{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"d60a3a45e0f678f4611fd747e115e3385fda110623292bef9b71ce9379921ee6","created":"2026-10-03T07:18:05Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"b51beea9b5b2c12628f7c1e7c4031d05fe92fedbcd92d9cec4c75c9154b7ee44","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"proof","assisted_by":[],"external_id":"mathph.bose-condensation.third-order-energy","kind":"well-posed","literature_status":"partially-resolved","n":"1","parents":[],"plain":"The energy of a dilute repulsive Bose gas is known rigorously to two terms in powers of the density. The next predicted term, with a logarithm, is proven only from above.","posed_since":"1959","precise":"With $\\hbar^{2}/(2m) = 1$, the ground-state energy density is predicted to be $e(\\rho) = 4 \\pi a \\rho^2 [1 + (128/(15 \\sqrt{\\pi})) (\\rho a^3)^{1/2} + 8 (4 \\pi/3 - \\sqrt{3}) \\rho a^3 \\operatorname{log}(\\rho a^3) + o(\\rho a^3 \\operatorname{log}(\\rho a^3))]$ (Wu, 1959). Prove the matching lower bound for general short-range $v \\ge 0$. Answer: a proof of the two-sided expansion.","problem_ref":null,"references":"","settled_by":"A lower bound on $e(\\rho)$ matching the Wu term.","status_note":"The two-term Lee-Huang-Yang formula is proven (Yau and Yin 2009, upper bound; Fournais and Solovej 2020, lower bound); a third-order upper bound in the thermodynamic limit was proven by Brooks, Oldenburg, Saint Aubin and Schlein (arXiv 2506.04153, 2025).","title":"Third-order term in the dilute Bose gas ground-state energy","topic_ref":"8e2da899d10157a0800115bf0ebd7035da124ea4f6901ed1432508c7509786fe"},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"15249fcc69fc61f22774f16c72c055f50f86ce9259881666583d2bba031a9574","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"eb3330422ae9610e0c87bbb2ea10b72760f646297f7003ff308cabccde26eb10","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"xiuzpGhVGWWXwQURTpZ_nLby_nb--6cw4h5hFQ7TT_64DOPjLqEXnQ0NiyhOmq6jdp1Ncrz2D3A1ay24QW9UDw"},"schema":"pubphys.envelope/1"},"record_hash":"15249fcc69fc61f22774f16c72c055f50f86ce9259881666583d2bba031a9574","leaf_index":1601}