{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"353b97ed18dc22224785ea8d8377f9135074618b5bb1ad8dd4406ddad4f1aaf6","created":"2026-10-15T23:43:22Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["15249fcc69fc61f22774f16c72c055f50f86ce9259881666583d2bba031a9574","ce59fb6cd812fc05906d0c60a71c75ea22bc686bcf38e7c3ef7010834d98ee7b"],"salt":"a2c3d34606c1ca0e63c9eb64b7bc61819f995bb2a7dace9cfaea2651f330fa60","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":"2","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 $e(\\rho) = 4 \\pi a \\rho^2 [1 + (128/(15 \\sqrt{\\pi})) (\\rho a^3)^{1/2} + 8 (4 \\pi/3 - \\sqrt{3}) \\rho a^3 \\log(\\rho a^3) + o(\\rho a^3 \\log(\\rho a^3))]$ (Wu, 1959). Prove the matching lower bound for general short-range $v \\ge 0$. Answer: a proof of the two-sided expansion.","problem_ref":"15249fcc69fc61f22774f16c72c055f50f86ce9259881666583d2bba031a9574","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":null},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"ddc0b7425b6e04ffc036508987f2d2eb71b15e20823aa95909c820bfa9d7f6d6","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"c419d4ec0aa039aa14e82d1d86e7dbd5da9dfb6d3d275af8a16a65cd84a20f99","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"IM2hx5SuhKBRwRlaX1qvuf4sUrig60ey7LCQ1nGpdI-XFIhWxhYPPLpcl26u11_qVEE9KEqcCxMUjSpXSvgsBQ"},"schema":"pubphys.envelope/1"},"record_hash":"ddc0b7425b6e04ffc036508987f2d2eb71b15e20823aa95909c820bfa9d7f6d6","leaf_index":3406}