Inherent Structure Landscape Connection between Liquids, Granular Materials, and the Jamming Phase Diagram

We provide a comprehensive picture of the jamming phase diagram by connecting the athermal, granular ensemble of jammed states and the equilibrium fluid through the inherent structure paradigm for a system of hard disks confined to a narrow channel. The $J$ line is shown to be divided into packings that are either accessible or inaccessible from the equilibrium fluid. The $J$ point itself is found to occur at the transition between these two sets of packings and is located at the maximum of the inherent structure distribution. We also present a general thermodynamic argument that suggests the density of the states at the maximum of the configurational entropy represents a lower bound on the $J$-point density in hard sphere systems. Finally, we show that the granular system, modeled using the Edwards ensemble, and the fluid sample the same set of thermodynamically accessible states over the full range of thermodynamic state points, but only occupy the same set of inherent structures, under the same thermodynamic conditions, at two points, corresponding to zero and infinite pressures, where they sample the $J$-point states and the most dense packing, respectively.

Figure 1

Analytical quench connecting equilibrium fluid configurations of disks $i$, $m$, $n$, $j$ to the most dense (1, 3) and least dense (2, 4) local packing arrangements. Equilibrium configuration of four disks are initially mapped to a tangent disk configuration by compression in the $x$ axis. The local arrangement of the disks contained in the product of the kernel $K(y_i,y_j)$ and the dot product of vectors [Eq. (4)] maps the central disks to their jammed configuration. See text for more details.

Figure 2

$S_c/Nk$ versus ${\varphi }_J$ for $H/\sigma =1.866$ showing features of the $J$ line. The thermodynamically accessible packings have densities between the $J$ point, with ${\varphi }_J^{*}=0.659$ (green square) and the most dense jammed state (black circle) with ${\varphi }_J=0.842$. ${\varphi }_{th}\sim 0.72$ (orange diamond) is located at a higher density than the $J$ point. The thermodynamically inaccessible packings densities between the $J$ point and the least dense state with ${\varphi }_J=0.561$ (red triangle).

Figure 3

EOS for $H/\sigma =1.866$. EOS for granular ensemble: $\beta P_L{h}_0$ as a function of ${\varphi }_J$ (blue dot-dashed line). EOS for equilibrium thermal fluid: $\beta P_L{h}_0$ as a function of $\varphi$ (black solid line). ${\varphi }_J$ of basins sampled by equilibrium fluid (red dashed line). EOS for fluid compressed using the LS method with different $\partial \sigma /\partial t$ (dotted lines). ${\varphi }_d\approx 0.48$ obtained from MD simulations. Insert: ${\varphi }_J$ as a function of $\partial \sigma /\partial t$.