Gaussian fluctuation limits for hard-sphere gases
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For the deterministic grand-canonical three-dimensional hard-sphere gas with smooth spatially summable Gaussian-bounded initial density and gradient, do exactly centered empirical fluctuations converge to a Gaussian linear fluctuating Boltzmann field? The target is convergence for each finite collection of times and tests on every prescribed finite interval where the classical Boltzmann solution has uniform Gaussian velocity decay.
Why it matters and what counts as progress
Why it matters: It asks whether nonequilibrium kinetic fluctuations remain Gaussian over the regular Boltzmann lifespan, beyond short-time or equilibrium limits.
What counts as progress: Prove Gaussian fluctuation convergence beyond short times on the stated regular kinetic lifespan, with exact centering and the prescribed finite-time tests.
How the problem developed
- 24 Aug 2020 · Partial result [1]
Thierry Bodineau, Isabelle Gallagher, Laure Saint-Raymond, and Sergio Simonella Derived Gaussian fluctuations of the empirical measure around a non-equilibrium Boltzmann solution in the Boltzmann–Grad limit for short times.
- 12 Jan 2022 · Partial result [2]
Thierry Bodineau, Isabelle Gallagher, Laure Saint-Raymond, and Sergio Simonella Derived the equilibrium Gaussian fluctuation process for arbitrarily long times in the low-density limit.
- 23 Sept 2026 · Reported claim [3]
OpenAI The manuscript reports a finite-dimensional non-equilibrium Gaussian fluctuation limit for a deterministic grand-canonical hard-sphere gas on any finite interval where the classical Boltzmann solution has uniform Gaussian velocity decay, with exact microscopic centering.
AI contribution
Hard-sphere fluctuations on the regular Boltzmann lifespan
For the stated deterministic grand-canonical three-dimensional hard-sphere gas with smooth spatially summable Gaussian-bounded initial density and gradient, exactly centered empirical fluctuations converge at every fixed finite set of times and tests to a Gaussian linear fluctuating Boltzmann field on each prescribed finite interval where the classical Boltzmann solution has uniform Gaussian velocity decay.
Sources
- Bodineau, Gallagher, Saint-Raymond, and Simonella, Statistical dynamics of a hard sphere gas: fluctuating Boltzmann equation and large deviations
- Bodineau, Gallagher, Saint-Raymond, and Simonella, Long-time derivation at equilibrium of the fluctuating Boltzmann equation
- OpenAI, Hard-sphere fluctuations on the regular Boltzmann lifespan
- Hard-sphere fluctuations on the regular Boltzmann lifespan
- OpenAI, Hard-sphere fluctuations: statement and assumptions
- Deng, Hani and Ma, Prior kinetic-law result
- OpenAI mathematics release overview
Website timeline
Publication dates and editorial changes.
OpenAI reports hard-sphere fluctuations on a regular Boltzmann lifespan