Hard-Disk Packing & Jamming Transition

Controls

N disks: 80 Packing fraction φ: 0.60 Polydispersity δr/r: 0.10 Compression steps: 500

φ = Nπr²/L² (packing fraction)
φ_RCP ≈ 0.640 (monodisperse)
φ_crystal = π/√12 ≈ 0.9069
Jamming: Z = 2d (isostatic)
Z = 4 in 2D at jamming
Δp ~ (φ−φ_J)^1 near J-point

φ_actual: —

Contacts Z: —

State: —

Hard disks cannot overlap. Rapid compression leads to random close packing at φ≈0.64 (2D). At jamming, each disk has exactly Z=4 contacts on average (isostatic condition). Near this point, pressure P ~ (φ−φ_J) and the contact network becomes rigid.