FIG. 5: (A) The mechanical coordination number Z$_{j}$ versus the volume fraction φ$_{j}$ with different system sizes (500 and 10,000 particles) and algorithms (Lubachevsky-Stillinger LS, force-biased FBA, and split SA algorithms). The results show that the transition does not depend on the system size and the algorithm used. (B) Comparison between geometrical coordination number z$_{g}$ and mechanical coordination number Z$_{j}$ . Along the disordered branch, z$_{g}$ closely follows the mechanical coordination number Z$_{j}$ . We expect that in the thermodynamical limit the gap between both coordinations may diminish. z$_{g}$ and Z$_{j}$ start to diverge at RCP, which is an indication of increasing geometric degeneracies in the contact network at the onset of crystallization. within the system error. The pressure of the packings at the jamming point studied in this paper is 100 ± 8 KPa. The difference between the volume fraction φ of these packings and the critical volume fraction φ$_{j}$ from power-law fitting in Eq. (4) is about 10 − $^{3}$. The same preparation protocol is repeated by using the force-biased algorithm (FBA) of [13] as initial protocol. The force-biased algorithm is a variant of the original method of W. S. Jodrey and E. M. Tory, Phys. Rev. A 32 , 2347 (1985), We also generate packings following the split algorithm of [7] starting with low initial volume fractions at φ$_{i}$ = 0 . 3 below RLP. The mechanical coordination number, Z$_{j}$ , versus the final jamming density, φ$_{j}$ , is plotted in Fig. 5A for all the obtained jammed packings which total 720. The plot signals the existence of a transition at RCP. It is analogous to the equilibrium liquid-solid transition in hard-spheres, if we replace Z$_{j}$ by the kinetic pressure of the fluid [6]. 18 (A) ❐ ❒ ❮ ❰ GLYPH<a207> GLYPH<a208> GLYPH<a209> GLYPH<a210> GLYPH<a211> GLYPH<a212> GLYPH<a213> GLYPH<a214> GLYPH<a215> GLYPH<a216> GLYPH<a217> GLYPH<a218> GLYPH<a219> GLYPH<a220> GLYPH<a221> GLYPH<a222> GLYPH<a223> GLYPH<a224> GLYPH<a225> GLYPH<a226> GLYPH<a227> GLYPH<a228> GLYPH<a229> GLYPH<a230> GLYPH<a231> GLYPH<a232> GLYPH<a233> GLYPH<a234> GLYPH<a235> GLYPH<a236> GLYPH<a237> GLYPH<a238> GLYPH<a239> GLYPH<a240> GLYPH<a241> GLYPH<a242> GLYPH<a243> GLYPH<a244> GLYPH<a245> GLYPH<a246> GLYPH<a247> GLYPH<a248> GLYPH<a249> GLYPH<a250> GLYPH<a251> GLYPH<a252> GLYPH<a253> GLYPH<a254> GLYPH<a255> ✁ GLYPH<a0> ✂ ✄ ☎ ✆ ✝ ✞ ✟ ✑ ✠ ✡ ☛ ☞ ✌ [ ✍ ] ✎ ✏ ✒ ✓ ✔ ✕ ✖✗ ✘ ✙ ✚ ✛ ✜ ✢ ✣ ✤✥ ✦ ✧ ★ ✩ ✪ ✫ ✬ ✭✮ ✯ ✰✱ ✲ ✳ ✴ ✵ ✶ ✷ ✸✹ ✺ ✻ ✼ ✽ ✾ ✿ ❀❁ ❂ ❃❄ ❅ ❆ ❇ ❈ ❉ ❊❋ ● ❍ ■ ❏ ❑ ▲ ▼ ◆ ❖ GLYPH<a80> ◗ ❘ ❙ ❚ ❯ ❱ ❲ ❳ ❨ Z j j (B) 0.56 0.60 0.64 0.68 0.72 4 6 8 10 12 ❩ ❬ ❭ ❪ ❫ ❴ ❵ ❛ ❜ ❝ ❞ ❡ ❢ ❣ ❤ ✐ ❥ ❦ ❧ ♠ ♥ ♦ ♣ GLYPH<a113> GLYPH<a114> GLYPH<a115> GLYPH<a116> ✉ ✈ ✇ j