Jamming Transition — Soft Spheres

The jamming transition occurs at a critical packing fraction φ_J where a disordered packing of soft spheres suddenly develops a rigid network of contacts. Below φ_J spheres can rearrange freely; above it, they overlap and generate repulsive forces. At the transition, the mean contact number Z jumps to Z_c = 2d (isostaticity — exactly enough contacts to constrain all degrees of freedom). The pressure grows as P ~ (φ−φ_J)^α with α≈1 for harmonic spheres, α≈3/2 for Hertzian contact. In 2D with a bidisperse mixture, φ_J ≈ 0.84; for monodisperse disks φ_J ≈ 0.906 (hexagonal close packing). The jamming point is a critical point with diverging length scales and anomalous low-frequency modes (excess phonons).