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

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.