The rigorous statistical theory of the deterministic hard-sphere gas in the Boltzmann–Grad limit, developed by Bodineau, Gallagher, Saint-Raymond and Simonella, derives for short times both a central limit theorem for fluctuations around the Boltzmann equation (the fluctuating Boltzmann equation) and a large-deviation characterization of atypical trajectories. The complete proofs (Annals of Mathematics, 2023) still hold only on a time interval that is a fraction of Lanford's validity time, only for hard spheres, and the large-deviation result is restricted to regular paths because global convexity of the rate function is lacking. Later work resolved the equilibrium case globally in time and extended Lanford's law of large numbers to arbitrary times (Deng–Hani–Ma), but the derivation of the fluctuation theory and large-deviation asymptotics for long times away from thermal equilibrium, the full path-space large-deviation principle, and the extension to smooth short-range interactions all remain open.

Background

A classical gas of NN identical hard spheres of diameter ε\varepsilon on a torus, in the Boltzmann–Grad scaling Nεd−1=1N\varepsilon^{d-1}=1 (so that the mean free path is of order one), is the canonical deterministic microscopic model behind kinetic theory. Lanford's theorem (1975) states that for such a gas, started from approximately independent (chaotic) initial data, the empirical phase-space measure converges as ε→0\varepsilon\to 0 to the solution of Boltzmann's equation ∂tf+v⋅∇xf=C(f,f)\partial_t f+v\cdot\nabla_x f=C(f,f), where CC is Boltzmann's quadratic collision operator — but only on a short time interval, a restriction tied to the tree-expansion proof via the BBGKY hierarchy (the hierarchy of equations for the rescaled nn-particle correlation functions). This law of large numbers discards all information carried by correlations, in particular the fluctuations of typical trajectories and the probabilities of atypical ones. Bodineau, Gallagher, Saint-Raymond and Simonella, first in a 2020 announcement and then with complete proofs in the Annals of Mathematics (2023), recovered part of this lost information for short times: (1) a central limit theorem — the fluctuation field ζtε=με(πtε−Eεπtε)\zeta^\varepsilon_t=\sqrt{\mu_\varepsilon}(\pi^\varepsilon_t-\mathbb{E}_\varepsilon\pi^\varepsilon_t), with με\mu_\varepsilon the average number of particles, converges to a Gaussian process governed by the fluctuating Boltzmann equation dζt=Ltζt dt+dηtd\zeta_t=\mathcal{L}_t\zeta_t\,dt+d\eta_t, where Lt\mathcal{L}_t is Boltzmann's operator linearized around the solution ftf_t and ηt\eta_t is a transport-collision noise; and (2) a large-deviation statement — probabilities that the empirical measure stays close to a profile φ≠f\varphi\neq f over a time interval are exponentially small in με\mu_\varepsilon, with a rate functional obtained by Legendre transform of a cumulant generating function (cumulants are the connected parts of the correlation functions, extracted by cluster-type expansions) and characterized by a Hamilton–Jacobi equation. The method controls dynamical cumulants through uniform a priori bounds and pseudo-trajectory (backward collision-tree) combinatorics. Three restrictions survive the complete 2023 proofs and are explicitly listed by the authors as open: the time of validity T⋆T^\star is only a fraction of Lanford's already-short time; the analysis is specific to hard spheres, the authors deeming the extension to smooth, compactly supported short-range interaction potentials plausible but considerably more involved; and, because the rate functional lacks global convexity, the large-deviation lower bound is proved only for trajectories in regularity classes R\mathcal{R} and R^\hat{\mathcal{R}}, so no full large-deviation principle over path space is available. Subsequent work has resolved neighboring cases: at thermal equilibrium the fluctuating Boltzmann equation holds globally in time and fluctuating hydrodynamics is derived in the diffusive regime; and, at the level of the law of large numbers only, Deng, Hani and Ma (2024) extended Lanford's convergence to arbitrarily long times via a long-time cumulant ansatz, which also completes the derivation of fluid equations (Hilbert's sixth problem for rarefied gases). At the level of the limiting variational objects, a global-in-time solution of the associated functional Hamilton–Jacobi equation has been constructed (Qi, 2024) and the large-time landscape of the Boltzmann rate function, including dynamical phase transitions, has been analyzed (Basile–Benedetto–Bertini–Heydecker, 2025). None of these, however, derives the microscopic fluctuation or large-deviation theory itself beyond short times away from equilibrium, removes the regular-path restriction, or handles non-hard-sphere interactions.

Problem Statement

Establish a rigorous Boltzmann–Grad statistical theory of the deterministic dilute gas that removes the three restrictions left by Bodineau–Gallagher–Saint-Raymond–Simonella's short-time theory, with the common objective of giving a complete (in time, in path space, and in interaction class) characterization of fluctuations and atypical behavior of the hard-sphere (or short-range) gas around the Boltzmann equation. Concretely, for NN particles of diameter ε\varepsilon in the Boltzmann–Grad scaling Nεd−1=1N\varepsilon^{d-1}=1 with chaotic initial data: (1) prove that the central limit theorem for the fluctuation field, governed by the fluctuating Boltzmann equation, and the large-deviation asymptotics of the empirical measure hold beyond the short Lanford window — for arbitrarily long times away from thermal equilibrium, that is, on any time interval on which the Boltzmann solution ftf_t exists (with the linearized dynamics Lt\mathcal{L}_t around ftf_t, which defines the limiting fluctuation equation, well-posed on it); (2) prove a full dynamical large-deviation principle over trajectory space, removing the restriction to the regular-path classes R\mathcal{R} and R^\hat{\mathcal{R}} that the current lower bound requires due to the lack of global convexity of the rate functional; and (3) extend the fluctuation theory and the large-deviation characterization from hard spheres to gases interacting through smooth, compactly supported short-range potentials. The three parts are jointly required and must hold simultaneously on the common domain of arbitrarily long times, trajectories outside R\mathcal{R} and R^\hat{\mathcal{R}}, and smooth compactly supported short-range potentials; resolving one part alone leaves the other restrictions in place. Each part should be understood relative to the same microscopic model, initial-data conventions, and rate functional (equivalently, the functional Hamilton–Jacobi equation) of the 2020/2023 theory; results that hold only at equilibrium or only at the law-of-large-numbers level are strictly weaker than what is asked.

References

  1. Thierry Bodineau, Isabelle Gallagher, Laure Saint-Raymond, Sergio Simonella, "Fluctuation theory in the Boltzmann–Grad limit", Journal of Statistical Physics 180 (2020) 873–895, DOI 10.1007/s10955-020-02549-5, arXiv:2004.00311, https://arxiv.org/abs/2004.00311
  2. Thierry Bodineau, Isabelle Gallagher, Laure Saint-Raymond, Sergio Simonella, "Statistical dynamics of a hard sphere gas: fluctuating Boltzmann equation and large deviations", Annals of Mathematics 198 (2023), DOI 10.4007/annals.2023.198.3.3, arXiv:2008.10403, https://arxiv.org/abs/2008.10403
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