Packing Hyperspheres in High-Dimensional Euclidean Spaces

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 ${\varphi }_{MRJ}\simeq 0.46$, 0.31 and 0.20, respectively. To a good approximation, the MRJ density obeys the scaling form ${\varphi }_{MRJ}=c_1/2^d+(c_{2}d)/2^d$, where $c_1=-2.72$ and $c_2=2.56$, which appears to be consistent with 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 ${\varphi }_{MRJ}\simeq 0.64$), 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 $\simeq 0.4$. 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, ${\varphi }_F\simeq 0.32$ and ${\varphi }_M\simeq 0.39$, respectively, for $d=4$, and ${\varphi }_F\simeq 0.19$ and ${\varphi }_M\simeq 0.24$, respectively, for $d=5$.

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

Cited by in corpus (14)

• A Landscape Analysis of Constraint Satisfaction Problems
• A deductive statistical mechanics approach for granular matter
• Point processes in arbitrary dimension from fermionic gases, random matrix theory, and number theory
• Geometric interpretation of pre-vitrification in hard sphere liquids
• Geometrical Frustration: A Study of 4d Hard Spheres
• Theory of the superglass phase
• Solution of the Percus-Yevick equation for hard hyperspheres in even dimensions
• Estimates of the optimal density and kissing number of sphere packings in high dimensions
• Gaussian core model phase diagram and pair correlations in high Euclidean dimensions
• Virial series for fluids of hard hyperspheres in odd dimensions
• Structural transitions in granular packs: statistical mechanics and statistical geometry investigations
• Percus-Yevick theory for the structural properties of the seven-dimensional hard-sphere fluid
• Counterintuitive ground states in soft-core models
• Comment to "Packing Hyperspheres in High-Dimensional Euclidean Space"