Edwards entropy and compactivity in a model of granular matter

Formulating a statistical mechanics for granular matter remains a significant challenge, in part due to the difficulty associated with a complete characterization of the systems under study. We present a fully characterized model of a granular material consisting of $N$ two-dimensional, frictionless hard disks, confined between hard walls, including a complete enumeration of all possible jammed structures. We show that the properties of the jammed packings are independent of the distribution of defects within the system and that all the packings are isostatic. This suggests that the assumption of equal probability for states of equal volume, which provides one possible way of constructing the equivalent of a microcanonical ensemble, is likely to be valid for our model. An application of the second law of thermodynamics involving two subsystems in contact shows that the expected spontaneous equilibration of defects between the two is accompanied by an increase in entropy and that the equilibrium, obtained by entropy maximization, is characterized by the equality of compactivities. Finally, we explore the properties of the equivalent to the canonical ensemble for this system.

Figure 1

(a) A jammed configuration. The dashed lines join the centers of contacting disks and are labeled 0 for the most dense, or 1 for the least dense (defect) arrangement. The shaded regions represent the volumes associated with each. (b) An arrangement with two neighboring defects (-1-1-) is not jammed because the shaded disk does not fulfill the local jamming condition and is able to move vertically.

Figure 2

(Top panel) The volume per particle for each subsystem $V_1/N_1$ (dashed-dotted line), $V_2/N_2$ (dashed line), and the composite system $V_T/N$ (dashed-dotted line). (Bottom panel) The entropy per particle $S_1/N_1$ (dashed-dotted line), $S_2/N_2$ (dashed line), and the composite system $S_T/N$ (dashed-dotted line). The vertical dotted line locates the maximum in $S_T$, $N_1=N_2=100$, ${\sigma }_1=1$, ${\sigma }_2=1.1{\sigma }_1$, $H=1.8{\sigma }_1$, ${\theta }_1^{\prime }=0$, and ${\theta }_2^{\prime }=0.27$.

Figure 3

(a) The entropy $S$ and (b) the volume relative to the volume of the most dense state $V/V_{\text{min}}$ as a function $X$ for a system with particles $H/\sigma =1.86$ (solid line) and $H/\sigma =1.50$ (dashed line).