Open problem, Mathematical physics, posed 1900

Hilbert's sixth problem

Partially solvedHilbert #6

Treat in the same manner as geometry, by means of axioms, those physical sciences in which mathematics already plays an important part — first and foremost the theory of probabilities and mechanics — and develop mathematically the limiting processes, such as Boltzmann's kinetic theory of gases, that lead from the atomistic view of $N$ colliding particles as $N \to \infty$ to the continuum laws of motion of fluids.

Posed by, 1900David Hilbert

Research frontier as of 2026

As of 2026 Hilbert's sixth problem is partially solved: its probability branch is completely settled by Kolmogorov's 1933 measure-theoretic axioms, and its classical kinetic-theory branch — deriving continuum fluid mechanics from atomistic Newtonian dynamics — achieved a landmark breakthrough in 2024–2025. For fifty years after Lanford's 1975 theorem, rigorous derivations of the Boltzmann equation from $N$ hard spheres in the Boltzmann–Grad limit $N\varepsilon^{d-1}=1$ were trapped below a fraction of a mean free time because recollision histories proliferate factorially. In preprints posted in November 2024 and March 2025 (arXiv:2503.01800), Yu Deng, Zaher Hani, and Xiao Ma introduced a layered cluster-expansion and tree-pruning scheme that controls recollisions on $\mathbb{T}^3$ for arbitrary macroscopic time intervals — as long as the target Boltzmann solution remains smooth — and chained this with hydrodynamic scaling limits to obtain the compressible Euler and incompressible Navier–Stokes–Fourier equations directly from Newton's laws, explaining mathematically how macroscopic time-irreversibility emerges from time-reversible particle collisions. Meanwhile, the broader ambition of axiomatizing all of physics (in particular four-dimensional interacting quantum field theories and quantum gravity) remains open.

Best known results

Measure-theoretic axiomatization of probability theory on $(\Omega, \mathcal{F}, \mathbb{P})$ (Kolmogorov, 1933) and Hilbert-space axiomatization of quantum mechanics (von Neumann, 1932).
Short-time rigorous derivation of the Boltzmann equation from Newtonian hard spheres in the Boltzmann–Grad limit $N\varepsilon^{d-1} = 1$ (Lanford, 1975).
Long-time derivation of the Boltzmann equation and hydrodynamic limit to the compressible Euler and incompressible Navier–Stokes–Fourier equations from hard spheres on $\mathbb{T}^3$ (Deng–Hani–Ma, 2025 preprint, arXiv:2503.01800).

Tools and where they stop

Tool	Achieved	Where it stops
Measure-theoretic probability and operator algebras	Provides a clean, universal axiomatic framework for classical probability (Kolmogorov, 1933) and for non-relativistic quantum mechanics and algebraic quantum field theory (von Neumann, Wightman, Haag–Kastler).	Writing down axioms does not by itself prove that nontrivial four-dimensional interacting models (like 4D Yang–Mills) actually exist and satisfy them.
BBGKY hierarchy and Duhamel series (Lanford's method)	Expresses $k$ -particle marginals via collision trees and proves convergence to the Boltzmann equation on a short time interval $t < t_0$ in the Boltzmann–Grad limit.	The number of collision trees of depth $n$ grows like $n!$ , causing the naive Duhamel series to diverge after roughly one-fifth of a mean free time — too short to take a hydrodynamic limit to fluid equations.
Layered cluster expansions and collision-history pruning (Deng–Hani–Ma)	Decomposes long time intervals into thin time layers, cumulant-expands correlations between layers, and bounds bad recollision molecule graphs to reach arbitrary macroscopic times and fluid limits on $\mathbb{T}^3$ (2025 preprint).	Requires periodic boundary conditions ( $\mathbb{T}^3$ ), hard-sphere collisions, and a regime where the limit Boltzmann/fluid solution stays smooth; singular fluid regimes and general long-range forces remain open.

Open questions

Can the long-time derivation of the Boltzmann and fluid equations be extended from hard spheres on $\mathbb{T}^3$ to domains with physical boundaries and to general short- or long-range interaction potentials?
Can four-dimensional interacting quantum field theories and relativistic gravitation be placed on a unified, mathematically rigorous axiomatic footing?

References

David Hilbert (1900). Mathematische Probleme
Andrey N. Kolmogorov (1933). Grundbegriffe der Wahrscheinlichkeitsrechnung
Oscar E. Lanford III (1975). Time evolution of large classical systems · DOI:10.1007/3-540-07171-7_1
Yu Deng, Zaher Hani, Xiao Ma (2025). Hilbert's sixth problem: derivation of fluid equations via Boltzmann's kinetic theory · arXiv:2503.01800 [preprint, not peer-reviewed]