MATHPH
In the literature: open
Crystalline phase of hard spheres at high packing fraction
In plain words
Hard balls with no attraction freeze into a crystal when packed densely enough, as computer simulations show above about 55 percent filling. Prove that such a crystalline equilibrium state exists.
Precise statement
For hard spheres of diameter $a$ in $R^3$ at packing fraction $\eta$ close to the close-packing value $\pi/\sqrt{18}$, about 0.7405, prove existence of a non-translation-invariant Gibbs measure, and more sharply a first-order freezing transition; simulations place fluid-solid coexistence at $\eta$ about 0.494 and 0.545.
What would settle it
A rigorous proof of crystalline long-range order for hard spheres at some $\eta$ below close packing.
Related problems
- Special case of Broken translation symmetry for a 3D continuum particle system