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

OpenAI · October 5, 2026

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Abstract

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 $8/(3\sqrt\pi)$, giving the depletion fraction $\frac{8}{3\sqrt\pi}\sqrt{\rho a^3}+o(\sqrt{\rho a^3})$ for density ρ and hard-sphere exclusion distance a.

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Cite

@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}
}

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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.

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