{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"7c8288b451ff157d1197c79d754f4451b0ac8d625871d7b139ca803176098003","created":"2026-10-03T07:18:05Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["281467524de81f66573dcbbed47192e3c0423267cd2a96b34feb574cb93de781","5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"840728defd61f0caa397c2000818b16165ad71a0492e61c5e12d81a3a99ffe1d","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"value","assisted_by":[],"external_id":"mathph.bose-condensation.tc-shift","kind":"well-posed","literature_status":"open","n":"1","parents":[{"note":"","parent_revision":"281467524de81f66573dcbbed47192e3c0423267cd2a96b34feb574cb93de781","relation":"special_case"}],"plain":"Weak repulsion between bosons raises their condensation temperature slightly, in proportion to the interaction strength. The coefficient is known only from computer simulations.","posed_since":"","precise":"For the dilute 3D Bose gas, prove that $(T_{c} - T_{c}^{0})/T_{c}^{0} = c\\rho^{1/3}a + o(\\rho^{1/3}a)$, with $T_{c}^{0}$ the ideal-gas value, and determine c (lattice Monte Carlo gives c approximately 1.3, Arnold and Moore; Kashurnikov, Prokof'ev and Svistunov, 2001). Answer: the existence of the phase transition and the value of c.","problem_ref":null,"references":"","settled_by":"A proof of a phase transition at $T_{c}$ with rigorous bounds on $c$.","status_note":"","title":"Shift of the condensation temperature caused by weak repulsion","topic_ref":"8e2da899d10157a0800115bf0ebd7035da124ea4f6901ed1432508c7509786fe"},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"e3cb2174b2801545074b6e6433dcbc83fbde17c3075c51ea3979c3e78e6d7556","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"5d50660ffd00f1e94118d4c32b08e55703b26d589a36ccf368299a2fbc9a5349","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"RBOURJ-i0wIw50C5XY4WHq0Y4DCl3iZhOBN44oyMTHUpBkfkoi1LKfONCNZMrk5i-G6obGIEMfifSlVPe_oaCw"},"schema":"pubphys.envelope/1"},"record_hash":"e3cb2174b2801545074b6e6433dcbc83fbde17c3075c51ea3979c3e78e6d7556","leaf_index":1600}