Open extensions of Hilbert's sixth-problem program: long-time Newton-to-Boltzmann-to-fluid derivation in dimensions $d\geq 4$, longer time windows, and interactions beyond hard spheres

In a 2025 preprint, Deng, Hani and Ma completed Hilbert's sixth-program route from Newtonian mechanics to fluid mechanics for a specific regime: they derived the Boltzmann equation, valid on long time intervals, from a grand-canonical ensemble of elastic hard spheres of diameter $\varepsilon$ on the torus $\mathbb{T}^d$ with $d\in\{2,3\}$ under the Boltzmann-Grad scaling $N\varepsilon^{d-1}=\alpha$, and then composed this with known hydrodynamic limits to obtain the incompressible Navier-Stokes-Fourier and compressible Euler equations. Their Theorem 1, however, is restricted to $d\in\{2,3\}$, to parameters satisfying $\max(1,\alpha)\cdot\max(1,A)\cdot\max(1,t_{\mathrm{fin}})\ll(\log|\log\varepsilon|)^{1/2}$, to initial data carrying one extra spatial derivative, and to hard-sphere collisions. The authors explicitly flag these restrictions as open (for $d\geq 4$ they state they believe the result should still be true), and a 2026 Bourbaki seminar exposition by Bodineau, Gallagher, Saint-Raymond and Simonella confirms that the low-density limit for non-compactly-supported potentials remains fully open in the field. The task is to lift these regime restrictions of the same Newton-to-Boltzmann-to-fluid chain: prove the analogue of Theorem 1 on $\mathbb{T}^d$ for $d\geq 4$ with explicit estimates, extend the admissible time/scaling window beyond the $(\log|\log\varepsilon|)^{1/2}$ constraint, and extend the full derivation to collision laws beyond elastic hard spheres, most centrally long-range potentials.

Background

Hilbert's sixth problem (from his 1900 address) asks for an axiomatic treatment of the physical sciences; in his elaboration he singled out one concrete question, to develop rigorously the limiting processes leading, in his words, from the atomistic view to the laws of motion of continua. The accepted mathematical rendering of this program is a two-step chain. Step one: derive Boltzmann's kinetic equation from Newton's laws for a gas of $N$ particles of diameter $\varepsilon$ undergoing elastic hard-sphere collisions, in the Boltzmann-Grad (low-density) limit $N\to\infty$, $\varepsilon\to 0$ with $N\varepsilon^{d-1}=\alpha$ held fixed ($\alpha$ is the collision rate; the relation expresses that each particle meets of order one collision per unit time, i.e. the dilute-gas regime). The one-particle density should then converge to the solution $n(t,x,v)$ of the Boltzmann equation $(\partial_t+v\cdot\nabla_x)n=\alpha\,\mathcal{Q}(n,n)$, where $\mathcal{Q}$ is the bilinear hard-sphere collision operator. Lanford proved in 1975 that this convergence holds, but only for a time interval of a small fraction of the mean free time (the average time between successive collisions of a particle), because correlations generated by recollisions — particles meeting again after a collision — can no longer be controlled beyond that. Step two is the hydrodynamic limit: as $\alpha\to\infty$ (mean free path tending to zero), suitable solutions of the Boltzmann equation converge to solutions of fluid equations, such as the incompressible Navier-Stokes-Fourier system or the compressible Euler equations. The two steps interlock badly: known hydrodynamic limits require the Boltzmann equation to be valid on time scales of order $\alpha$ or longer, far beyond Lanford's short-time window. Overcoming exactly this obstacle was the content of the recent breakthrough by Deng, Hani and Ma. Their 2024 work first derived the Boltzmann equation from hard-sphere dynamics in $\mathbb{R}^3$ for the full life span of any regular Boltzmann solution, using a cluster (molecular) expansion of the correlations together with a combinatorial cutting algorithm that decomposes interaction graphs into elementary pieces; the size of each piece is quantified by an excess estimate, roughly a gain of a power $\varepsilon^{d-1-}$ per controlled recollision cluster, and long free-flight segments (long bonds) between collisions provide the key improvement. Their 2025 paper extended the method to 2D and 3D tori (the periodic domain $\mathbb{T}^d=[0,1]^d$ with opposite faces identified) with explicit quantitative estimates and combined it with hydrodynamic limit results (notably the convergence of smooth solutions from Boltzmann to Navier-Stokes due to Gallagher and Tristani) to derive the incompressible Navier-Stokes-Fourier and compressible Euler equations from the particle system, thereby completing Hilbert's program in this setting. The main theorem (Theorem 1) states: fix $d\in\{2,3\}$ and a velocity weight $\beta>0$; if the parameters $(\alpha,A,t_{\mathrm{fin}})$ satisfy $\max(1,\alpha)\cdot\max(1,A)\cdot\max(1,t_{\mathrm{fin}})\ll(\log|\log\varepsilon|)^{1/2}$, and the (Gaussian-weighted, bounded) Boltzmann solution together with its initial datum satisfies a regularity bound $A$ in which the initial profile $n_0$ carries one extra spatial derivative, then for the grand-canonical initial ensemble (an ensemble in which the particle number is itself random, distributed by an equilibrium-type law) the $s$-particle correlation functions $f_s$ satisfy, uniformly for $t\in[0,t_{\mathrm{fin}}]$ and $s\leq|\log\varepsilon|$, the $L^1$ estimate $\|f_s-\prod_{j=1}^{s}n(t,z_j)\|_{L^1}\leq\varepsilon^{\theta}$ for some $\theta>0$. The authors note three explicit limitations (Remark 1.5 and Remark 1.9). First, the restriction $d\in\{2,3\}$: they believe the result should still hold for $d\geq 4$, but a nondegeneracy property of the collision geometry used to reach the near-optimal excess $\varepsilon^{d-1-}$ fails in higher dimensions, and the combinatorial algorithm must be refined to produce enough good components. Second, the parameter constraint: improving the $(\log|\log\varepsilon|)^{1/2}$ bound, which directly limits the range of hydrodynamic scalings in the fluid limits, seems hard with their methods. Third, the one-derivative loss on $n_0$ can be removed only at the price of freezing the parameters $(\alpha,A,t_{\mathrm{fin}})$ independently of $\varepsilon$. In addition, the whole construction is specific to elastic hard-sphere collisions; the extension to other interaction laws, in particular non-compactly-supported (long-range) potentials for which the angular collision cross-section is non-integrable and the gain and loss parts of the collision operator only cancel in combination, is regarded by the kinetic-theory community as a central open problem in its own right.

Problem Statement

Extend the rigorous Newton-to-Boltzmann-to-fluid derivation completed by Deng, Hani and Ma beyond the regime in which they proved it. Their Theorem 1 derives the Boltzmann equation, on long time intervals with explicit error estimates, from a grand-canonical ensemble of $N$ elastic hard spheres of diameter $\varepsilon$ on the torus $\mathbb{T}^d$ under the Boltzmann-Grad scaling $N\varepsilon^{d-1}=\alpha$, for $d\in\{2,3\}$, under the parameter constraint $\max(1,\alpha)\cdot\max(1,A)\cdot\max(1,t_{\mathrm{fin}})\ll(\log|\log\varepsilon|)^{1/2},$ and assuming one extra spatial derivative on the initial Boltzmann profile, concluding an $L^1$ convergence $\|f_s-\prod_j n(t,z_j)\|\leq\varepsilon^{\theta}$ of the $s$-particle correlation functions for $s\leq|\log\varepsilon|$; composition with hydrodynamic limits then yields the incompressible Navier-Stokes-Fourier and compressible Euler equations. The common scientific objective of the open problem is to lift the regime restrictions of this single theorem chain. Concretely, the following independent sub-problems each ask, at the level of rigor and generality of the source theorem, for a quantitative theorem comparing particle dynamics to the Boltzmann solution on hydrodynamically relevant time scales:

(1) Higher dimensions: the analogue of Theorem 1 on $\mathbb{T}^d$ for $d\geq 4$ (the authors state they believe it should hold), with explicit control of the correlation/excess estimates, together with the induced hydrodynamic consequences.

(2) Longer times and wider scaling window: the same convergence with the constraint $\max(1,\alpha)\cdot\max(1,A)\cdot\max(1,t_{\mathrm{fin}})\ll(\log|\log\varepsilon|)^{1/2}$ replaced by a strictly larger admissible range as a function of $\varepsilon$, which equivalently enlarges the window of hydrodynamic scalings $\delta$ in the Euler and Navier-Stokes-Fourier limits.

(3) Beyond hard spheres: the full Newton-to-Boltzmann-to-fluid derivation for interaction laws other than elastic hard spheres, most centrally non-compactly-supported (long-range) potentials, for which even the low-density limit to the nonlinear Boltzmann equation is open. In particular, establishing the derivation for the non-compactly-supported (long-range) case settles the non-compact-range requirement by itself.

A secondary refinement explicitly acknowledged by the authors: remove the one-derivative loss on the initial datum while retaining the $\varepsilon$-dependent parameter growth. A rigorous disproof of an extension (for example, a construction of admissible data in some $d\geq 4$ for which long-time Boltzmann validity fails) is an admissible alternative answer to the corresponding part. The problem is fully resolved when the dimension, time-window, and interaction-law restrictions are all lifted.

References

1. Yu Deng, Zaher Hani, Xiao Ma, Hilbert's sixth problem: derivation of fluid equations via Boltzmann's kinetic theory, preprint (2025), arXiv:2503.01800, DOI 10.48550/arXiv.2503.01800, https://arxiv.org/abs/2503.01800
2. Yu Deng, Zaher Hani, Xiao Ma, Long time derivation of the Boltzmann equation from hard sphere dynamics, preprint (2024, v3), arXiv:2408.07818, https://arxiv.org/abs/2408.07818
3. Thierry Bodineau, Isabelle Gallagher, Laure Saint-Raymond, Sergio Simonella, Derivation of the Boltzmann equation from hard-sphere dynamics (after Y. Deng, Z. Hani, and X. Ma), Bourbaki Seminar exposé, preprint (2026, arXiv submission 4 February 2026), arXiv:2602.04407, https://arxiv.org/abs/2602.04407
4. Shan Gao, Comment on 'Hilbert's Sixth Problem: Derivation of Fluid Equations via Boltzmann's Kinetic Theory' by Deng, Hani, and Ma, preprint (2025), arXiv:2504.06297, DOI 10.48550/arXiv.2504.06297, https://arxiv.org/abs/2504.06297
5. I. Gallagher, I. Tristani, On the convergence of smooth solutions from Boltzmann to Navier-Stokes, Ann. Henri Lebesgue 3 (2020), 561-614, DOI 10.5802/ahl.40, https://doi.org/10.5802/ahl.40
6. O. E. Lanford, Time evolution of large classical systems, in Dynamical Systems, Theory and Applications, Lecture Notes in Physics vol. 38, Springer (1975), 1-111, DOI 10.1007/3-540-07171-7_1, https://doi.org/10.1007/3-540-07171-7_1
7. David Hilbert, Mathematical Problems, Bulletin of the American Mathematical Society 8 (10) (1902), 437-479 (lecture delivered 1900), DOI 10.1090/S0002-9904-1902-00923-3, https://doi.org/10.1090/S0002-9904-1902-00923-3, zbMATH JFM 32.0084.02
8. Mario Pulvirenti, Sergio Simonella, Some considerations on the fluid-dynamical limit of particle systems, Mathematics and Mechanics of Complex Systems 14 (2) (2026), 143-154, DOI 10.2140/memocs.2026.14.143, https://doi.org/10.2140/memocs.2026.14.143
9. Corentin Le Bihan, Long time validity of the linearized Boltzmann uncut-off and the linearized Landau equations from the Newton Law, preprint (2024), arXiv:2408.03597, https://arxiv.org/abs/2408.03597