Hard Sphere Jamming
Random packing approaching the jamming point — contact network and pressure
Jamming Transition (Liu & Nagel 1998): Random packings of hard spheres undergo a jamming transition at a critical packing fraction φ_J≈0.64 (random close packing in 3D; ~0.84 in 2D). Below φ_J, the system flows freely. At φ_J, the contact network becomes mechanically rigid — every sphere has z=2d contacts on average (isostaticity: Maxwell counting). Pressure P∝(φ−φ_J)^1 diverges, and the bulk modulus shows critical scaling. The contact network forms a force chain structure. This simulation grows sphere radii slowly, relaxing overlaps via gradient descent, and tracks φ, pressure, and contact number z as jamming approaches.