Collective density variables rho (k) have proved to be useful tools in the study of many-body problems in a variety of fields that are concerned with structural and kinematic phenomena. In spite of their broad applicability, mathematical understanding of collective density variables remains an underexplored subject. In this paper, we examine features associated with collective density variables in two dimensions using numerical exploration techniques to generate particle patterns in the classical ground state. Particle pair interactions are governed by a continuous, bounded potential. Our approach involves constraining related collective parameters C (k) , with wave vector k magnitudes at or below a chosen cutoff, to their absolute minimum values. Density fluctuations for those k 's thus are suppressed. The resulting investigation distinguishes three structural regimes as the number of constrained wave vectors is increased-disordered, wavy crystalline, and crystalline regimes-each with characteristic distinguishing features. It should be noted that our choice of pair potential can lead to pair correlation functions that exhibit an effective hard core and thus signal the formation of a hard-disk-like equilibrium fluid. In addition, our method creates particle patterns that are hyperuniform, thus supporting the notion that structural glasses can be hyperuniform as the temperature T-->0 .
Real collective density variables C(k) [cf. Eq. 1 3 ] in many-particle systems arise from nonlinear transformations of particle positions, and determine the structure factor S(k) , where k denotes the wave vector. Our objective is to prescribe C(k) and then to find many-particle configurations that correspond to such a target C(k) using a numerical optimization technique. Numerical results reported here extend earlier one- and two-dimensional studies to include three dimensions. In addition, they demonstrate the capacity to control S(k) in the neighborhood of |k|=0. The optimization method employed generates multiparticle configurations for which S(k) proportional, |k|alpha, |k|
Sphere packings in high dimensions have been the subject of recent theoretical interest. Employing numerical and theoretical methods, we investigate the structural characteristics of random sequential addition (RSA) of congruent spheres in d-dimensional Euclidean space R d in the infinitetime or saturation limit for the first six space dimensions (1 ≤ d ≤ 6). Specifically, we determine the saturation density, pair correlation function, cumulative coordination number and the structure factor in each of these dimensions. We find that for 2 ≤ d ≤ 6, the saturation density φ s scales with dimension as φ s = c 1 /2 d + c 2 d/2 d , where c 1 = 0.202048 and c 2 = 0.973872. We also show analytically that the same density scaling persists in the high-dimensional limit, albeit with different coefficients. A byproduct of this high-dimensional analysis is a relatively sharp lower bound on the saturation density for any d given by φ s ≥ (d + 2)(1 − S 0 )/2 d+1 , where S 0 ∈ [0, 1] is the structure factor at k = 0 (i.e., infinite-wavelength number variance) in the high-dimensional limit.We prove rigorously that a Palàsti-like conjecture (the saturation density in R d is equal to that of the one-dimensional problem raised to the dth power) cannot be true for RSA hyperspheres.We demonstrate that the structure factor S(k) must be analytic at k = 0 and that RSA packings for 1 ≤ d ≤ 6 are nearly "hyperuniform." Consistent with the recent "decorrelation principle," we find that pair correlations markedly diminish as the space dimension increases up to six. We also obtain kissing (contact) number statistics for saturated RSA configurations on the surface of a ddimensional sphere for dimensions 2 ≤ d ≤ 5 and compare to the maximal kissing numbers in these dimensions. We determine the structure factor exactly for the related "ghost" RSA packing in R d and show that its distance from "hyperuniformity" increases as the space dimension increases, approaching a constant asymptotic value of 1/2. Our work has implications for the possible existence of disordered classical ground states for some continuous potentials in sufficiently high dimensions.
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