{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"d635f1918d7744650ca655ec3fe82aacc67de12e493bb37500527fffc80ff918","created":"2026-10-03T07:18:05Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"3b75bec44bb7562a8acc9bac15877b9545b09b1fd918b44d25eb84ecd99bb9fa","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"yes-no","assisted_by":[],"external_id":"mathph.crystal-order-temperature.continuum-3d-crystal","kind":"well-posed","literature_status":"open","n":"1","parents":[],"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.","posed_since":"","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.","problem_ref":null,"references":"","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.","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.","title":"Broken translation symmetry for a 3D continuum particle system","topic_ref":"ea8bde04090399eb2ce07a2138c4a5deafce80c26031655b989bc3381e0879d5"},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"e1fabcc9d99cc1fbb81083f240267eb6d9ab9d915051d745f3a047049e5172fd","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"38f72a9fa1151614d2656fcef2787877b1deeb67fd25f1a56ff51f3402280dfa","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"qLS-E70Fleg5HE3ullHdFhZwOqsm2QWeo26IC6KpKazMoazghEnS4OzFqGcb_f7UD5hG4jlHM8r7LMUL9ri8Ag"},"schema":"pubphys.envelope/1"},"record_hash":"e1fabcc9d99cc1fbb81083f240267eb6d9ab9d915051d745f3a047049e5172fd","leaf_index":1602}