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ResolvedMath Physics·2025

Hilbert's Sixth Problem

Rigorously derived Euler and Navier-Stokes equations from hard-sphere particle dynamics.

Formula

Boltzmann equation
∂tf+v⋅∇xf=Q(f,f)\partial_t f + v\cdot\nabla_x f = Q(f,f)

The kinetic equation governing the distribution of particles in a dilute gas — the central bridge in Hilbert's sixth problem.

Hydrodynamic limit
fε→ε→0Mρ,u,T⟹(ρ,u,T) solve Euler or Navier–Stokesf^\varepsilon \xrightarrow{\varepsilon\to 0} M_{\rho,u,T}\quad\Longrightarrow\quad (\rho,u,T)\text{ solve Euler or Navier–Stokes}

In the small mean-free-path limit, the Boltzmann distribution converges to a local Maxwellian whose parameters satisfy fluid equations.

Lanford's theorem (1975)
fN(1)(t)→f(t)as N→∞,t<15tmfpf_N^{(1)}(t)\to f(t)\quad\text{as }N\to\infty,\quad t < \tfrac{1}{5}t_{\text{mfp}}

Rigorous derivation of the Boltzmann equation from Newtonian N-particle dynamics, valid for short times.

Summary

Hilbert's 6th problem (1900) asks for the axiomatization of physics. Deng, Hani, and Ma rigorously derived the fundamental equations of fluid mechanics — compressible Euler and incompressible Navier-Stokes-Fourier — from Newtonian particle dynamics via the Boltzmann-Grad limit. A 125-year challenge.

Sources

Hilbert's sixth problem: derivation of fluid equations via Boltzmann's kinetic theory

arXiv:2503.018002025

Time evolution of large classical systems

1975

Hilbert's sixth problem: derivation of fluid equations via Boltzmann's kinetic theory

arXiv:2408.078182024

Videos

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