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Positive-temperature Bose–Einstein condensation and exact quantum depletion

Proves condensation in every ground state of the dilute hard-sphere Bose gas; claims thermal and depletion laws.

5 papers September 24, 2026 Explained by AI

Bose–Einstein condensation means a positive share of all particles sits in one single state. For interacting gases at fixed density, no one had proved it happens, even in the simplest model of hard balls. One paper shows a fixed positive share condenses in every ground state of a dilute hard-sphere gas, and Lean checks this. Four more papers claim condensation at small positive temperature and the exact share knocked out of the condensate.

First step
1925
Satyendra Nath Bose and Albert Einstein, condensation prediction for the ideal Bose gas
Significance
Major
Condensation in an interacting gas at fixed density is a famous open problem; a 2025 survey calls it a major open problem.
Trust
Formal proof listed
Field
Mathematical physics
5 papers

The explanation

Put N bosons, particles that may share a state, in a large box. Make them hard balls of diameter a: no two centres can come closer than a. The density ρ is particles per unit volume. The key number is the gas parameter ρa³, a rough measure of how crowded the gas is. Take ρa³ = 10⁻⁶, a very thin gas. Without interactions, at zero temperature, every particle would sit in the same flat, constant wave. Collisions knock some out of it. That missing share is called the depletion.

Bose–Einstein condensation means a single one-particle state holds a macroscopic part of the system. Penrose and Onsager gave the test: the occupation of one state must stay a fixed fraction of N. Lee, Huang and Yang predicted a depleted fraction of order √(ρa³). For our gas that is of order 0.001, so about 99.9% should stay in the condensate.

The energy of this gas has been worked out in detail. But those energy results do not settle single-state occupation, because the energy gap between states shrinks as the box grows. A tiny gap means many states have almost the same energy. Particles could spread over them without costing much. Solovej's 2025 survey calls fixed-density ground-state condensation a major open problem.

The fifth paper settles the ground-state case. For a small enough fixed gas parameter, the constant wave holds a positive fraction of the particles as the box grows. It holds for every ground state, and one fraction works for all small ρa³. It does not find the best fraction.

The other papers claim more. For hard spheres at zero temperature, the share outside the condensate is claimed to be (8/(3√π))√(ρa³), plus smaller terms. The factor 8/(3√π) is about 1.5045. At ρa³ = 10⁻⁶ the depletion is about 0.0015, or 0.15%. The same law is claimed for bounded, repulsive, short-range interactions. At positive temperature, one paper claims condensation persists up to a fixed temperature that does not depend on the box size. For bounded repulsive interactions, another claims a condensate bound for all temperatures from zero up to ρ².

How sure is this? Lean, a computer proof checker, proves the ground-state result. It covers every pure ground state and every mixture of them, with one constant for all small gas parameters. It does not cover positive temperature or the depletion law. Those rest on four long papers, and no specialist review of them has been published.

What it doesn't show. This does not show condensation at every temperature below the critical one, or in dense gases such as liquid helium.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Major if it holds. Ground-state condensation for hard spheres has a Lean proof; the positive-temperature and exact depletion claims rest on unreviewed papers.

Words used

Boson
A kind of particle that can share the same quantum state with others of its kind.
Bose–Einstein condensation
A fixed positive share of all particles sitting in one single quantum state.
Hard-sphere gas
A model where particles are hard balls of diameter a that cannot overlap and otherwise do not interact.
Gas parameter ρa³
Density times the cube of the ball size; small values mean a dilute gas.
Thermodynamic limit
Letting the number of particles and the box volume grow together at fixed density.
Depletion
The share of particles outside the condensate state.
Ground state
The lowest-energy state of the whole system, at zero temperature.

The original papers (5)

Lean notes

All quoted lines (11)
  1. Bose–Einstein condensation means that a single one-particle state carries a macroscopic part of a many-particle system.
    Ground-state condensation in the dilute hard-sphere gas Back to text
  2. For an interacting gas, the density-matrix criterion of Penrose and Onsager identifies condensation through a macroscopic one-particle occupation.
    Ground-state condensation in the dilute hard-sphere gas Back to text
  3. For hard spheres, Lee, Huang and Yang's pseudopotential calculation predicts a depleted fraction of order ρa3\sqrt{\rho a^3}.
    Ground-state condensation in the dilute hard-sphere gas Back to text
  4. These energy asymptotics do not determine single-orbital occupation as the one-particle spectral gap vanishes in large volume.
    Ground-state condensation in the dilute hard-sphere gas Back to text
  5. Solovej's 2025 survey identifies fixed-density ground-state condensation as a major open problem.
    Ground-state condensation in the dilute hard-sphere gas Back to text
  6. At every sufficiently small fixed gas parameter, the constant orbital contains a positive fraction of the particles in the thermodynamic limit.
    Ground-state condensation in the dilute hard-sphere gas Back to text
  7. We prove a positive fraction; we do not estimate the optimal depletion or show that the fraction tends to one as ρa3\rho a^3 tends to zero.
    Ground-state condensation in the dilute hard-sphere gas Back to text
  8. 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\rho and hard-sphere exclusion distance aa.
    Quantum Depletion and Momentum Distribution in the Dilute Hard-Sphere Bose Gas Back to text
  9. 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 aa.
    Quantum Depletion for Fixed Bounded Repulsive Potentials Back to text
  10. 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.
    Bose–Einstein condensation at positive temperature in the dilute hard-sphere gas Back to text
  11. 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.
    A density-uniform condensate bound for dilute Bose gases Back to text

Other sources

  1. openai/math, Lean scope note for family 267 The formalization proves ground-state Bose–Einstein condensation in the dilute hard-sphere gas.
  2. openai/math release notes (2026): model provenance and review status manuscripts and supporting proof artifacts produced by an internal OpenAI model

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