Packing hyperspheres in high-dimensional Euclidean spaces

Abstract

We present a 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 phi(MRJ) approximately 0.46, 0.31, and 0.20, respectively. To a good approximation, the MRJ density obeys the scaling form phi(MRJ)=c1/2(d)+(c2d)/2d, where c1=-2.72 and c2=2.56, which appears to be consistent with the high-dimensional asymptotic limit, albeit with different coefficients. Calculations of the pair correlation function g2(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 other things, states that unconstrained correlations diminish as the dimension increases and vanish entirely in the limit d-->infinity. As in three dimensions (where phi(MRJ) approximately 0.64), the packings show no signs of crystallization, are isostatic, and have a power-law divergence in g2(r) at contact with power-law exponent approximately 0.4. Across dimensions, the cumulative number of neighbors equals the kissing number of the conjectured densest packing close to where g2(r) has its first minimum. Additionally, we obtain estimates for the freezing and melting packing fractions for the equilibrium hard-sphere fluid-solid transition, phi(F) approximately 0.32 and phi(M) approximately 0.39, respectively, for d=4, and phi(F) approximately 0.20 and phi(M) approximately 0.25, 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.