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The Bogoliubov quantum-depletion law for the dilute Bose gas in the thermodynamic limit

Bogoliubov's theory (1947), carried to the hard-sphere gas by Lee, Huang and Yang (1957), predicts that at zero temperature a dilute three-dimensional Bose gas with scattering length aa and density ρ\rho has a fraction 83πρa3\frac{8}{3\sqrt\pi}\sqrt{\rho a^3} of its particles outside the condensate, distributed in momentum by an explicit Bogoliubov profile on the scale 8πρa\sqrt{8\pi\rho a}. The energy to Lee-Huang-Yang order is now proved, but energy bounds do not control occupation of modes whose energy vanishes with the volume, and the depletion had been derived only in Gross-Pitaevskii-type scalings or for approximate equations. In the thermodynamic limit at fixed density, followed by ρa3→0\rho a^3\to0, is the ground-state depletion 83πρa3+o(ρa3)\frac{8}{3\sqrt\pi}\sqrt{\rho a^3}+o(\sqrt{\rho a^3})?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Quantum many-body theory; dilute Bose gases
Posed by
Prediction of N. N. Bogoliubov (1947) and T. D. Lee, K. Huang and C. N. Yang (1957)
Year posed
1957
Years open
69y
Solved
2026-10-05
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
35 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

For hard spheres (October 5): for every bounded continuous ff and ε>0\varepsilon>0, at all sufficiently small fixed ρ\rho, every thermodynamic accumulation value of 1Nρa3∑k≠0f(k/8πρa) n(k)\frac{1}{N\sqrt{\rho a^3}}\sum_{k\ne0}f(k/\sqrt{8\pi\rho a})\,n(k) is within ε\varepsilon of ∫f dνBog\int f\,d\nu_{\mathrm{Bog}}, uniformly over ground states; taking f=1f=1 gives depletion 83πρa3+o(ρa3)\frac{8}{3\sqrt\pi}\sqrt{\rho a^3}+o(\sqrt{\rho a^3}). A companion proves the same depletion asymptotic for fixed bounded, nonnegative, radial, finite-range potentials with positive scattering length. Not covered: positive temperature, the next-order correction, or unbounded non-hard-core potentials.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family has five manuscripts (September 24 to October 5, 2026). The positive-temperature paper says it builds on, and reproduces the needed arguments of, two earlier manuscripts in the same family, so later results rest on earlier model output.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Corollary 1.2 of the hard-sphere depletion manuscript and the main theorem of the bounded-potential companion (both October 5) were read against the Bogoliubov and Lee-Huang-Yang prediction. Not refereed; no Lean formalization for either. The order of limits is thermodynamic first at fixed density, then dilute; the result is uniform over all ground states and does not assume a unique thermodynamic limit. The argument uses the published Lee-Huang-Yang energy upper bound for hard spheres (Basti, Brooks, Cenatiempo, Olgiati, Schlein) and operator estimates from Fournais, Junge, Girardot, Morin, Olivieri and Triay.

Sources

Changelog1 change

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