively. The black and red dotted lines illustrate the large-$R$ growth rate of the local variances.
• Figure 3: Spectral density and local variance computed from the dipole correlation function $c_m(r)$ of an equilibrium water model in Ref. Sh07. (a) Dimensionless generalized spectral density $\tilde{\chi}_{\bm \mu}(k) / [\rho \expval{\abs{\bm {\mu}}^2}]$ as a function of a wave number $k\rho^{-1/3}$. (b) Dimensionless local variance $\sigma_{\bm \mu} ^2(R)/ \expval{\abs{{\bm \mu}}^2}$ scaled by the dimensionless window volume $\rho v_1(R)$ as a function of a dimensionless window radius $R\rho^{1/3}$.
• Figure 4: Theoretical prediction of a 1D Poisson point process weighted with the Voronoi volume. (a) Dimensionless weight-averaged total correlation function $\rho^2 h_v(r)$ as a function of the dimensionless radius $r\rho$; see Eq. \ref{['eq:hv-1D-Poisson']}. (b) Dimensionless weight-averaged spectral density $\tilde{\chi}_v(k)$ as a function of the dimensionless wave number $k/\rho$ on a log-log scale; see Eq. \ref{['eq:chivk-1D-Poisson']}. (c) Dimensionless weight-averaged local variance $\rho^2 \sigma_v^2(R)$ as a function of the dimensionless radius $R\rho$ in a linear scale; see Eq. \ref{['eq:sigma-1D-Poisson']}.
• Figure 5: Representative images of 2D weighted point configurations derived from (a) HIP, (b) RSA (with a packing fraction $\phi=0.30$), and (c) SHU (with a stealthiness parameter $\chi=0.35$) using the dimensionless Voronoi cell-areas $\rho v$ as weights. Each Voronoi cell is colored according to the area of its Voronoi cell.