Long-time convergence to the fluctuating Boltzmann equation for a two-dimensional hard-disk gas at equilibrium, and extension to time windows growing with the mean free path

For a gas of hard spheres at equilibrium in the Boltzmann-Grad (low-density) limit, Bodineau, Gallagher, Saint-Raymond and Simonella proved that the fluctuation field of the empirical density converges, as a process and for arbitrarily long fixed times, to the Gaussian solution of the fluctuating Boltzmann equation — but only in spatial dimension $d\ge 3$, and with quantitative estimates that hold on time windows growing no faster than $O(\log\log|\log\varepsilon|)$. The open problem is to extend this process-level, long-time derivation to the two-dimensional hard-disk gas, where the long-time covariance convergence is already known but the Gaussian (Wick-rule) structure and full process convergence are not, and to push the quantitative convergence to time windows $\Theta(\varepsilon)\to\infty$ growing strictly faster than the current sub-logarithmic reach.

Background

A hard-sphere gas is a system of $N$ identical particles of diameter $\varepsilon$ moving freely on the periodic torus $\mathbb{T}^d$ with specular reflection at contact — the deterministic Newtonian dynamics of a dilute gas. In the Boltzmann-Grad limit, the particle diameter shrinks while the density grows so that the mean free path stays of order one ($\mu_\varepsilon=\varepsilon^{-(d-1)}$ scaling). At equilibrium the gas is distributed according to a grand-canonical Gibbs measure with Maxwellian velocity distribution $M(v)$. The empirical density $\pi_t^\varepsilon$ then concentrates on $M$, and the object of interest is the rescaled fluctuation field $\zeta_t^{\varepsilon,\mathrm{eq}}=\sqrt{\mu_\varepsilon}\,(\pi_t^\varepsilon-\mathbb{E}_\varepsilon^{\mathrm{eq}}\pi_t^\varepsilon)$, describing deviations of size $1/\sqrt{\mu_\varepsilon}$ from equilibrium.

Physicists predicted, by a fluctuation-dissipation argument, that in this limit the fluctuation field should obey the fluctuating Boltzmann equation: the linear SPDE

$d\zeta_t=\mathcal{L}\zeta_t\,dt+d\eta_t,$

where $\mathcal{L}$ is the Boltzmann collision operator linearized around $M$ and $d\eta_t$ is a stationary Gaussian noise, white in time and space but correlated in velocities, with covariance determined by the local collision kernel. Rigorously, van Beijeren, Lanford, Lebowitz and Spohn (1980) computed the equilibrium time-covariances in the low-density limit, and Spohn (1981) established the covariance convergence around non-equilibrium states and formulated the fluctuating-Boltzmann conjecture. The first process-level results (convergence in law of the full field, tightness in a Skorokhod space — the space of càdlàg, i.e. right-continuous with left limits, paths — of paths valued in a Sobolev space $H^{-k}$ of distributions with $k$ degrees of negative regularity, and Gaussianity) were obtained only for short kinetic times of order a fraction of the mean free time — the same obstruction as in Lanford's validity theorem for the nonlinear Boltzmann equation (his derivation of the Boltzmann equation from hard-sphere dynamics, likewise proved only on such a short time interval). Bodineau, Gallagher, Saint-Raymond and Simonella later proved the full fluctuation theory (process convergence and large deviations via a cumulant/cluster expansion over pseudo-trajectories — cumulants being the correlation-type statistics that vanish for independent variables, and pseudo-trajectories the piecewise-free-flight path carriages stitched at collisions that replace true hard-sphere trajectories) for such short times, in any dimension $d\ge 2$ and also away from equilibrium (Ann. of Math. 2023), and the long-time covariance convergence at equilibrium in two dimensions (Ann. PDE 2017, leading to the Stokes-Fourier equations, the coupled diffusion equations for temperature and mass velocity that govern the gas on long diffusive scales) and in $d\ge 3$ (Comm. Pure Appl. Math. 2023).

The source paper (Ann. Probab. 2024) removed the short-time restriction at the process level: at equilibrium in $d\ge 3$, all moments of the fluctuation field factorize according to the Wick pairing rule (the Gaussian moment rule: a moment of order $P$ equals the sum, over all pairings of the $P$ indices, of products of second moments) for arbitrarily long fixed times, so the field converges as a process to the fluctuating Boltzmann equation. The proof combines the cumulant expansion with a weak-convergence method exploiting the invariant measure: a conditioning discards atypical dynamics, and an iteration over separated time scales ($\delta$ for trajectory pullback, $\tau$ for intermediate sampling, $\Theta$ for observation) eliminates recollisions and superexponential clustering. Two limitations are explicit in the paper. First, by Remark 1.2, the geometric arguments that control pathological trajectories require a specific treatment in two space dimensions and are deliberately left out, so the theorem is stated only for $d\ge 3$. Second, by Remark 1.3 and the constraints of the main quantitative estimate (Proposition 2.1, requiring $\Theta^{3(P-1)}\tau\ll 1$ and $\Theta/(\tau\log|\log\varepsilon|)\ll 1$), the moment bounds remain valid only on time windows diverging no faster than $\Theta=O(\log\log|\log\varepsilon|)$; in the authors' survey this is contrasted with the physically relevant diffusive scales, which remain out of reach. Note that in $d=2$ only the covariance level is settled at long times: the Wick-rule/Gaussian structure of the limit, and convergence of the field as a process over long times, are open precisely in the two-dimensional case, which is the setting of the earlier Stokes-Fourier covariance theory.

Problem Statement

Extend the long-time equilibrium derivation of the fluctuating Boltzmann equation beyond its current reach, in the two components left open by Bodineau–Gallagher–Saint-Raymond–Simonella (Ann. Probab. 2024):

(a) Two-dimensional case. Consider a gas of hard disks of diameter $\varepsilon$ on the periodic torus $\mathbb{T}^2$, at grand-canonical equilibrium, in the Boltzmann-Grad limit $\mu_\varepsilon=\varepsilon^{-1}\to\infty$. Prove that the equilibrium fluctuation field $\zeta^{\varepsilon,\mathrm{eq}}_t$ converges in law, as a process on the Skorokhod space $D([0,\Theta],H^{-k})$ for every fixed $\Theta>0$, to the stationary Gaussian solution of the fluctuating Boltzmann equation $d\zeta_t=\mathcal{L}\zeta_t\,dt+d\eta_t$ (linearized collision operator around the Maxwellian, fluctuation-dissipation noise), i.e. extend Theorem 1.3 of the source paper from $d\ge 3$ to $d=2$; equivalently, prove that all moments of the field factorize according to the Wick pairing of covariances uniformly on $[0,\Theta]$, together with the tightness of the field. This requires a specific geometric treatment, in the plane, of the pathological trajectories, recollisions and overlaps that the $d\ge 3$ arguments exclude.

(b) Longer time windows. Establish quantitative moment/cumulant estimates for the convergence (in any dimension, including $d\ge 3$) that remain valid on observation windows $\Theta(\varepsilon)\to\infty$ growing strictly faster than the current reach $\Theta=O(\log\log|\log\varepsilon|)$ implied by the constraints $\Theta^{3(P-1)}\tau\ll 1$, $\Theta/(\tau\log|\log\varepsilon|)\ll 1$ of Proposition 2.1 of the source paper.

Both components serve the single objective of the source paper — the long-time, process-level derivation of the fluctuating Boltzmann equation at equilibrium. The further directions named in the source (long-time convergence from non-equilibrium initial data, which the authors' survey identifies as requiring a new stability notion for the nonlinear Boltzmann equation, and simplification of the conditioning/cumulant machinery) are recorded as adjacent open directions.

References

1. Thierry Bodineau, Isabelle Gallagher, Laure Saint-Raymond, Sergio Simonella — Long-time derivation at equilibrium of the fluctuating Boltzmann equation, The Annals of Probability 52(1), 2024. DOI 10.1214/23-aop1656, https://doi.org/10.1214/23-aop1656 (preprint arXiv:2201.04514, https://arxiv.org/abs/2201.04514)
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(3), 2023. DOI 10.4007/annals.2023.198.3.3, https://doi.org/10.4007/annals.2023.198.3.3
3. Thierry Bodineau, Isabelle Gallagher, Laure Saint-Raymond, Sergio Simonella — Long-time correlations for a hard-sphere gas at equilibrium, Communications on Pure and Applied Mathematics 76(12), 3852–3911, 2023. DOI 10.1002/cpa.22120, https://doi.org/10.1002/cpa.22120
4. Thierry Bodineau, Isabelle Gallagher, Laure Saint-Raymond — From Hard Sphere Dynamics to the Stokes–Fourier Equations: An Analysis of the Boltzmann–Grad Limit, Annals of PDE 3, Article 2, 2017. DOI 10.1007/s40818-016-0018-0, https://doi.org/10.1007/s40818-016-0018-0
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