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} = \alpha) fixed) yielding the Boltzmann equation, followed by a hydrodynamic limit ((\alpha \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 (\alpha) 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.