Revision 1

Accepted · first revision · record

Title: Broken translation symmetry for a 3D continuum particle system
Plain: Prove that some realistic system of particles in continuous 3D space has, at low temperature, a thermal equilibrium state in which particles keep preferred lattice positions over infinite distances.
Precise: For classical particles in $R^3$ with a stable, regular pair potential $V$ (for example Lennard-Jones, or a smooth short-range potential with a crystalline ground state), prove that for large $\beta = 1/T$ at suitable density there is an infinite-volume Gibbs point process that is not translation invariant, with density correlations showing long-range periodic order. One such potential suffices.
Settled by: A construction, for example by a multiscale or Pirogov-Sinai-type expansion around the crystal, of a non-translation-invariant Gibbs measure for one continuum potential in 3D.
Kind: well-posed
Answer type: yes-no
References: (empty)
Literature status: open
Status note: Proved for lattice models via Pirogov-Sinai theory, and positional order is ruled out for 2D continuum systems (Richthammer 2007); as of 2026 no continuum system in $d \ge 3$ has a proof of a crystalline Gibbs state.
Posed since: (empty)