FIG. 9: Radial distribution functions g ( r ) with different volume fractions φ$_{j}$ . The figure clearly show the increasing of long range spatial order above φ$_{j}$ = 0 . 64. In the coexistence region, g ( r ) has more peaks than those of random packings, but the magnitude of the peaks decays very fast as the distance r becomes larger. The long-range order increases with the volume fraction in the coexistence region. In the thermodynamics of jammed matter, the internal energy U is replaced by the volume W (usually called the volume function) [21]. Other thermodynamic potentials, such as enthalpy H , free energy F , Gibbs free energy G , are related to the volume function W as IV. THERMODYNAMIC VIEWPOINT OF THE RCP TRANSITION A. Free energy 27 H = W, (20) F = W − XS, (21) G = F = W − XS, (22) GLYPH<a253> GLYPH<a254> GLYPH<a255> ☎ ✆ ✝ ✞ ✁ ✂ GLYPH<a0> ✄ ✟ ✠ ✡ ☛ ☞ ✌ ✍ ✎ ✏ ✑ ✒ ✓ ✔ ✕ r ✖ ✗ ✘ ✙ ✚ g ( r ) ✛ ✜ ✢ ✣ ✤ ✥ ✦ ✧ ★ ✩