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 ${\varphi }_{\mathit{MRJ}}\approx 0.46$, 0.31, and 0.20, respectively. To a good approximation, the MRJ density obeys the scaling form ${\varphi }_{\mathit{MRJ}}=c_1∕2^d+\left(c_{2}d\right)∕2^d$, 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 $g_2\left(r\right)$ and structure factor $S\left(k\right)$ 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 $d\to \infty$. As in three dimensions (where ${\varphi }_{\mathit{MRJ}}\approx 0.64$), the packings show no signs of crystallization, are isostatic, and have a power-law divergence in $g_2\left(r\right)$ at contact with power-law exponent $\approx 0.4$. Across dimensions, the cumulative number of neighbors equals the kissing number of the conjectured densest packing close to where $g_2\left(r\right)$ has its first minimum. Additionally, we obtain estimates for the freezing and melting packing fractions for the equilibrium hard-sphere fluid-solid transition, ${\varphi }_F\approx 0.32$ and ${\varphi }_M\approx 0.39$, respectively, for $d=4$, and ${\varphi }_F\approx 0.20$ and ${\varphi }_M\approx 0.25$, 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.

Figure 1

A highly schematic plot of the subspace in the density-disorder $\varphi \text{−}\psi$ plane, where strictly jammed three-dimensional packings exist, as adapted from Ref. 1. Point A corresponds to the lowest-density jammed packing, and it is intuitive to expect that a certain ordering will be needed to produce low-density jammed packings. Point B corresponds to the most dense jammed packing, which is also expected to be the most ordered. Point MRJ represents the maximally random jammed state. The jamming region in the $\varphi \text{−}\psi$ plane will of course depend on the jamming category. The gray region is devoid of hard-sphere configurations.

Figure 2

(Color online) Reduced pressure $p$ as a function of density $\varphi$, for a range of system sizes (see legend), for (a) $d=4$ systems of spheres, initially in a $D_4$ lattice, and negative expansion rate $\gamma =-{10}^{-6}$ and (b) $d=5$ systems of spheres, initially in a $D_5$ lattice, and negative expansion rate $\gamma =-{10}^{-5}$. $N$ was chosen to make a perfect $D_d$ lattice with periodic boundary conditions, i.e., $N={\left(2n\right)}^d∕2$ for $n∊\mathbb{Z}$. Also plotted is the theoretical prediction of Luban and Michels (LM) for the equation of state [15]. Curves for larger system sizes lie farther to the right.

Figure 3

(Color online) Left panel: The estimated jamming packing fraction ${\tilde{\varphi }}_J$ as a function of density $\varphi$ for systems of 648 spheres for $d=4$ with various expansion rates [see legend and note that there are two samples labeled (a) and (b) for $\gamma ={10}^{-8}$]. For the curves showing no partial crystallization (i.e., $\gamma ={10}^{-5}$, ${10}^{-6}$, and ${10}^{-7}$), curves with smaller expansion rates have larger peak heights. For the curves that show partial crystallization [i.e., $\gamma ={10}^{-8}$ (a) and (b) and ${10}^{-9}$], curves with smaller expansion rate lie farther to the left. Right panel: The cumulative coordination $Z\left(r\right)$ (i.e., the number of contacts) for the perfect $D_4$ lattice and for the partially crystallized packings at $p>{10}^{12}$ obtained for expansion rates $\gamma ={10}^{-8}$ and $\gamma ={10}^{-9}$. The jamming packing fraction for the $\gamma ={10}^{-8}$ packing is $\varphi =0.511$, and the jamming packing fraction for the $\gamma ={10}^{-9}$ packing agreed up to 12 significant figures with the density of the $D_4$ lattice, $\varphi ={\pi }^2∕16\approx 0.617$.

Figure 4

(Color online) The estimated jamming packing fraction ${\tilde{\varphi }}_J$ as a function of density $\varphi$ for a system of 10 000 spheres for $d=4$ with various expansion rates. Curves with smaller expansion rates have larger peak heights. The curve labeled “mix” corresponds to the following sequence of expansion rates: $\gamma ={10}^{-2}$ until $p=10$, $\gamma ={10}^{-3}$ until $p={10}^4$, $\gamma ={10}^{-4}$ until $p={10}^6$, and $\gamma ={10}^{-5}$ until $p={10}^{12}$.

Figure 5

(Color online) The estimated jamming packing fraction ${\tilde{\varphi }}_J$ as a function of packing fraction $\varphi$ for $d=3$. Shown are systems of 4096 spheres with various expansion rates and systems of 10 976 spheres placed in an fcc lattice with negative expansion rates $\gamma =-{10}^{-4}$, $-{10}^{-5}$, and $-{10}^{-6}$ (last three curves). Also plotted are approximations to the equilibrium EOS for the fluid phase, the coexistence region, and the crystal phase [47], as well as the Percus-Yevick (PY) EOS for the fluid phase. Compare this figure to the curves shown in Figs. 3, 4. For the curves showing no partial crystallization (i.e., $\gamma =32\times {10}^{-6}$, $64\times {10}^{-6}$, and $128\times {10}^{-6}$), curves with smaller expansion rates have larger peak heights. For the curves that show partial crystallization (i.e., $\gamma ={10}^{-6}$, $4\times {10}^{-6}$, and $16\times {10}^{-6}$), curves with smaller expanion rates lie farther to the left. For the melting curves (i.e., $\gamma =-{10}^{-4}$, $-{10}^{-5}$, and $-{10}^{-6}$), curves with smaller compression rates lie farther to the right.

Figure 6

(Color online) Fit of the data for the product $2^d{\varphi }_{\mathit{MRJ}}$ to the linear form (6) for $3⩽d⩽6$ with $c_1=-2.72$ and $c_2=2.56$.

Figure 7

(Color online) The pair correlation function $g_2\left(r\right)$ for MRJ packings of ${10}^5$ hard spheres for $d=3$, 4, 5, and 6 at the respective densities reported in Table . Pair separation is plotted in units of the sphere diameter $D$. [For $d=6$, $g_2\left(r\right)$ was only calculated up to $r∕D=3$ due to the system size and periodic boundary conditions.] The delta-function contribution [cf. Eq. (9)] at contact, of course, is not shown. The inset shows $\mid h\left(r\right)\mid =\mid g_2\left(r\right)-1\mid$ on a logarithmic scale for $d=3$, 4, and 5. Each curve for $g_2\left(r\right)$ is obtained from a single packing realization (not time-averaged). Curves for higher dimensions are increasingly diminished.

Figure 8

(Color online) The structure factor $S\left(k\right)$ for jammed packings of ${10}^5$ spheres for $d=3$, 4, and 5 at the respective densities reported in Table . Inset: A comparison for $d=4$ of $S\left(k\right)$ for a jammed packing and for a fluid near the freezing point $(\varphi \approx 0.31)$. Each curve for $S\left(k\right)$ is obtained from a single packing realization (not time averaged).

Figure 9

(Color online) The near-contact cumulative coordination $Z\left(x\right)$ [cf. Eq. (12)] for ${10}^4$-sphere MRJ packings for $d=4$ (a) and for $d=5$ (b), with rattlers removed. The inset shows $Z\left(x\right)$ on a log-log scale along with power-law fits for intermediate interparticle gap $x-1$ beyond contact. ${10}^5$-sphere MRJ packings in $d=5$ with final expansion rates of $\gamma ={10}^{-4}$ give similar results; such packings with final expansion rates of $\gamma ={10}^{-5}$ are (presently) too computationally expensive. Compare these plots to the equivalent results for $d=3$ in Ref. 3 (cf. Fig. 8).