Packing Hyperspheres in High-Dimensional Euclidean Spaces

M. Skoge; A. Donev; F. H. Stillinger; S. Torquato

doi:10.1103/PhysRevE.74.041127

paper

Packing Hyperspheres in High-Dimensional Euclidean Spaces

arXiv:cond-mat/0608362 · doi:10.1103/PhysRevE.74.041127

Abstract

We present the first study of disordered jammed hard-sphere packings in four-, five- and six-dimensional 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 $ϕ_{M R J} ≃ 0.46$ , 0.31 and 0.20, respectively. To a good approximation, the MRJ density obeys the scaling form $ϕ_{M R J} = c_{1} / 2^{d} + (c_{2} d) / 2^{d}$ , where $c_{1} = - 2.72$ and $c_{2} = 2.56$ , 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 $d \to \infty$ . As in three dimensions (where $ϕ_{M R J} ≃ 0.64$ ), 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. We obtain estimates for the freezing and melting desnities 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 $d = 5$ .

28 pages, 9 figures. To appear in Physical Review E

Packing Hyperspheres in High-Dimensional Euclidean Spaces

Abstract

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