Ensemble Theory for Stealthy Hyperuniform Disordered Ground States

Abstract

It has been shown numerically that systems of particles interacting with isotropic "stealthy" bounded long-ranged pair potentials (similar to Friedel oscillations) have classical ground states that are (counterintuitively) disordered, hyperuniform, and highly degenerate. Disordered hyperuniform systems have received attention recently because they are distinguishable exotic states of matter poised between a crystal and liquid that are endowed with novel thermodynamic and physical properties. The task of formulating an ensemble theory that yields analytical predictions for the structural characteristics and other properties of stealthy degenerate ground states in d-dimensional Euclidean space R d is highly nontrivial because the dimensionality of the configuration space depends on the number density ρ and there is a multitude of ways of sampling the ground-state manifold, each with its own probability measure for finding a particular ground-state configuration. The purpose of this paper is to take some initial steps in this direction. Specifically, we derive general exact relations for thermodynamic properties (energy, pressure, and isothermal compressibility) that apply to any ground-state ensemble as a function of ρ in any d, and we show how disordered degenerate ground states arise as part of the ground-state manifold. We also derive exact integral conditions that both the pair correlation function g 2 ðrÞ and structure factor SðkÞ must obey for any d. We then specialize our results to the canonical ensemble (in the zero-temperature limit) by exploiting an ansatz that stealthy states behave remarkably like "pseudo"-equilibrium hard-sphere systems in Fourier space. Our theoretical predictions for g 2 ðrÞ and SðkÞ are in excellent agreement with computer simulations across the first three space dimensions. These results are used to obtain order metrics, local number variance, and nearest-neighbor functions across dimensions. We also derive accurate analytical formulas for the structure factor and thermal expansion coefficient for the excited states at sufficiently small temperatures for any d. The development of this theory provides new insights regarding our fundamental understanding of the nature and formation of low-temperature states of amorphous matter. Our work also offers challenges to experimentalists to synthesize stealthy ground states at the molecular level.

Evolutionary‐Optimized Photonic Network Structure in White Beetle Wing Scales

“…This fact is further corroborated by investigating the local variation of the volume fraction of scatterers as a function of a spherical probe of radius R. Figure 3f shows that the scaled local volume fraction variance R 3 σ 2 (R) varies linearly with R and plateaus for large R. Hyperuniformity requires a faster decay of σ 2 (R) for R → ∞. [28,32] The results of Figure 3 therefore clearly demonstrate the absence of hyperuniformity and that the two size distributions of Figure 3b,c are close to Gaussian. In addition, we have employed a variety of other structural descriptors to spatially characterize the beetle ultrastructure, as detailed in the Supporting Information.…”

Structural Transition in a Fluid of Spheroids: A Low-Density Vestige of Jamming

“…Our algorithm for the detection of particle center positions and orientational angles is based on the PLuTARC algorithm, as also on the well-known algorithm of Crocker and Grier [41], previously used for tracking of colloidal spheres [39][40]. The algorithm, implemented in C/C++, takes as an input a three-dimensional stack of confocal slice images through the sample.…”

“…To explore the role of particle anisotropy for the fluid structure beyond the first coordination shell, we use the gðrÞ to define a positional order parameter [39]: τ ≡ R ∞ 0 jgðrÞ − 1j 2 r 2 dr, where r is in units of L. By definition, τ ¼ 0 for an ideal gas, where all correlations are missing. At a given ϕ, the strongest positional correlations occur for the spheres, where the orientations are completely disordered [see Fig.…”

Transport, geometrical, and topological properties of stealthy disordered hyperuniform two-phase systems

“…As χ increases above 0.5, longrange translational and rotational order begins to emerge and eventually the system crystallizes. We have previously studied these disordered ground states, and computed their pair correlation functions, [18][19][20]25,26 structure factors, [18][19][20]25,26 Voronoi cell volume distribution, and particle-exclusion probabilities.…”

“…Materials that are simultaneously disordered and hyperuniform can be regarded to be exotic states of matter that lie between a crystal and a liquid; a) Electronic mail: torquato@electron.princeton.edu they behave more like crystals in the manner in which they suppress large-scale density fluctuations, and yet they also resemble typical statistically isotropic liquids and glasses with no Bragg peaks. 18 An important class of disordered hyperuniform manyparticle systems is comprised of the classical ground states of "stealthy potentials," [18][19][20][21] which are bounded, long-range, pairwise additive potentials designed in Fourier space. These classical ground states are of particular fundamental interest because they can be degenerate and noncrystalline.…”