Jamming Below Upper Critical Dimension

Extensive numerical simulations in the past decades proved that the critical exponents of the jamming of frictionless spherical particles are the same in two and three dimensions. This implies that the upper critical dimension is $d_u=2$ or lower. In this Letter, we study the jamming transition below the upper critical dimension. We investigate a quasi-one-dimensional system: disks confined in a narrow channel. We show that the system is isostatic at the jamming transition point as in the case of standard jamming transition of the bulk systems in two and three dimensions. Nevertheless, the scaling of the excess contact number shows the linear scaling. Furthermore, the gap distribution remains finite even at the jamming transition point. These results are qualitatively different from those of the bulk systems in two and three dimensions.

Figure 1

A configuration at ${\phi }_J$ for $N=32$ and $L_y=2{\sigma }_{\max }$. Gray circles represent particles, and the solid lines denote the contacts.

Figure 2

$L_y$ dependence of (a) the jamming transition point ${\phi }_J$ and (b) the contact number per particle at the jamming transition point $z_J$. Markers denote numerical results, and solid lines denote the guide to the eye. The dashed lines denote the linear fits ${\phi }_J=0.84-0.28{\sigma }_{\max }/L_y$ and $z_J=4-1.4{\sigma }_{\max }/L_y$.

Figure 3

$L_y$ dependence of the number of constraints per particle at the jamming transition point $c_J$. Markers denote the numerical results, and the dashed line denotes the isostatic number $c_{\mathrm{iso}}=2$.

Figure 4

(a) $\delta \phi$ dependence of the energy per particle $V_N/N$. Maker denote numerical results, and the solid line denotes $\delta {\phi }^2$. (b) $\delta \phi$ dependence of the pressure $p$. Maker denote numerical results, and the solid line denotes $\delta \phi$.

Figure 5

(a) $\delta c$ as a function of $\delta \phi$. Markers denote numerical results. The solid and dashed lines denote $\delta c\sim \delta {\phi }^{1/2}$ and $\delta c\sim \delta \phi$, respectively. (b) Scaling plot for the same data.

Figure 6

(a) $z$ as a function of $\delta \phi$. Markers denote numerical results. (b) $\delta z=z-z_J$ as a function of $\delta \phi$. The solid and dashed lines denote $\delta z\sim \delta {\phi }^{1/2}$ and $\delta z\sim \delta \phi$, respectively. (c) Scaling plot for the same data.

Figure 7

(a) CDF of the gap function $h$. Markers denote numerical results. The solid and dashed lines denote $h^{1-\gamma }$ and $h^1$, respectively. (b) Scaling plot for the same data.

Figure 8

Results for the quasi-two-dimensional system and bulk three dimensional system. (a) $\delta c$ as a function of $\delta \phi$. Markers denote numerical results. The solid and dashed lines denote $\delta c\sim \delta {\phi }^{1/2}$ and $\delta c\sim \delta \phi$, respectively. (b) CDF of the gap function $h$. Markers denote numerical results. The solid and dashed lines denote $\mathrm{CDF}\sim h^{1-\gamma }$ and $\mathrm{CDF}\sim h^1$, respectively.