Absence of Hyperuniformity in Amorphous Hard-Sphere Packings of Nonvanishing Complexity

We relate the structure factor $S(\mathbf{k}\to \mathbf{0})$ in a system of jammed hard spheres of number density $\rho$ to its complexity per particle $\mathrm{\Sigma }(\rho )$ by the formula $S(\mathbf{k}\to \mathbf{0})=-1/[{\rho }^2{\mathrm{\Sigma }}^{″}(\rho )+2\rho {\mathrm{\Sigma }}^{\prime }(\rho )]$. We have verified this formula for the case of jammed disks in a narrow channel, for which it is possible to find $\mathrm{\Sigma }(\rho )$ and $S(\mathbf{k})$ analytically. Hyperuniformity, which is the vanishing of $S(\mathbf{k}\to \mathbf{0})$, will therefore not occur if the complexity is nonzero. An example is given of a jammed state of hard disks in a narrow channel which is hyperuniform when generated by dynamical rules that produce a nonextensive complexity.

Figure 1

Local packing arrangements of hard disks of diameter $\sigma$ in a channel of width $H_d$. Every disk is in contact with the wall and with two neighboring disks. The shaded disk is not jammed. Red dashed lines (“bonds”) join the centers of neighboring disks that are in contact with opposite walls and blue lines join the centers of disks that are in contact with the same wall; the latter can be regarded as “defects” in the zigzag order. The bonds have projections $\delta$ and $\sigma$ along the channel.

Figure 2

Complexity per particle $\mathrm{\Sigma }$ versus packing fraction $\varphi$ for the case $\delta /\sigma =3/5$. Also shown is the “compressibility” $\kappa \equiv d\rho /df$; this quantity is equal to $S(k\to 0)$ and is nonzero for all ${\varphi }_{\min }<\varphi <{\varphi }_{\max }$, implying that the jammed states at these densities are not hyperuniform.

Figure 3

Structure factor $S(k)$ versus wave number $k$ for $\delta /\sigma =3/5$ at a packing fraction $\varphi =0.649$, which corresponds to the maximum in the compressibility shown in Fig. 2. For small $k$, $S(k)\to \text{const}+\mathcal{O}(k^2)$.