Browse › Mathematical physics › 267 Positive-temperature Bose–Einstein condensation and exact quantum depletion
A density-uniform condensate bound for dilute Bose gases
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Abstract
For each fixed bounded measurable, nonnegative, radial interaction of finite range in three dimensions that is not zero almost everywhere, we prove a positive lower bound on the constant-orbital condensate fraction that is uniform over all sufficiently small densities and all temperatures between zero and the square of the density. The interaction and density remain fixed in the thermodynamic limit.
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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 gives a lower bound on the condensate density for continuous Bose gases with superstable interactions under a one-particle gap assumption; this paper's main theorem is stated for fixed bounded repulsive potentials on the torus.
- W. König, The free energy of the interacting Bose gas: a variational description with loops and interlacements (arXiv:2604.14318, April 2026). This paper treats the interacting Bose gas at positive temperature in the thermodynamic limit through a Brownian-loop (Feynman–Kac) representation, the same kind of path measure this paper's comparison argument is built on.
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:A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026,
author = {{OpenAI}},
title = {{A density-uniform condensate bound for dilute Bose gases}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/paper.pdf}{OAI:A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026}},
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
- Ground-state condensation in the dilute hard-sphere gas · September 24, 2026