Long-time nonequilibrium fluctuations of the hard-sphere gas (finite-dimensional laws)
Beyond the Boltzmann equation, the fluctuations of the empirical measure of a dilute hard-sphere gas are expected to be Gaussian and governed by the linear fluctuating Boltzmann equation. Bodineau, Gallagher, Saint-Raymond and Simonella proved this for short times away from equilibrium and for long times at equilibrium. After deriving the Boltzmann equation on every regular kinetic interval, Deng, Hani and Ma identified long-time nonequilibrium fluctuations as a further application of their method. Do the fluctuations of a hard-sphere gas converge to the Gaussian solution of the linear fluctuating Boltzmann equation throughout every interval on which the Boltzmann solution is regular, without an equilibrium or small-data assumption?
- Result
- Proved(see note)
- Status
- Partial result
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Kinetic theory; fluctuation theory
- Posed by
- Yu Deng, Zaher Hani and Xiao Ma (arXiv 2408.07818, Section 1.4.3)
- Year posed
- 2024
- Years open
- 2y
- Solved
- 2026-09-23
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 24 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
For a deterministic grand-canonical hard-sphere gas in with smooth initial density having summable Gaussian bounds on it and its gradient, on every where the Boltzmann solution has uniform Gaussian velocity decay, the empirical measure centered at its exact microscopic expectation and scaled by has finite-dimensional distributions converging to those of the Gaussian solution of the linear fluctuating Boltzmann equation. It does not prove process-level convergence or tightness, does not treat periodic domains (the BGSS setting) or potentials, and assumes the regular kinetic solution exists.
What the AI did
The release README says the results were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. 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, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author. Companion of the stable-potential Boltzmann-Grad paper of the same date; it retains the Deng-Hani-Ma layered collision histories.
Verification
No independent mathematician has checked this yet. Checked here: the main theorem of the manuscript was read against the further application as the paper reports it from Deng-Hani-Ma Section 1.4.3; the Deng-Hani-Ma text itself was not read. The theorem is a finite-dimensional central limit theorem in whole space, not convergence of the fluctuation process (no path-space tightness), which is why the entry is partial. The proof was not refereed. No Lean formalization exists for this family.