The rigorous derivation of the Boltzmann equation from Newtonian particle dynamics was extended by Deng, Hani, and Ma (2024) from Lanford's short-time regime to arbitrarily long times — as long as the Boltzmann solution stays regular — for hard spheres on Rd\mathbb{R}^d. Their companion paper (2025) closed the torus case and derived the fluid equations, completing Hilbert's sixth problem for a rarefied hard-sphere gas. Two gaps deliberately left open by this program remain: (1) extending the long-time derivation from hard spheres to gases interacting through compactly supported (short-range) repulsive potentials in the sense of Gallagher–Saint-Raymond–Texier, where collisions are no longer instantaneous binary events; and (2) sharpening the quantitative output of the hard-sphere theorem — the validity time currently diverges only like log⁡∣log⁡ε∣\log|\log\varepsilon|, the correlation order is restricted to s≤∣log⁡ε∣s\le|\log\varepsilon|, and the convergence rate εθ\varepsilon^\theta is non-optimal. This problem asks for a resolution of both gaps, which together complete the long-time Newton-to-Boltzmann derivation in its expected physical generality and quantitative strength.

Background

The Boltzmann equation governs the evolution of the one-particle density f(t,x,v)f(t,x,v) of a rarefied gas and is derived from Newtonian mechanics under the assumption that colliding particles are statistically independent — Boltzmann's molecular chaos hypothesis. A rigorous justification requires passing to the Boltzmann–Grad (low-density) limit: for a gas of NN hard spheres of diameter ε\varepsilon with Nεd−1=1N\varepsilon^{d-1}=1 (the inverse Knudsen number, i.e. the ratio of macroscopic length scale to mean free path, held equal to one), the ss-particle correlation functions fsf_s should factorize asymptotically, fs(t,zs)≈∏j=1sf(t,zj)f_s(t,z^s)\approx\prod_{j=1}^s f(t,z_j), where zj=(xj,vj)z_j=(x_j,v_j). Lanford's theorem (1975) established this propagation of molecular chaos, and hence convergence to the Boltzmann solution, but only on a time interval of the order of a fraction of the mean free time — far too short for the hydrodynamic regime.

Deng, Hani, and Ma (arXiv:2408.07818) proved the convergence for arbitrarily long times: for every finite time interval on which the Boltzmann solution is sufficiently regular, the correlation functions satisfy, uniformly in t∈[0,tfin]t\in[0,t_{\mathrm{fin}}] and s≤∣log⁡ε∣s\le|\log\varepsilon|,

∥fs(t,zs)−∏j=1sf(t,zj)⋅1Ds(zs)∥L1(R2ds)≤εθ,\Big\|f_s(t,z^s)-\prod_{j=1}^s f(t,z_j)\cdot\mathbf{1}_{D_s}(z^s)\Big\|_{L^1(\mathbb{R}^{2ds})}\le\varepsilon^\theta,

with θ>0\theta>0 depending only on the dimension; here 1Ds\mathbf{1}_{D_s} is the indicator of the hard-sphere exclusion domain (configurations in which no two sphere centers overlap). The proof propagates a long-time cumulant ansatz — cumulants, i.e. joint cumulants of the correlation functions, quantify deviations from statistical independence and retain the full memory of the particles' collision history — and reduces the required smallness of the cumulants to combinatorial estimates on diagrammatic graphs encoding collision histories, called molecules, proved via an elaborate cutting algorithm. A companion paper by the same authors (arXiv:2503.01800) extends the derivation to 2D and 3D tori and, combining it with known hydrodynamic limits of the Boltzmann equation (the fast-relaxation asymptotics leading to the compressible Euler and incompressible Navier–Stokes–Fourier equations), derives the fluid equations directly from the hard-sphere particle system — thereby completing Hilbert's sixth problem (the axiomatization of physics, as it pertains to deriving macroscopic equations from microscopic mechanics) for the rarefied hard-sphere gas.

Two limitations survive. First, the result covers hard spheres, whereas physically realistic gases interact through potentials. For short-range potentials — compactly supported, repulsive interaction potentials Φ\Phi of the Gallagher–Saint-Raymond–Texier class (radial, supported in the unit ball, smooth away from the origin and singular near it), where particles accelerate under −ε−1∇Φ((xi−xj)/ε)-\varepsilon^{-1}\nabla\Phi((x_i-x_j)/\varepsilon) — the short-time Lanford theorem is known from the work of Gallagher, Saint-Raymond, and Texier, but the long-time extension is unproved: interactions are no longer instantaneous binary collisions, so the collision-tree combinatorics must handle genuinely simultaneous multi-particle interaction zones. Second, the quantitative parameters of the hard-sphere theorem are explicitly non-optimal: if the Boltzmann solution is global and regular, the proven validity time diverges only like tfin∼log⁡∣log⁡ε∣t_{\mathrm{fin}}\sim\log|\log\varepsilon| as ε→0\varepsilon\to 0 (the authors pose improvement to tfin∼∣log⁡ε∣t_{\mathrm{fin}}\sim|\log\varepsilon| as an interesting question); the correlation order s≤∣log⁡ε∣s\le|\log\varepsilon| could, per the authors, be pushed to s≤ε−κs\le\varepsilon^{-\kappa} by their method but this is not proved; and for s=1s=1 the optimal convergence exponent should be θ=1−\theta=1-. Quantifying the divergence of the validity time matters because the scaling laws leading from the particle system to hydrodynamic equations via the Boltzmann regime require explicit ε\varepsilon-dependent time rates, as emphasized in the 2026 Bourbaki survey of the Deng–Hani–Ma work by Bodineau, Gallagher, Saint-Raymond, and Simonella.

Problem Statement

Complete the long-time derivation of the Boltzmann equation from Newtonian dynamics, in the generality and quantitative strength expected of it, by resolving the two gaps explicitly left open by Deng–Hani–Ma (arXiv:2408.07818). The two components serve one common objective — bringing the long-time Newton-to-Boltzmann theorem to physical generality and quantitative sharpness. The four components below carry independent standing — the short-range long-time theorem (1) and the three hard-sphere quantitative sharpenings (2)(a), (2)(b), (2)(c) — and a full resolution establishes all of them.

(1) Short-range potential case (Section 1.4.2 of the source). Consider the particle system on Rd\mathbb{R}^d (d≥2d\ge 2) governed by

dxidt=vi,dvidt=−1ε∑j≠i∇Φ(xi−xjε),\frac{dx_i}{dt}=v_i,\qquad \frac{dv_i}{dt}=-\frac{1}{\varepsilon}\sum_{j\ne i}\nabla\Phi\Big(\frac{x_i-x_j}{\varepsilon}\Big),

where Φ\Phi is a short-range repulsive potential in the setting of Gallagher–Saint-Raymond–Texier — radial, nonnegative, nonincreasing, supported in the unit ball, of class C2C^2 in {0<∣x∣<1}\{0<|x|<1\} and unbounded near the origin, going to zero at ∣x∣=1|x|=1 with bounded derivatives, and with ∇Φ\nabla\Phi vanishing only on {∣x∣=1}\{|x|=1\} (Assumption 2.1 of their monograph), and satisfying the monotonicity condition ρ Φ′′(ρ)+2 Φ′(ρ)≥0\rho\,\Phi''(\rho)+2\,\Phi'(\rho)\ge 0 (their Lemma 3.1) ensuring that the deflection angle is one-to-one in the impact parameter, so that the scattering cross-section is well defined — in the Boltzmann–Grad scaling Nεd−1=1N\varepsilon^{d-1}=1, with random initial data given by a grand-canonical ensemble of initially (mod exclusion) independent, identically distributed particles. Prove the analogue of the Deng–Hani–Ma theorem: for every finite time interval [0,tfin][0,t_{\mathrm{fin}}] on which the solution ff of the corresponding Boltzmann equation (with the natural collision kernel generated by Φ\Phi through the two-body scattering map, i.e. the scattering cross-section of Gallagher–Saint-Raymond–Texier, Chapter 7; since Φ\Phi is compactly supported, the total cross-section is finite and no truncation of grazing collisions is imposed) satisfies the same regularity and Maxwellian-decay bounds, i.e. Gaussian-type decay of ff in the velocity variable, as in the hard-sphere theorem, the ss-particle correlation functions satisfy

∥fs(t,zs)−∏j=1sf(t,zj)∥L1(R2ds)≤εθ,\Big\|f_s(t,z^s)-\prod_{j=1}^s f(t,z_j)\Big\|_{L^1(\mathbb{R}^{2ds})}\le\varepsilon^{\theta},

uniformly in t∈[0,tfin]t\in[0,t_{\mathrm{fin}}] and s≤∣log⁡ε∣s\le|\log\varepsilon|, for some θ>0\theta>0; and, as in the hard-sphere case, the result must extend to a diverging horizon tfin(ε)→∞t_{\mathrm{fin}}(\varepsilon)\to\infty at the quantitative rate targeted in (2)(a), namely tfin(ε)∼∣log⁡ε∣t_{\mathrm{fin}}(\varepsilon)\sim|\log\varepsilon|, when the Boltzmann solution is global and uniformly regular. The essential new difficulty, flagged by the authors, is that continuous-force potentials allow genuinely simultaneous interactions of more than two particles, so the binary-collision-tree combinatorics and the hard-sphere recollision analysis — recollisions being repeated collisions of the same pair of particles — must be reworked.

(2) Quantitative sharpening of the hard-sphere theorem (Remark 1.5(2)–(3) of the source). (a) Improve the divergence rate of the validity time: assuming the Boltzmann solution is global and satisfies the uniform bounds required by the theorem, prove that the convergence holds for tfin(ε)t_{\mathrm{fin}}(\varepsilon) growing like ∣log⁡ε∣|\log\varepsilon| (in place of the proven log⁡∣log⁡ε∣\log|\log\varepsilon|). (b) Prove the convergence estimate for correlation orders s≤ε−κs\le\varepsilon^{-\kappa} for some sufficiently small κ>0\kappa>0 (which may depend on tfint_{\mathrm{fin}}), which the authors state their method should cover but do not prove. (c) Prove that for s=1s=1 the optimal convergence exponent θ=1−\theta=1- is attained.

The hydrodynamic-limit and torus components of the source’s open-directions list are excluded because they were resolved by the companion paper arXiv:2503.01800, and the fluctuation/large-deviation continuation is a distinct open direction not included in this problem.

References

  1. Yu Deng, Zaher Hani, Xiao Ma, “Long time derivation of the Boltzmann equation from hard sphere dynamics”, arXiv:2408.07818 (2024), DOI 10.48550/arXiv.2408.07818, https://arxiv.org/abs/2408.07818
  2. Yu Deng, Zaher Hani, Xiao Ma, “Hilbert's sixth problem: derivation of fluid equations via Boltzmann's kinetic theory”, arXiv:2503.01800 (2025), DOI 10.48550/arXiv.2503.01800, https://arxiv.org/abs/2503.01800
  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)”, Séminaire Bourbaki, January 2026, arXiv:2602.04407, https://arxiv.org/abs/2602.04407
  4. Isabelle Gallagher, Laure Saint-Raymond, Benjamin Texier, “From Newton to Boltzmann: Hard Spheres and Short-range Potentials”, EMS Zurich Lectures in Advanced Mathematics, DOI 10.4171/129 (2014), https://doi.org/10.4171/129
  5. Oscar E. Lanford, “Time evolution of large classical systems”, in Dynamical Systems, Theory and Applications (Battelle Seattle 1974), Lecture Notes in Physics 38, Springer (1975), pp. 1–111, DOI 10.1007/3-540-07171-7_1, https://doi.org/10.1007/3-540-07171-7_1
  6. 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 (2) 198 (2023), no. 3, DOI 10.4007/annals.2023.198.3.3, https://doi.org/10.4007/annals.2023.198.3.3
  7. Corentin Le Bihan, “Long time validity of the linearized Boltzmann uncut off and the linearized Landau equations from the Newton Law”, arXiv:2408.03597 (2024), https://arxiv.org/abs/2408.03597