{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"1e5608bfd928afa67905e9874474c29c7670b7c548da1c896032101bf5018ea1","created":"2026-10-03T07:18:05Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"05cbb3eb60dd91abed4253e8f7f5d851a80bd5b8370afdbe5366f0fc62431cac","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":"1","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 $R^{3}$, $H = \\operatorname{sum}_i -\\operatorname{Lap}_i + \\operatorname{sum}_{i<j} v(x_i - x_j)$ 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":null,"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":"8e2da899d10157a0800115bf0ebd7035da124ea4f6901ed1432508c7509786fe"},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"281467524de81f66573dcbbed47192e3c0423267cd2a96b34feb574cb93de781","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"5562829607a7322394fb22f5de0487ccfbbc0f16e2370d43f9c3b8902848850f","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"E5rYaZHeB7QYc0DGLKqSEV1tbN7KrnpZlFi3768pOgTAauY8NAv7kCbhlyE6Vj_M0K-HiIP5g-O6Og-M09WnBw"},"schema":"pubphys.envelope/1"},"record_hash":"281467524de81f66573dcbbed47192e3c0423267cd2a96b34feb574cb93de781","leaf_index":1597}