Nearest-Neighbor Functions for Disordered Stealthy Hyperuniform Many-Particle Systems

Abstract

Disordered stealthy many-particle systems in ${\mathbb{R}}^d$ are exotic states of matter that suppress single scattering events for a finite range of wavenumbers around the origin in reciprocal space. We derive analytical formulas for the nearest-neighbor functions of disordered stealthy systems. First, we analyze asymptotic small-$r$ approximations and bounding expressions of the nearest-neighbor functions based on the pseudo-hard-sphere ansatz. We then determine how many of the standard $n$-point correlation functions are needed to determine the nearest neighbor functions, and find that a finite number suffice. Via theoretical and computational methods, we compare the large-$r$ behavior of these functions for disordered stealthy systems to those belonging to crystalline lattices. Such ordered and disordered stealthy systems have bounded hole sizes. However, we find that the approach to the critical-hole size can be quantitatively different. We argue that the probability of finding a hole close to the critical-hole size should decrease as a power law with an exponent only dependent on the space dimension $d$ for ordered systems, but that this probability decays asymptotically faster for disordered systems. This implies that holes close to the critical-hole size are rarer in disordered systems. The rarity of observing large holes in disordered systems creates substantial numerical difficulties in sampling the nearest neighbor distributions near the critical-hole size. This motivates both the need for new computational methods for efficient sampling and the development of novel theoretical methods. We also devise a simple analytical formula that accurately describes these systems in the underconstrained regime for all $r$. These results provide a foundation for the analytical description of the nearest-neighbor functions of stealthy systems in the disordered, underconstrained regime.

arXiv comment41 pages, 17 figures

References in corpus (31)

• Hyperuniform States of Matter
• Designer disordered materials with large complete photonic band gaps
• Unexpected Density Fluctuations in Jammed Disordered Sphere Packings
• Avian photoreceptor patterns represent a disordered hyperuniform solution to a multiscale packing problem
• Isotropic Band Gaps and Freeform Waveguides Observed in Hyperuniform Disordered Photonic Solids
• Hyperuniformity in point patterns and two-phase random heterogeneous media
• Random Sequential Addition of Hard Spheres in High Euclidean Dimensions
• Classical Disordered Ground States: Super-Ideal Gases, and Stealth and Equi-Luminous Materials
• Transport Phase Diagram and Anderson Localization in Hyperuniform Disordered Photonic Materials
• Hyperuniform long-range correlations are a signature of disordered jammed hard-particle packings
• Photonic band gap in isotropic hyperuniform disordered solids with low dielectric contrast
• The Glass-like Universe: Real-space correlation properties of standard cosmological models
• Ensemble Theory for Stealthy Hyperuniform Disordered Ground States
• Precise Algorithm to Generate Random Sequential Addition of Hard Hyperspheres at Saturation
• Point processes in arbitrary dimension from fermionic gases, random matrix theory, and number theory
• Suppressed compressibility at large scale in jammed packings of size disperse spheres
• Transport, Geometrical and Topological Properties of Stealthy Disordered Hyperuniform Two-Phase Systems
• Hyperuniformity of Quasicrystals
• Incompressibility of polydisperse random close packed colloidal particles
• Statistical properties of determinantal point processes in high-dimensional Euclidean spaces
• Ground states of stealthy hyperuniform potentials: I. Entropically favored configurations
• Collective Coordinate Control of Density Distributions
• Crystalline ground states for classical particles
• The Perfect Glass Paradigm: Disordered Hyperuniform Glasses Down to Absolute Zero
• Anomalous local coordination, density fluctuations, and void statistics in disordered hyperuniform many-particle ground states
• Equilibrium Phase Behavior and Maximally Random Jammed State of Truncated Tetrahedra
• Generation of Primordial Cosmological Perturbations from Statistical Mechanical Models
• Generalized stealthy hyperuniform processes : maximal rigidity and the bounded holes conjecture
• Ground states of stealthy hyperuniform potentials. II. Stacked-slider phases
• Characterizing the mesh size of polymer solutions via the pore size distribution
• Characterization of Maximally Random Jammed Sphere Packings: II. Correlation Functions and Density Fluctuations