Packing hyperspheres in high-dimensional Euclidean spaces

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)approximate to 0.46, 0.31, and 0.20, respectively. To a good approximation, the MRJ density obeys the scaling form phi(MRJ)=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 the 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 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)approximate to 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 approximate to 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, phi(F)approximate to 0.32 and phi(M)approximate to 0.39, respectively, for d=4, and phi(F)approximate to 0.20 and phi(M)approximate to 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.