{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"b48eb2af8b481301300a5f2463cf20badf5f7f78835b99c36a8c5e5d60cdc0d5","created":"2026-10-15T23:43:22Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["281467524de81f66573dcbbed47192e3c0423267cd2a96b34feb574cb93de781","ce59fb6cd812fc05906d0c60a71c75ea22bc686bcf38e7c3ef7010834d98ee7b"],"salt":"af1a9f381effb7a09225998ebbc5053daeed63edce74a56fe2fa8003ef48bb37","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"proof","assisted_by":[],"external_id":"mathph.bose-condensation.bec-thermo-limit","kind":"well-posed","literature_status":"open","n":"2","parents":[],"plain":"Prove that a gas of mutually repelling bosons at low temperature puts a finite fraction of its particles into one quantum state when the container is infinitely large at fixed density. Without interactions this is a textbook calculation; with them there is no accepted proof.","posed_since":"","precise":"$N$ bosons in a box of side $L$ in $R^{3}$, $H = \\sum_i -\\operatorname{Lap}_i + \\sum_{i<j} v(x_i - x_j)$ with $v \\ge 0$ compactly supported, density $\\rho = N/L^{3}$ fixed, $L \\to \\infty$. Prove at $T = 0$ or at small $T > 0$ that the one-body density matrix has off-diagonal long-range order, lim_{|x-y| -> infinity} gamma^(1)(x,y) = $\\rho$_0 > 0. Answer: a proof for some open range of rho and $T$.","problem_ref":"281467524de81f66573dcbbed47192e3c0423267cd2a96b34feb574cb93de781","references":"","settled_by":"A proof of a nonzero condensate density in the thermodynamic limit for one continuum model.","status_note":"Proven in the Gross-Pitaevskii scaling (Lieb and Seiringer, 2002) and for lattice hard-core bosons at half filling (Dyson, Lieb and Simon, 1978), not for continuum gases in the thermodynamic limit; a preprint by Suto (arXiv 1710.04441, 2017) claims a proof via Feynman-Kac permutation cycles and lists no journal publication.","title":"Condensation of a repulsive Bose gas in the thermodynamic limit","topic_ref":null},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"31286813f49c88becda7831fb93f095ac05af79f4ef75926c87d3414d000f465","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"ee2e5275b2b594097befe7c8f69f13a46a885e341658fac2d3df808ff733f718","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"xw1Od8bqwM40x2sIbtXPKbjKZ1A_fhvHiX2vdhY468GUDdfMhVaBwwuxPVPVjNCV5lpTNYdjnLdF0n9c6mCYCg"},"schema":"pubphys.envelope/1"},"record_hash":"31286813f49c88becda7831fb93f095ac05af79f4ef75926c87d3414d000f465","leaf_index":3402}