Structural Characterization of Many-Particle Systems on Approach to Hyperuniform States

Abstract

We explore quantitative descriptors that herald when a many-particle system in $d$-dimensional Euclidean space ${\mathbb{R}}^d$ approaches a hyperuniform state as a function of the relevant control parameter. We establish quantitative criteria to ascertain the extent of hyperuniform and nonhyperuniform distance-scaling regimes n terms of the ratio $B/A$, where $A$ is "volume" coefficient and $B$ is "surface-area" coefficient associated with the local number variance ${\sigma }^2(R)$ for a spherical window of radius $R$. To complement the known direct-space representation of the coefficient $B$ in terms of the total correlation function $h(\mathbf{r})$, we derive its corresponding Fourier representation in terms of the structure factor $S(\mathbf{k})$, which is especially useful when scattering information is available experimentally or theoretically. We show that the free-volume theory of the pressure of equilibrium packings of identical hard spheres that approach a strictly jammed state either along the stable crystal or metastable disordered branch dictates that such end states be exactly hyperuniform. Using the ratio $B/A$, the hyperuniformity index $H$ and the direct-correlation function length scale ${\xi }_c$, we study three different exactly solvable models as a function of the relevant control parameter, either density or temperature, with end states that are perfectly hyperuniform. We analyze equilibrium hard rods and "sticky" hard-sphere systems in arbitrary space dimension $d$ as a function of density. We also examine low-temperature excited states of many-particle systems interacting with "stealthy" long-ranged pair interactions as the temperature tends to zero. The capacity to identify hyperuniform scaling regimes should be particularly useful in analyzing experimentally- or computationally-generated samples that are necessarily of finite size.

19 pages, 14 figures

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• Existence of Nonequilibrium Glasses in the Degenerate Stealthy Hyperuniform Ground-State Manifold
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