X$_{c}$ Δ s$_{fus}$ = 0 . 09 V$_{g}$ while the compactivity stays constant. This process corresponds to the typical latent heat in exothermic first-order transitions. 

Systems jammed at RCP need to overcome a volume barrier Δ h$_{fus}$ for crystal formation or, equivalently, particle displacements Δ r$_{fus}$ ≈ 0 . 45 R . From a thermodynamic perspective, the requirement is equivalent to bringing a random packing in contact with a compactivity bath at X < X$_{c}$ = 0 . 031 V$_{g}$ . The fundamental idea is to surround a random packing above X$_{c}$ with a crystal lattice below X$_{c}$ and perturb the system to equilibrate. A shear cycling experiment—which conserves the shape of the box containing the particles—suffices to explore the crystal branch [26, 27]. Shear-induced crystallization has been observed [26, 27] when the maximum angle of horizontal shear is above θ ≈ 10 ◦ . This value is of the same order as our estimate of the shear amplitude to crystallize at φ$_{j}$ = 68% based on the entropy of fusion, which gives θ ≡ tan − $^{1}$(Δ r$_{fus}$/ 2 R ) ≈ 13 ◦ . Furthermore, recent shear cycling experiments [8, 28] appear to be in reasonable agreement with the present results. We also expect that 2d equal-sized disks may have a near zero entropy of fusion owning to their tendency to easily crystallize while Δ s$_{fus}$ may sharply increase in 4d and above [6]. 

The behavior of the free energy density shown in Figs. 4C and 4D summarizes the mechanism to achieve RCP. The free energy in Fig. 4C increases as X decreases from RLP to freezing at RCP. At X$_{c}$ , the system transitions to the phase with the lower free energy through an entropy discontinuity given by s = − ∂f/∂X . The system may also enter the metastable branch as indicated in Fig. 4C and in Figs. 1 and 3B from a → b . In the spirit of Landau mean field theory of phase transitions, we relate the distribution of the local order parameter to the free energy functional, F , and X as P ( q$_{6}$ ) ≈ exp[ −F ( q$_{6}$ ) /X ] [20]. Figure 4D shows F ( q$_{6}$ ) displaying the minima of F ( q$_{6}$ ) defining the order and disorder phases at different φ$_{j}$ . We find that the location of the minimum at q min 6 ≈ 0 . 425 remains constant from RLP up to the freezing point as expected in the disordered phase. The value of F ( q min 6 ) is very deep for φ$_{j}$ = 0 . 55 and becomes less deep as the freezing point is approached. The value of F ( q min 6 ) at the freezing and melting points become approximately equally deep indicating the phase coexistence at X$_{c}$ . Within a statistical mechanics framework, these results are a natural consequence and give support to such an underlying statistical picture. 

In conclusion, treating granular packings from the perspective of theoretical physics de- veloped by Boltzmann and Gibbs has the potential to answer basic questions in the field of disordered media. State variables like the compactivity can be introduced with the potential 

