Statistics of conserved quantities 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 compute the averages, variances, and correlations of conserved quantities (stress ${\Gamma }_{\mathcal{C}}$, force-tile area $A_{\mathcal{C}}$, Voronoi volume $V_{\mathcal{C}}$, number of particles $N_{\mathcal{C}}$, and number of small particles $N_{s\mathcal{C}}$) on compact subclusters of particles $\mathcal{C}$, as a function of the cluster size and the global system stress. We find several significant differences depending on whether the cluster $\mathcal{C}$ is defined by a fixed radius $R$ or a fixed number of particles $M$. We comment on the implications of our findings for maximum entropy models of jammed packings.

Figure 1

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 2

Average packing fraction $\langle \varphi \rangle$ vs total system stress per particle $\tilde{p}\equiv {\Gamma }_N/N$. Error bars represent the width of the distribution rather than the statistical error in the average.

Figure 3

$[\langle A_N\rangle /\left({\Gamma }_N^2\langle \frac{1}{V}\rangle \right)]-1$ vs $\tilde{p}\equiv {\Gamma }_N/N$, where $A_N$ is the total force-tile area, ${\Gamma }_N$ is the total system stress, and $V$ is the total system volume. We expect this quantity to vanish at all $\tilde{p}$. Our system has a total of $N=8192$ particles.

Figure 4

Ratio of intensive quantities defined on a cluster of radius $R$, $(X_R/\pi R^2)$, to the corresponding quantity defined on the total system, $(X_N/V)$, vs cluster radius $R$ for $X$ equal to the (a) stress $\Gamma$; (b) force-tile area $A$; (c) Voronoi volume $V$; (d) number of particles $N$; (e) number of small particles $N_s$. Three different values of the total system stress per particle $\tilde{p}$ are shown, represented by three different symbol shapes. Our system has a total of $N=8192$ particles.

Figure 5

Variance of quantities defined on a cluster of radius $R$, $\mathrm{var}\left(X_R\right)/\pi R^2$ vs $R$, for $X$ equal to the (a) stress $\Gamma$; (b) force-tile area $A$; (c) Voronoi volume $V$; (d) number of particles $N$; (e) number of small particles $N_s$. Three different values of the total system stress per particle $\tilde{p}$ are shown, represented by three different symbol shapes. Solid lines are fits to the form $c_1+c_2/R$; for $V_R$, $c_1=0$. Our system has a total of $N=8192$ particles.

Figure 6

$\mathrm{var}\left({\Gamma }_R\right)/\pi R^2$ and $\mathrm{var}\left(A_R\right)/\pi R^2$, in the large $R$ limit, vs total stress per particle $\tilde{p}$. Solid lines are fits to power laws and give $\sim {\tilde{p}}^{1.9}$ and $\sim {\tilde{p}}^{3.9}$, respectively. Our system has a total of $N=8192$ particles.

Figure 7

Scatter plots showing configuration specific values of $({\Gamma }_R-\langle {\Gamma }_R\rangle )/{\sigma }_{{\Gamma }_R}$ vs $(X_R-\langle X_R\rangle )/{\sigma }_{X_R}$ for $X_R$ equal to the (a) square root of the force-tile area $A_R^{1/2}$; (b) Voronoi volume $V_R$; (c) number of particles $N_R$; (d) number of small particles $N_{sR}$. Here ${\sigma }_{X_R}$ is the standard deviation of variable $X_R$ and results are for the specific case $R=5.4$ and $\tilde{p}=0.00078$.

Figure 8

Covariance between rescaled stress ${\widehat{\Gamma }}_R$ and other variables vs cluster radius $R$, for three different values of total system stress per particle $\tilde{p}$. The rescaled variables are defined by ${\widehat{X}}_R\equiv (X_R-\langle X_R\rangle )/{\sigma }_{X_R}$, with ${\sigma }_{X_R}$ the standard deviation of $X_R$, and the plot shows results for $X_R$ equal to the square root of the force-tile area $A_R^{1/2}$, Voronoi volume $V_R$, number of particles $N_R$, and number of small particles $N_{sR}$. Our system has a total of $N=8192$ particles.

Figure 9

Scatter plots showing configuration specific values of $(V_R-\langle V_R\rangle )/{\sigma }_{V_R}$ vs $(X_R-\langle X_R\rangle )/{\sigma }_{X_R}$ for $X_R$ equal to the (a) square root of the force-tile area $A_R^{1/2}$; (b) number of particles $N_R$; (c) number of small particles $N_{sR}$. Here ${\sigma }_{X_R}$ is the standard deviation of variable $X_R$ and results are for the specific case $R=5.4$ and $\tilde{p}=0.00078$.

Figure 10

Covariance between rescaled Voronoi volume ${\widehat{V}}_R$ and other variables vs cluster radius $R$, for three different values of total system stress per particle $\tilde{p}$. The rescaled variables are defined by ${\widehat{X}}_R\equiv (X-\langle X_R\rangle )/{\sigma }_{X_R}$, with ${\sigma }_{X_R}$ the standard deviation of $X_R$. We show results for $X_R$ equal to the stress ${\Gamma }_R$, square root of the force-tile area $A_R^{1/2}$, number of particles $N_R$, and number of small particles $N_{sR}$. Our system has a total of $N=8192$ particles.

Figure 11

Ratio of intensive quantities defined on a cluster of fixed number of particles $M$, $(X_M/M)$, to the corresponding quantity defined on the total system, $(X_N/N)$, vs $M$ for $X$ equal to the stress $\Gamma$, force-tile area $A$, Voronoi volume $V$, and number of small particles $N_s$. Three different values of the total system stress per particle $\tilde{p}$ are shown, represented by three different symbol shapes. Solid lines are fits to the form $1+c/M$. Our system has a total of $N=8192$ particles.

Figure 12

Variance of quantities defined on a cluster of fixed number of particles $M$, $\mathrm{var}\left(X_M\right)/M$ vs $M$, for $X$ equal to the (a) stress $\Gamma$; (b) force-tile area $A$; (c) Voronoi volume $V$; (d) number of small particles $N_s$. Three different values of the total system stress per particle $\tilde{p}$ are shown, represented by three different symbol shapes. Solid lines are fits to the form $c_1+c_2/\sqrt{M}$. Our system has a total of $N=8192$ particles.

Figure 13

$\mathrm{var}\left({\Gamma }_M\right)/M$ and $\mathrm{var}\left(A_M\right)/M$, in the large $M$ limit, vs total stress per particle $\tilde{p}$. Solid lines are fits to power laws and give $\sim {\tilde{p}}^{1.9}$ and $\sim {\tilde{p}}^{3.9}$, respectively. Our system has a total of $N=8192$ particles.

Figure 14

Concentration of small particles $x_s\left(\langle N_R\rangle \right)$ and $x_s\left(M\right)$ for clusters of fixed radius $R$ and fixed number of particles $M$, respectively. In the first case, $x_s\left(\langle N_R\rangle \right)$ is plotted vs $\langle N_R\rangle$, the average number of particles in the cluster. For $x_s\left(\langle N_R\rangle \right)$ we show results from the direct computation of Eq. (14) (open symbols) as well as the prediction of Eq. (17) (solid symbols). Solid lines are fits to the forms shown. Three different values of the total system stress per particle $\tilde{p}$ are shown, represented by three different symbol shapes.

Figure 15

Scatter plots showing configuration specific values of $({\Gamma }_M-\langle {\Gamma }_M\rangle )/{\sigma }_{{\Gamma }_R}$ vs $(X_M-\langle X_M\rangle )/{\sigma }_{X_M}$ for $X_M$ equal to the (a) square root of the force-tile area $A_M^{1/2}$; (b) Voronoi volume $V_M$; (c) number of small particles $N_{sM}$. Here ${\sigma }_{X_M}$ is the standard deviation of variable $X_M$ and results are for the specific case $M=66$ and $\tilde{p}=0.00078$ (a cluster with $M=66$ has an average radius of $\langle R\rangle \approx 5.4$).

Figure 16

Covariance between rescaled stress ${\widehat{\Gamma }}_M$ and other variables vs the number of particles in the cluster $M$. Results are shown for three different values of total system stress per particle $\tilde{p}$, as indicated by different symbol shapes. The rescaled variables are defined by ${\widehat{X}}_M\equiv (X_M-\langle X_M\rangle )/{\sigma }_{X_M}$, with ${\sigma }_{X_M}$ the standard deviation of $X_M$, and the plot shows results for $X_M$ equal to the square root of the force-tile area $A_M^{1/2}$, Voronoi volume $V_M$, and number of small particles $N_{sM}$. Our system has a total of $N=8192$ particles.

Figure 17

Scatter plots showing configuration specific values of $(V_M-\langle V_M\rangle )/{\sigma }_{V_R}$ vs $(X_M-\langle X_M\rangle )/{\sigma }_{X_M}$ for $X_M$ equal to the (a) square root of the force-tile area $A_M^{1/2}$; (b) number of small particles $N_{sM}$. Here ${\sigma }_{X_M}$ is the standard deviation of variable $X_M$ and results are for the specific case $M=66$ and $\tilde{p}=0.00078$ (a cluster with $M=66$ has an average radius of $\langle R\rangle \approx 5.4$).

Figure 18

Covariance between rescaled stress ${\widehat{V}}_M$ and other variables vs the number of particles in the cluster $M$. Results are shown for three different values of total system stress per particle $\tilde{p}$, as indicated by different symbol shapes. The rescaled variables are defined by ${\widehat{X}}_M\equiv (X_M-\langle X_M\rangle )/{\sigma }_{X_M}$, with ${\sigma }_{X_M}$ the standard deviation of $X_M$, and the plot shows results for $X_M$ equal to the stress ${\Gamma }_M$, square root of the force-tile area $A_M^{1/2}$, and number of small particles $N_{sM}$. Our system has a total of $N=8192$ particles.