Packing hyperspheres in high-dimensional Euclidean spaces

Abstract

We present the first study of disordered jammed hard-sphere packings in four-, five-and sixdimensional Euclidean spaces. Using a collision-driven packing generation algorithm, we obtain the first estimates for the packing fractions of the maximally random jammed (MRJ) states for space dimensions d = 4, 5 and 6 to be φ M RJ ≃ 0.46, 0.31 and 0.20, respectively. To a good approximation, the MRJ density obeys the scaling formand c 2 = 2.56, which appears to be consistent with high-dimensional asymptotic limit, albeit with different coefficients. Calculations of the pair correlation function g 2 (r) and structure factor S(k) for these states show that short-range ordering appreciably decreases with increasing dimension, consistent with a recently proposed "decorrelation principle," which, among othe things, states that unconstrained correlations diminish as the dimension increases and vanish entirely in the limit d → ∞. As in three dimensions (where φ M RJ ≃ 0.64), the packings show no signs of crystallization, are isostatic, and have a power-law divergence in g 2 (r) at contact with power-law exponent ≃ 0.4. Across dimensions, the cumulative number of neighbors equals the kissing number of the conjectured densest packing close to where g 2 (r) has its first minimum. Additionally, we obtain estimates for the freezing and melting packing fractions for the equilibrium hard-sphere fluid-solid transition, φ F ≃ 0.32 and φ M ≃ 0.39, respectively, for d = 4, and φ F ≃ 0.19 and φ M ≃ 0.24, respectively, for d = 5. Although our results indicate the stable phase at high density is a crystalline solid, nucleation appears to be strongly suppressed with increasing dimension. * Electronic address: torquato@electron.princeton.edu 2

Microstructure and macroscopic properties of polydisperse systems of hard spheres

“…He noted that a disordered packing has a limiting (critical) density, i.e., a packing with higher density inevitably contains crystalline regions. Physical experiments give an estimate of 0.637-0.64 for this density [1,62,63], computer simulations provide 0.637-0.649 [13,14,[64][65][66][67]. This density is substantially lower than the maximum packing fraction attainable in the densest crystalline structures, and depends on history [68], and other material parameters.…”

“…Initial velocities are set randomly in order to keep the system dynamic and random, for details see Ref. [67]. When the target volume fraction is reached, the growth process is stopped.…”

“…Starting from a random uniform distribution of points in a cubical box, the radius of the particles grows linearly with time. We use non-overlapping spheres [81], with an EventDriven code [67], whose growing rate conforms to and conserves a prescribed size distribution. Initial velocities are set randomly in order to keep the system dynamic and random, for details see Ref.…”

“…For rigid spheres that do not interact except via an infinite repulsion on contact, i.e., with zero interaction / contact duration, the total energy is given by E = E k , whereas for soft spheres the potential energy also has to be considered. The temperature in equilibrium hard-sphere systems is not a relevant parameter, since it does not affect the equilibrium configuration [67], it only scales time (or the free energy [28]). As a consequence, there is only one independent thermodynamic state variable, which can be either the reduced (dimensionless) pressure (so-called compressibility factor) Z = P V/Nk B T or the density (volume fraction) ν = 4π a 3 i /(3V ), related through the equation of state (EOS), where k B T = 2E/(3N) with the Boltzmann constant k B .…”

“…Fluid and jammed configurations of hard sphere mixtures are examined for various size distributions at slow compression rates [66,67,161,162]. Several authors proposed theories to compute the amorphus jamming density of binary and polydisperse hard sphere mixtures, see Refs.…”