{"id":"openai-math/Bose-Einstein-condensation-at-positive-temperature-in-the-dilute-hard-sphere-gas","pdf":"https://github.com/openai/math/blob/main/preprints/Bose-Einstein-condensation-at-positive-temperature-in-the-dilute-hard-sphere-gas-October-5-2026/positive-temperature-hard-spheres.pdf","url":"https://github.com/openai/math/tree/main/preprints/Bose-Einstein-condensation-at-positive-temperature-in-the-dilute-hard-sphere-gas-October-5-2026","tags":["statistical-mechanics","bose-einstein-condensation","mathematical-physics"],"score":0,"title":"Bose–Einstein condensation at positive temperature in the dilute hard-sphere gas","domain":"Mathematical physics","family":"267","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.MP","summary":"We prove Bose–Einstein condensation at positive temperature in the three-dimensional dilute hard-sphere gas. For each fixed exclusion distance and every sufficiently small fixed density, there is a strictly positive temperature, independent of the volume, at which the exact canonical Gibbs state has a positive condensate fraction in the thermodynamic limit. The condensate occupies the constant orbital.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-10-05","reactions":{},"formalized":false,"family_title":"Positive-temperature Bose–Einstein condensation and exact quantum depletion"}