Search for hyperuniformity in mechanically stable packings of frictionless disks above jamming

We numerically simulate mechanically stable packings of soft-core, frictionless, bidisperse disks in two dimensions, above the jamming packing fraction ${\varphi }_J$. For configurations with a fixed isotropic global stress tensor, we investigate the fluctuations of the local packing fraction $\varphi \left(\mathbf{r}\right)$ to test whether such configurations display the hyperuniformity that has been claimed to exist exactly at ${\varphi }_J$. For our configurations, generated by a rapid quench protocol, we find that hyperuniformity persists only out to a finite length scale and that this length scale appears to remain finite as the system stress decreases towards zero, i.e., towards the jamming transition. Our result suggests that the presence of hyperuniformity at jamming may be sensitive to the specific protocol used to construct the jammed configurations.

Figure 1

Shaded volume represents the volume that contributes to ${\varphi }_R$ according to (a) the definition I of $\varphi \left(\mathbf{r}\right)$ in Eq. (5) and (b) the definition II of $\varphi \left(\mathbf{r}\right)$ in Eq. (7).

Figure 2

Geometry of our system box. $L_x$ and $L_y$ are the lengths in the $\widehat{\mathbf{x}}$ and $\widehat{\mathbf{y}}$ directions, and $\gamma$ is the skew ratio. Lees-Edwards boundary conditions are used.

Figure 3

Average global packing fraction $\langle \varphi \rangle$ vs stress per particle $\tilde{p}$, for systems with $N=8192$ to $65536$ particles on (a) linear-linear and (b) log-log scales. Solid lines are a fit to a quadratic function $\langle \varphi \rangle ={\varphi }_J+a_1\tilde{p}+a_2{\tilde{p}}^2$.

Figure 4

Average excess contact number $\langle \mathrm{\Delta }z\rangle =\langle z\rangle -z_{\mathrm{iso}}$ vs stress per particle $\tilde{p}$, for systems with $N=8192$ to $65536$ particles on (a) linear-linear and (b) log-log scales. Solid lines are a fit to $\langle \mathrm{\Delta }z\rangle ={\tilde{p}}^x(a_0+a_1\tilde{p}+a_2{\tilde{p}}^2+a_2{\tilde{p}}^3)$, where we find $x\approx 0.546$.

Figure 5

Density of frequencies of small elastic vibrations $D\left(\omega \right)$ vs $\omega$. Results are shown for a system of $N=8192$ particles for a range of stress per particle $\tilde{p}=0.0001373$ to $0.0183105$. Each curve is an average over three independent energy minimized configurations.

Figure 6

Structure function $S\left(q\widehat{\mathbf{y}}\right)$ vs $q$, giving the fluctuation in the density of particles at wave vector $q\widehat{\mathbf{y}}$. Results are shown for a system with $N=32768$ particles, for several different values of the stress per particle, $\tilde{p}=0.0001373$ to $0.0183105$. $S\left(q\widehat{\mathbf{y}}\right)$ approaches a constant as $q\to 0$. The error bars shown in the figure represent one standard deviation of estimated statistical error.

Figure 7

Fluctuation in packing fraction $\chi \left(q\widehat{\mathbf{y}}\right)$ vs $q$, using the definition I of Eq. (5) for the local packing fraction $\varphi \left(\mathbf{r}\right)$. Results are shown for a system with $N=32768$ particles, for several different values of the stress per particle, $\tilde{p}=0.0001373$ to $0.0183105$. As $q$ decreases, we find that $\chi \left(q\widehat{\mathbf{y}}\right)$ decreases roughly linearly but then reaches a minimum below which it increases, thus implying the absence of hyperuniformity on large length scales. As $\tilde{p}$ decreases, the curves of $\chi \left(q\widehat{\mathbf{y}}\right)$ approach a common limiting curve. The error bars shown in the figure represent one standard deviation of estimated statistical error.

Figure 8

Fluctuation in packing fraction $\chi \left(q\widehat{\mathbf{y}}\right)$ vs $q$, using the definition I of Eq. (5) for the local packing fraction $\varphi \left(\mathbf{r}\right)$. Results are shown for several different system sizes, $N=8192$ to $65536$, for our smallest and largest values of the stress per particle, $\tilde{p}=0.0001373$ and $0.0183105$. The error bars shown in the figure represent one standard deviation of estimated statistical error.

Figure 9

Comparison of the different wave vector-dependent measures of hyperuniformity: “definition I” and “definition II” refer to the fluctuation of local packing fraction $\chi \left(\mathbf{q}\right)$ with $\varphi \left(\mathbf{r}\right)$ determined from the definitions of Eq. (5) and Eq. (7), respectively, while “definition III” refers to the quantity $nT{\chi }_T\left(\mathbf{q}\right)$ defined in Eq. (15). Results are shown for a system with $N=32768$ particles, for our smallest and largest values of the stress per particle, $\tilde{p}=0.0001373$ and $0.0183105$. The error bars shown in the figure represent one standard deviation of estimated statistical error.

Figure 10

Comparison of the fluctuation in local packing fraction, $\mathrm{var}\left({\varphi }_R\right)$ vs $R$, using definition I of Eq. (5) and definition II of Eq. (7) for the local packing fraction $\varphi \left(\mathbf{r}\right)$. Results are shown for a system with $N=32768$ particles, for our smallest and largest values of the stress per particle, $\tilde{p}=0.0001373$ and $0.0183105$. Thick solid lines represent the dependencies $1/R^2$ and $1/R^3$, as indicated.

Figure 11

$R^2\mathrm{var}\left({\varphi }_R\right)$ vs window radius $R$ for systems with $N=8192$ to $65536$ particles, at stress (a) $\tilde{p}=0.0001373$ and (b) $\tilde{p}=0.0183105$. The power-law decay at small $R$ is indicated. Definition I of Eq. (5) for the local packing fraction $\varphi \left(\mathbf{r}\right)$ is used.

Figure 12

$R^2\mathrm{var}\left({\varphi }_R\right)$ vs window radius $R$ for systems with $N=8192$ to $65536$ particles, at stress (a) $\tilde{p}=0.0001373$ and (b) $\tilde{p}=0.0183105$. The power-law decay at small $R$ is indicated. Definition II of Eq. (7) for the local packing fraction $\varphi \left(\mathbf{r}\right)$ is used.

Figure 13

Plot of $f^2\left(y\right)$ vs $y$, with the function $f\left(y\right)$ as defined in Eq. (14). The envelope of the oscillations at large $y$ decays as $1/y^3$.

Figure 14

Packing fraction fluctuation $\chi \left(q\widehat{\mathbf{y}}\right)$ as computed using definition I of Eq. (5), and definition II of Eq. (7), as well as the structure function $S\left(q\widehat{\mathbf{y}}\right)$, for a wide range of wave vectors up to $q=10$. Results are shown for a system of $N=32768$ particles at our lowest stress $\tilde{p}=0.0001373$.

Figure 15

$R^2\mathrm{var}\left({\varphi }_R\right)$ vs $R$, comparing the data of Figs. 11 and 12 (direct computation) with the result from summing $\chi \left(q\widehat{\mathbf{y}}\right)$ over $q$ in Eq. (28), for both definition I and definition II of the local packing fraction $\varphi \left(\mathbf{r}\right)$. Results are for a system with $N=32768$ particles at our smallest stress $\tilde{p}=0.0001373$.

Figure 16

Fluctuation in packing fraction $\chi \left(q\widehat{\mathrm{y}}\right)$ vs $q$, using the definition I of Eq. (5) for the local packing fraction $\varphi \left(\mathbf{r}\right)$, for the case of a Hertzian interaction $[\alpha =5/2$ in Eq. (22)]. Results are shown for several different values of the stress per particle $\tilde{p}$, for a system with $N=32768$ particles. The inset shows the average packing fraction $\langle \varphi \rangle$ as a function of $\tilde{p}$.

Figure 17

Fluctuation in packing fraction $\chi \left(q\widehat{\mathrm{y}}\right)$ vs $q$, using the definition I of Eq. (5) for the local packing fraction $\varphi \left(\mathbf{r}\right)$, for the case of the harmonic interaction in an ensemble at fixed global packing fraction $\varphi =0.8422$. Results are shown for a system with $N=32768$ particles.

Figure 18

Fraction of the particles that are rattlers, $\langle N_{\mathrm{rattlers}}\rangle /N$ vs stress per particle $\tilde{p}={\mathrm{\Gamma }}_N/N$. For the harmonic interaction we show results for systems with $N=8192$ to $65536$ particles; the points for different $N$ in the figure overlap each other. For the Hertzian interaction we show results for $N=32768$.

Figure 19

Packing fraction fluctuation $\chi \left(q\widehat{\mathbf{y}}\right)$ vs $q$, using definition II of Eq. (7) for a system of $N=65536$ particles, at two different values of the stress per particle $\tilde{p}$. We compare results where the weight of each particle is taken as the circular area of the isolated particle (denoted as “counting overlaps twice”) versus where the weight of each particle is taken as the nonoverlapping part of that circular area, as illustrated by the shaded region for particle $i$ in the inset (denoted as “counting overlaps once”). A small difference is seen between these two sets of weights at the larger value of $\tilde{p}$, but not at the smaller value.

Figure 20

(a) Relative fluctuations in the system box lengths $L_x$ and $L_y$ vs stress per particle $\tilde{p}$, for system sizes $N=8192$ to $65536$. Solid symbols show the fluctuations in $L_x$, while open symbols show $L_y$. (b) Fluctuations in the dimensionless box skew parameter $\gamma$ versus $\tilde{p}$. See Fig. 2 for the definition of parameters $L_x,L_y,\gamma$. In both cases the fluctuations scale as $1/\sqrt{N}$.

Figure 21

Average box skew $\langle \gamma \rangle$ vs stress per particle $\tilde{p}$, for system sizes $N=8192$ to $65536$. Error bars represent one standard deviation of estimated statistical error, showing that $\langle \gamma \rangle =0$ within the estimated errors.

Figure 22

Accuracy parameters (a) ${\delta }_1$, (b) ${\delta }_2$, and (c) ${\delta }_3$ of Eqs. (A2), (A3), and (A4), that measure the relative deviations of the stress tensor ${\mathrm{\Sigma }}_{\alpha \beta }$ from the target isotropic ${\mathrm{\Gamma }}_N{\delta }_{\alpha \beta }$, versus stress per particle $\tilde{p}={\mathrm{\Gamma }}_N/N$, for system sizes $N=8102$ to $65536$.

Figure 23

Average magnitude of the residual net force on a particle ${\mathbf{F}}_i$, normalized by the average magnitude of the contact force ${\mathbf{F}}_{ij}$, at the termination of our minimization procedure. Results are plotted versus the stress per particle $\tilde{p}={\mathrm{\Gamma }}_N/N$ for systems sizes $N=8192$ to $65536$.

Figure 24

Distribution $\mathcal{P}\left(\right|{\mathbf{F}}_i|/\langle |{\mathbf{F}}_{ij}|\rangle )$ of the residual net force on particles $|{\mathbf{F}}_i|$, normalized by the average magnitude of the contact force $|{\mathbf{F}}_{ij}|$, versus $|{\mathbf{F}}_i|/\langle |{\mathbf{F}}_{ij}|\rangle$, for different system sizes $N=8102$ to $65536$, at our (a) smallest $\tilde{p}=0.0001373$ and (b) largest $\tilde{p}=0.0183105$.

Figure 25

Distribution $\mathcal{P}\left(\right|{\mathbf{F}}_i|/\langle |{\mathbf{F}}_{ij}|\rangle )$ of the residual net force on particles $|{\mathbf{F}}_i|$, normalized by the average magnitude of the contact force $|{\mathbf{F}}_{ij}|$, versus $|{\mathbf{F}}_i|/\langle |{\mathbf{F}}_{ij}|\rangle$, for different values of the accuracy parameter $\varepsilon$ that terminates our minimization of $\tilde{U}$. Results are for a system of size $N=65536$ at $\tilde{p}=0.0001373$. Data labeled “${10}^{-10}*$” are results for a minimization of $\tilde{U}$ to accuracy $\varepsilon ={10}^{-10}$, followed by an additional minimization of interaction energy $U$ to accuracy ${10}^{-10}$ while holding the box parameters constant.

Figure 26

Fluctuation in packing fraction $\chi \left(q\widehat{\mathbf{y}}\right)$ vs $q$, using definition I of Eq. (5) for the local packing fraction $\varphi \left(\mathbf{r}\right)$. Results are shown for system size $N=65536$, for (a) our smallest value of the stress per particle, $\tilde{p}=0.0001373$, and (b) our largest value $\tilde{p}=0.0183105$. We compare the values of $\chi \left(q\widehat{\mathbf{y}}\right)$ obtained when using different values of the parameter $\varepsilon$ that determines the stopping criterion for our minimization of $\tilde{U}$. Data labeled “${10}^{-10}*$” in panel (a) are results for a minimization of $\tilde{U}$ to accuracy $\varepsilon ={10}^{-10}$, followed by an additional minimization of interaction energy $U$ to accuracy ${10}^{-10}$ while holding the box parameters constant.