Jamming Transition in Soft Matter

Jamming transition: A collection of frictionless soft spheres transitions from a flowing liquid to a rigid amorphous solid at the jamming point φ_J ≈ 0.64 (random close packing in 2D) when compressed at zero temperature.

Critical scaling: Just above φ_J, the mean contact number Z - Z_iso ~ (φ - φ_J)^0.5 and pressure p ~ (φ - φ_J)^α with α ≈ 1.5 for harmonic contacts. The system is at a critical point — scale-free force networks, diverging length scales.

Isostaticity: At φ_J, Z = Z_iso = 2d = 4 (in 2D) — exactly the minimum number of contacts needed for mechanical stability (Maxwell counting). Below Z_iso, the packing is floppy; above, it is overconstrained.

Point J: The jamming point sits at T=0, σ=0 (no applied stress), φ=φ_J. It controls the properties of the jammed phase — unusually flat density of states (boson peak), anomalous transport, and non-affine deformation.