Mean-field model for the density of states of jammed soft spheres

We propose a class of mean-field models for the isostatic transition of systems of soft spheres, in which the contact network is modeled as a random graph and each contact is associated to $d$ degrees of freedom. We study such models in the hypostatic, isostatic, and hyperstatic regimes. The density of states is evaluated by both the cavity method and exact diagonalization of the dynamical matrix. We show that the model correctly reproduces the main features of the density of states of real packings and, moreover, it predicts the presence of localized modes near the lower band edge. Finally, the behavior of the density of states $D\left(\omega \right)\sim {\omega }^{\alpha }$ for $\omega \to 0$ in the hyperstatic regime is studied. We find that the model predicts a nontrivial dependence of $\alpha$ on the details of the coordination distribution.

Figure 1

Pictorial representation of a small overjammed system ($N=256$) of soft spheres in $d=3$ dimensions with its corresponding contact network. The overjammed configuration has been obtained assuming periodic boundary conditions. To simplify the network figure the contacts across the boundary are not shown.

Figure 2

An instance of the ${\mathcal{G}}_{4,3}$ model for $N=200$ with a low-frequency eigenmode represented on it. The intensity of the color is proportional to the amplitude of the corresponding eigenmode on each site. It is evident that the eigenmode is localized on a site with $z=4$, the lowest possible coordination.

Figure 3

DOS and participation ratio [Eq. (13)] for the ${\mathcal{G}}_{5,0}$ model (a) and the ${\mathcal{G}}_{4,1}$ model (b) in the hypostatic case, $\overline{z}=5$, using the cavity method (black dots) and ED (color symbols). The numerical integration of the cavity method equations has been performed using $\varepsilon ={10}^{-8}$ and a population of at least ${10}^7$ fields for $\omega <0.2$, and a population of ${10}^6$ fields for the rest of the interval. The ED results were obtained for $N=500$ (red squares), $N=1000$ (blue circles), and $N=2000$ (green crosses).

Figure 4

Detail of the DOS $D\left(\omega \right)$, the cumulative function $\mathrm{\Phi }\left(\omega \right)$, and the participation ratio $Y\left(\omega \right)$ [Eq. (13)] at low frequencies for the ${\mathcal{G}}_{5,0}$ model (left) and the ${\mathcal{G}}_{4,1}$ model (right) using the cavity method (black) and ED (color).

Figure 5

Cumulative function $\mathrm{\Phi }$ obtained using the cavity method in the ${\mathcal{G}}_{4,\overline{\zeta }}$ model for different values of $\overline{z}=4+\overline{\zeta }<6$. We observe that the value of ${\omega }_0$ decreases and the gap closes as soon as $\overline{z}\to 6$. The smooth lines are represented as guides for the eye. On the right panel, the same data are plotted with the $x$ axis rescaled by $6-\overline{z}$ (the distance from the isostatic transition).

Figure 6

DOS, localization indicator $\widehat{Y}$, and participation ratio $\frac{1}{Y}$ for the ${\mathcal{G}}_{6,0}$ model (a) and the ${\mathcal{G}}_{4,2}$ model (b) in the isostatic case, $\overline{z}=6$, using the cavity method (black), the method of moments (red), and ED. The numerical integration of the cavity method equation has been performed using $\varepsilon ={10}^{-8}$ and a population of at least ${10}^7$ fields for $\omega <0.2$, and a population of ${10}^6$ fields for the rest of the interval. For the method of moments, 50 moments were used, averaging over 25 instances of a matrix $\mathsf{M}$ with $N={10}^6$. The method of moments gives very highly fluctuating results for $\omega <0.1$ that have not been represented. ED results for this region are shown in Fig. 7.

Figure 7

Detail of the DOS $D\left(\omega \right)$, the cumulative function $\mathrm{\Phi }\left(\omega \right)$, and the participation ratio at low frequency for the ${\mathcal{G}}_{6,0}$ (left) and the ${\mathcal{G}}_{4,2}$ (right) models. The results were obtained using ED (color) and the cavity method (black dots).

Figure 8

Distribution of the imaginary part of the resolvent ${\theta }_{\omega }:=\omega {\eta }_{\omega }$ in the isostatic case for both the ${\mathcal{G}}_{6,0}$ model (left) and the ${\mathcal{G}}_{4,2}$ model (right).

Figure 9

Average coordination $\overline{\langle z\rangle }$ as a function of $\overline{\omega }$ in the low-frequency region for the ${\mathcal{G}}_{4,2}$ model (left) and the ${\mathcal{G}}_{4,3}$ model (right), obtained using ED.

Figure 10

DOS, localization indicator $\widehat{Y}$, and participation ratio $\frac{1}{Y}$ for the ${\mathcal{G}}_{7,0}$ model (a) and the ${\mathcal{G}}_{4,3}$ model (b) in the hyperstatic case $\overline{z}=7$, using the cavity method (black), the method of moments (red), and ED. The numerical integration of the cavity method equation has been performed using $\varepsilon ={10}^{-8}$ and a population of at least ${10}^7$ fields for $\omega <0.2$, and a population of ${10}^6$ fields for the rest of the interval. For the method of moments, 50 moments were used, averaging over 25 instances of a matrix $\mathsf{M}$ with $N={10}^6$. The method of moments gives very highly fluctuating results for $\omega <0.2$ that have not been represented. ED results for this region are shown in Fig. 7.

Figure 11

ED and cavity method results for the hyperstatic regime. (a) Cumulative function (top) and the participation ratio (bottom) at low frequency for the ${\mathcal{G}}_{7,0}$ (left) and ${\mathcal{G}}_{4,3}$ (right) models. The arrows indicate the value of the average of the first nonzero frequency ${\overline{\omega }}_1:=\mathbb{E}\left[{\omega }_1\right]$ for each system size. (b) Cumulative function (top) for several systems with $\overline{z}=z_0+\overline{\zeta }=7$, namely, the ${\mathcal{G}}_{7,0}$, ${\mathcal{G}}_{6,1}$, ${\mathcal{G}}_{5,2}$, and ${\mathcal{G}}_{4,3}$ models. On the bottom, the average coordination $\overline{\langle z_k\rangle }$ as a function of the frequency of the $k\text{th}$ eigenvector [see Eq. (17)]. (c) Scaling of the average of the first mode frequency with $N$ in the hyperstatic case. The lines are fitted functions. The fits were performed excluding sizes $N<{10}^4$, since for smaller values of $N{\overline{\omega }}_1$ is typically located in the bulk and not in the low-frequency tail of the distribution [see Figs. 11 and 13]. The parameter $a$ of the fit $N^{-a}$ is related to the exponent $\alpha$ by $a={(\alpha +1)}^{-1}$ [see Eq. (18)], giving $\alpha =4.3\left(8\right)$, 0.67(8), 4.0(2), and 1.049(8) for ${\mathcal{G}}_{6,0.1}$, ${\mathcal{G}}_{4,2.1}$, ${\mathcal{G}}_{7,0}$, and ${\mathcal{G}}_{4,3}$, respectively.

Figure 12

Fraction of eigenvalues with different $\overline{\langle z\rangle }$ as a function of $\omega$ in the ${\mathcal{G}}_{4,3}$ model. Here we consider the system size $N=5000$.

Figure 13

Cumulative function $\mathrm{\Phi }\left(\omega \right)$ for the ${\mathcal{G}}_{6,0.1}$ model and for the ${\mathcal{G}}_{4,2.1}$ model using the cavity method (black) and ED (color). The numerical integration of the cavity method equation has been performed using $\varepsilon ={10}^{-8}$ and a population of ${10}^6$ fields. The arrows indicate the value of the average of the first nonzero frequency for each system size.

Figure 14

DOS for $d=4$ evaluated on a random regular graph topology for different values of coordination $\overline{z}$. In the inset, detail of the low-frequency regime. The data have been obtained using ED on the entire spectrum for small values of $N$. Smooth lines represent the cavity prediction in this case.