Result 267, Mathematical physics

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.

Lean formalization Proof

The bigger picture

Why it matters

A Bose gas can place a macroscopic share of its particles in one quantum state, even when particles repel each other. These manuscripts claim positive-temperature condensation and quantify the leading fraction left outside that shared state at zero temperature.

What changes?

For three-dimensional hard spheres, particles forbidden from approaching closer than a fixed exclusion distance, the manuscript reports condensation at every sufficiently small fixed density at some strictly positive temperature independent of volume. It treats the exact canonical Gibbs state, the thermal equilibrium state with fixed particle number. As volume and particle number grow at fixed density, a positive fraction occupies the constant orbital, a spatially uniform quantum state.

What does that help mathematicians do?

At zero temperature, the reported fraction outside that orbital is 8/(3 sqrt(pi)) times sqrt(rho a^3), plus a smaller-order remainder. Here rho is density; a is the hard-sphere exclusion distance or interaction's scattering length. The claim covers hard spheres and fixed bounded, nonnegative, radial, finite-range potentials with positive scattering length. Infinite volume comes before vanishing density. This identifies the precise leading size of interaction-driven depletion, rather than merely establishing that a condensate survives.

Are there practical applications?

The immediate value is foundational: the claims give precise mathematical support for Bogoliubov's description of depletion in these interacting gases. The asymptotic is uniform across pure and mixed ground states, even without a unique thermodynamic limit for the depletion fraction. For hard spheres, the claimed full scaled momentum distribution additionally describes how depleted particles occupy different momenta, going beyond their total number.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

5 manuscripts

Bose–Einstein condensation at positive temperature in the dilute hard-sphere gas

October 5, 2026 56 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Quantum Depletion and Momentum Distribution in the Dilute Hard-Sphere Bose Gas

October 5, 2026 79 pages

We prove the full scaled Bogoliubov momentum distribution for the depleted particles in the three-dimensional hard-sphere Bose gas at zero temperature. The thermodynamic limit is taken at each fixed density before the dilute limit, uniformly over all pure and mixed ground states. All thermodynamic accumulation values have the same dilute asymptotic: the limiting nonzero-momentum occupation measure has total mass 8/(3π)8/(3\sqrt\pi), giving the depletion fraction 83πρa3+o(ρa3)\frac{8}{3\sqrt\pi}\sqrt{\rho a^3}+o(\sqrt{\rho a^3}) for density ρ and hard-sphere exclusion distance a.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Quantum-Depletion-and-Momentum-Distribution-in-the-Dilute-Hard-Sphere-Bose-Gas-October-5-2026,
  author = {{OpenAI}},
  title = {{Quantum Depletion and Momentum Distribution in the Dilute Hard-Sphere Bose Gas}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Quantum-Depletion-and-Momentum-Distribution-in-the-Dilute-Hard-Sphere-Bose-Gas-October-5-2026/Quantum-Depletion-in-the-Dilute-Hard-Sphere-Bose-Gas.pdf}{OAI:Quantum-Depletion-and-Momentum-Distribution-in-the-Dilute-Hard-Sphere-Bose-Gas-October-5-2026}},
  year = {2026}
}

Quantum Depletion for Fixed Bounded Repulsive Potentials

October 5, 2026 49 pages

We prove the Bogoliubov quantum-depletion asymptotic for the ground state of a three-dimensional Bose gas with a fixed bounded, nonnegative, radial interaction of finite range and positive scattering length a. The thermodynamic limit is taken at each fixed density ρ before the dilute limit. Every thermodynamic accumulation value of the fraction outside the constant mode is 83πρa3+o(ρa3)\frac{8}{3\sqrt\pi}\sqrt{\rho a^3}+o(\sqrt{\rho a^3}) as ρ↓0\rho\downarrow0. The assertion is uniform over ground-state density matrices and does not require the occupation to have a unique thermodynamic limit.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Quantum-Depletion-for-Fixed-Bounded-Repulsive-Potentials-October-5-2026,
  author = {{OpenAI}},
  title = {{Quantum Depletion for Fixed Bounded Repulsive Potentials}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Quantum-Depletion-for-Fixed-Bounded-Repulsive-Potentials-October-5-2026/fixed-repulsion-quantum-depletion.pdf}{OAI:Quantum-Depletion-for-Fixed-Bounded-Repulsive-Potentials-October-5-2026}},
  year = {2026}
}

A density-uniform condensate bound for dilute Bose gases

September 27, 2026 42 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Ground-state condensation in the dilute hard-sphere gas

September 24, 2026 53 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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's note on what the formalization covers, from lean/docs/267.md.

Positive-temperature Bose–Einstein condensation and exact quantum depletion

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalization proves ground-state Bose–Einstein condensation in the dilute hard-sphere gas. There are absolute constants ε0,c0>0\varepsilon_0,c_0>0 such that, whenever the density ρ\rho and hard-sphere radius aa satisfy ρa3<ε0\rho a^3<\varepsilon_0, the condensate occupation fraction has limit inferior at least c0c_0 along every thermodynamic sequence with N/L3→ρN/L^3\to\rho.

The bound covers every pure ground state, every mixed state supported on the ground space, and the normalized ground-space projection. It is uniform in the gas parameter within this range and concerns ground states, without a positive-temperature assertion.

Comparator links

Result Comparator statement
Uniform ground-state condensation in the dilute hard-sphere gas HardSphere.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.