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Bose–Einstein condensation at positive temperature in the dilute hard-sphere gas
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Abstract
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.
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Possibly relevant literature (found by the agent, not checked)
- W. König, The free energy of the interacting Bose gas: a variational description with loops and interlacements (arXiv:2604.14318, April 2026). This paper analyzes the interacting Bose gas at positive temperature and fixed density in the thermodynamic limit through Brownian bridges and loop soups; this paper's proof also works with Brownian-bridge path laws for the canonical Gibbs state.
Digests and expositions
- Latent Seminar, Bose gases: condensation, quantum depletion, and momentum profiles (7 October 2026). Explains this family’s claims and proof mechanisms without independently verifying complete proofs. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 267
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Cite
@misc{OAI:Bose-Einstein-condensation-at-positive-temperature-in-the-dilute-hard-sphere-gas-October-5-2026,
author = {{OpenAI}},
title = {{Bose--Einstein condensation at positive temperature in the dilute hard-sphere gas}},
howpublished = {OpenAI Math Release preprint
\href{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}{OAI:Bose-Einstein-condensation-at-positive-temperature-in-the-dilute-hard-sphere-gas-October-5-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/267.md. The Lean statements have not been checked by people against the paper's claims.
Family 267: Positive-temperature Bose–Einstein condensation and exact quantum depletion
Proves Bose–Einstein condensation for the exact canonical Gibbs state of the three-dimensional hard-sphere gas: each fixed exclusion distance and sufficiently small fixed density admit a strictly positive temperature, independent of volume, with positive condensate fraction in the thermodynamic limit. At zero temperature, proves the Bogoliubov leading quantum-depletion law for hard spheres and fixed bounded nonnegative radial finite-range potentials of positive scattering length, taking the thermodynamic limit before the dilute limit.
Other papers in this family:
- Quantum Depletion and Momentum Distribution in the Dilute Hard-Sphere Bose Gas · October 5, 2026
- Quantum Depletion for Fixed Bounded Repulsive Potentials · October 5, 2026
- A density-uniform condensate bound for dilute Bose gases · September 27, 2026
- Ground-state condensation in the dilute hard-sphere gas · September 24, 2026