Large-scale structure of randomly jammed spheres

We numerically analyze the density field of three-dimensional randomly jammed packings of monodisperse soft frictionless spherical particles, paying special attention to fluctuations occurring at large length scales. We study in detail the two-point static structure factor at low wave vectors in Fourier space. We also analyze the nature of the density field in real space by studying the large-distance behavior of the two-point pair correlation function, of density fluctuations in subsystems of increasing sizes, and of the direct correlation function. We show that such real space analysis can be greatly improved by introducing a coarse-grained density field to disentangle genuine large-scale correlations from purely local effects. Our results confirm that both Fourier and real space signatures of vanishing density fluctuations at large scale are absent, indicating that randomly jammed packings are not hyperuniform. In addition, we establish that the pair correlation function displays a surprisingly complex structure at large distances, which is however not compatible with the long-range negative correlation of hyperuniform systems but fully compatible with an analytic form for the structure factor. This implies that the direct correlation function is short ranged, as we also demonstrate directly. Our results reveal that density fluctuations in jammed packings do not follow the behavior expected for random hyperuniform materials, but display instead a more complex behavior.

Figure 1

(a) Static structure factor at $\phi =$ 0.7, 0.68, 0.66, and 0.653 with the nonanalytic linear model with the parameters optimized in Ref. [2]. (b) A zoom on the low wave vector behavior of the top panel. The inset shows the second derivative of the static structure factor with a line highlighting the apparent linear behavior of $S\left(k\right)$. (c) The structure factor at $\phi =0.653$ fitted to various functional forms.

Figure 2

(a) Behavior of the coarse-grained density correlation function $P\left(r\right)$ calculated for a nonanalytic structure factor of the form $S\left(k\right)=A+B|\vec{k}|$, with $A$ and $B$ taken from the fit shown in Fig. 1. (b) A zoom on the negative correlation near $r\approx 5$ (a). The inset illustrates the $r^{-4}$ power law decay of $P\left(r\right)$ from negative values as $r\to \infty$.

Figure 3

The measured coarse-grained density correlation function $P\left(r\right)$ at $\phi =0.7$, 0.68, 0.66, 0.653, using three different representations to emphasize the peak at $r=0$ (a), the negative dip near $r\approx 5$ (b), and the positive correlation near $r\approx 10$ (c).

Figure 4

Behavior of $r^{4}P\left(r\right)$ for (a) the measured $r^{4}P\left(r\right)$ at $\phi =0.7$, 0.68, 0.66, and 0.653, and (b) the calculated $r^{4}P\left(r\right)$ from nonanalytic [Eq. (1)] and analytic [Eq. (5)] models of $S\left(k\right)$. The negative correlation predicted from the nonanalytic model is not observed in the measured $P\left(r\right)$, which exhibits a positive correlation which is well captured by the analytic models with $n\ge 4$.

Figure 5

Measured variance of density fluctuations in a spherical cavity of radius $R$ at $\phi =0.66$. $\langle \mathrm{\Delta }D\left(R\right)\rangle$ is the variance of the microscopic number density $\rho \left(\vec{x}\right)$, and $\langle \mathrm{\Delta }\mathrm{\Psi }\left(R\right)\rangle$ is the variance of the coarse-grained density $\psi \left(\vec{x}\right)$. The apparent anomalous scaling of $\langle \mathrm{\Delta }D\left(R\right)\rangle$ originates from short-ranged correlations which are filtered out in $\langle \mathrm{\Delta }\mathrm{\Psi }\left(R\right)\rangle$. This quantity obeys the scaling expected from the central limit theorem. This shows that density fluctuations are not suppressed at large scale in jammed packings. The vertical dashed line indicates half of the dimension of the simulation box.

Figure 6

Direct correlation function $C\left(r\right)$ obtained from numerical simulations at $\phi =0.66$ and calculated from various models for $S\left(k\right)$ from Eqs. (1) and (5). The analytic models with $n\ge 4$ reproduce the measured behavior very well.

Figure 7

The static structure factor at $\phi =$ 0.7 for different system sizes, showing that the upturn near $k\approx 0.3$ found for $N=512000$ is not a finite-size effect.

Figure 8

(a) System size dependence of the coarse-grained density correlation function $P_N\left(r\right)$ at $\phi =$ 0.7. (b) System size dependence of the coarse-grained density correlation function with the finite-size effect correction $P_N\left(r\right)+S\left(0\right)/N$ at $\phi =$ 0.7.

Figure 9

Dependence on the number of independent samples of (a) $S\left(k\right)$ at $\phi =0.7$ and (b) $r^{4}P\left(r\right)$ at $\phi =0.66$. These results show that the low-$k$ behavior of $S\left(k\right)$ and the large-$r$ behavior of $P\left(r\right)$ are statistically significant.