Liu-Nagel Jamming Phase Diagram

Jamming transition: volume fraction φ, temperature T, applied stress — J point at φ_c

φ=0.50 — unjammed
0.600
0.10
0.00
4
Liu-Nagel jamming diagram (1998): the J point at (φ_c ≈ 0.64 in 2D, T=0, σ=0) is a critical point. Above φ_c: system jams (rigid, Z ≥ Z_c contacts). Temperature unjams by thermal fluctuations; stress unjams by yielding. Near J: pressure P ~ (φ−φ_c)^β, contact number Z−Z_c ~ (φ−φ_c)^½.