Browse › Mathematical physics › 267 Positive-temperature Bose–Einstein condensation and exact quantum depletion
Ground-state condensation in the dilute hard-sphere gas
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Abstract
We prove Bose–Einstein condensation in every ground state of a dilute three-dimensional hard-sphere Bose gas. At every sufficiently small fixed gas parameter, the constant orbital contains a positive fraction of the particles in the thermodynamic limit. The fraction can be chosen independently of the gas parameter, and the conclusion holds for arbitrary complex ground states.
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Notes
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- OpenAI math release, Lean scope notes for family 267, Positive-temperature Bose–Einstein condensation and exact quantum depletion (lean/docs/267.md, 2026). The release's scope notes say the Lean formalization covers this paper's theorem: uniform ground-state condensation for $\rho a^3<\varepsilon_0$, including mixed ground-space states, with no positive-temperature assertion.
- OpenAI math release, HardSphere.lean comparator statement (lean/ComparatorChallenges/HardSphere.lean, 2026). The comparator file states the theorem
condensation_with_mixed(pure and mixed ground states, liminf occupation at least $c_0$) with the proof left assorry, as a challenge statement.
Possibly relevant literature (found by the agent, not checked)
- J. Lauwers, A. Verbeure, V. A. Zagrebnov, Proof of Bose-Einstein Condensation for Interacting Gases with a One-Particle Spectral Gap (arXiv:math-ph/0205037; J. Phys. A 36 (2003) L169–L174). This paper proves homogeneous condensation for interacting continuous Bose gases under a one-particle excitation gap, an assumption absent in the torus thermodynamic limit treated here.
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
- Emergent Mind: family 267
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Ground-state-condensation-in-the-dilute-hard-sphere-gas-September-24-2026,
author = {{OpenAI}},
title = {{Ground-state condensation in the dilute hard-sphere gas}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Ground-state-condensation-in-the-dilute-hard-sphere-gas-September-24-2026/paper.pdf}{OAI:Ground-state-condensation-in-the-dilute-hard-sphere-gas-September-24-2026}},
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:
- Bose–Einstein condensation at positive temperature in the dilute hard-sphere gas · October 5, 2026
- 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