Long-Range Anomalous Decay of the Correlation in Jammed Packings

We numerically study the structure of the interactions occurring in three-dimensional systems of hard spheres at jamming, focusing on the large-scale behavior. Given the fundamental role in the configuration of jammed packings, we analyze the propagation through the system of the weak forces and of the variation of the coordination number with respect to the isostaticity condition, $\mathrm{\Delta }Z$. We show that these correlations can be successfully probed by introducing a correlation function weighted on the density-density fluctuations. The results of this analysis can be further improved by introducing a representation of the system based on the contact points between particles. In particular, we find evidence that the weak forces and the $\mathrm{\Delta }Z$ fluctuations support the hypothesis of randomly jammed packings of spherical particles being hyperuniform by exhibiting an anomalous long-range decay. Moreover, we find that the large-scale structure of the density-density correlation exhibits a complex behavior due to the superimposition of two exponentially damped oscillating signals propagating with linearly depending frequencies.

Figure 1

Contacts and original RDFs (in blue and red, respectively) in $d=3$ as a function of the distance (expressed in the respective diameter units $r/{\sigma }^{\prime }$ and $r/\sigma$). Each peak (labeled with a different letter) occurs in correspondence to a different configuration of the fictive particles, as shown in the correspondent representations (which, however, do not exhaust all the possibilities), along with the distance at which each discontinuity appears (red lines). Inset: Schematic representation of the contacts-centered spheres model in $d=3$. Each fictive particle (dashed black lines) results from the contact of two HSs and has a diameter ${\sigma }^{\prime }=\sigma /2$, being $\sigma$ the diameter of the original HS.

Figure 2

Contacts density-density correlation $g(r/{\sigma }^{\prime })$ multiplied by the factor $e^{ar/{\sigma }^{\prime }}$ with $a\approx 0.3$. Two sets of distinct oscillations given by Eq. (4) appear to be superimposed. The first one, $f_m(r)$, propagating in the middle range $[4:9]r/{\sigma }^{\prime }$ and the second, $f_l(r)$, in the long-range region $[9:22]r/{\sigma }^{\prime }$. By fitting the resulting function (red line), we found that $p_l\approx 7.5$ and $p_m\approx 3.8$. The distance is expressed in diameter units.

Figure 3

(a) Weak force correlation function $C_f^{s=-1}(r)$ in $d=3$ and enlargement of the long-range region in log-log scale (inset). The density-density correlations propagating at long range have been filtered by introducing the average over the oscillation period $T$, $⟨C_f^s(r){⟩}_T$ (orange circles). The log-log scale in the inset points out a long-range power-law behavior (red line) decaying with an exponent ${\gamma }_f^{s=-1}=0.7\pm 0.3$ in the range [7:23]. (b) Weak forces correlation function $C_f^{s=-1/2}(r)$ in $d=3$. The fluctuations analysis points out a power-law decay with an exponent ${\gamma }_f^{s=-1/2}=4.1\pm 0.3$ in the range [7:23]. Notice that both insets show the modulus of the correlation function. The distances are expressed in diameter units.

Figure 4

Contact correlation function $C_{\mathrm{\Delta }{Z}^{\prime }}^{s=1}(r)$ in $d=3$. The density-density oscillations at short range (blue line) have been filtered by introducing the correlation averaged over the oscillation period $T$, $⟨C_{\mathrm{\Delta }{Z}^{\prime }}^{s=1}(r)⟩$ (orange circles). Inset: Enlargement of the mid- and long-range regions of the modulus of the $C_{\mathrm{\Delta }{Z}^{\prime }}^{s=1}(r)$ (blue line) in log-log scale. The correlation function exhibits a power-law decay (red line) propagating with a nontrivial exponent ${\gamma }_{\mathrm{\Delta }{Z}^{\prime }}=3.9\pm 0.2$. The distance is expressed in diameter units.