Comment on "Hilbert's Sixth Problem: Derivation of Fluid Equations via Boltzmann's Kinetic Theory" by Deng, Hani, and Ma

TL;DR

Hilbert's Sixth Problem seeks a rigorous, physically faithful derivation of continuum fluid dynamics from Newtonian mechanics. Deng, Hani, and Ma propose a two-step scheme on the torus, deriving the Boltzmann equation via a Boltzmann-Grad limit and then passing to a hydrodynamic limit to obtain Navier-Stokes-Fourier equations, using an iterated limit in $\varepsilon$ and $\alpha$. The critique identifies two key flaws: the Boltzmann-Grad scaling forces a vanishing volume fraction $\phi = N\varepsilon^{d}$, so the construction remains a dilute gas rather than a dense fluid, and molecular chaos fails in fluid-like regimes where $\phi = O(1)$, undermining the validity of the Boltzmann step. Consequently, the result is a mathematically structured but physically implausible rescaled gas, not a universal derivation of fluid dynamics from Newtonian mechanics; the paper underscores the need for alternative kinetic theories (e.g., Enskog) or mesoscopic approaches to address density-dependent phenomena and to advance Hilbert's program toward a generally valid micro-to-macro description.

Abstract

Deng, Hani, and Ma [arXiv:2503.01800] claim to resolve Hilbert's Sixth Problem by deriving the Navier-Stokes-Fourier equations from Newtonian mechanics via an iterated limit: a Boltzmann-Grad limit (\(\varepsilon \to 0\), \(N \varepsilon^{d-1} = α\) fixed) yielding the Boltzmann equation, followed by a hydrodynamic limit (\(α\to \infty\)) to obtain fluid dynamics. Though mathematically rigorous, their approach harbors two critical physical flaws. First, the vanishing volume fraction (\(N \varepsilon^d \to 0\)) confines the system to a dilute gas, incapable of embodying dense fluid properties even as \(α\) scales, rendering the resulting equations a rescaled gas model rather than a true continuum. Second, the Boltzmann equation's reliance on molecular chaos collapses in fluid-like regimes, where recollisions and correlations invalidate its derivation from Newtonian dynamics. These inconsistencies expose a disconnect between the formalism and the physical essence of fluids, failing to capture emergent, density-driven phenomena central to Hilbert's vision. We contend that the Sixth Problem remains open, urging a rethink of classical kinetic theory's limits and the exploration of alternative frameworks to unify microscale mechanics with macroscale fluid behavior.