Gao, Shan (2025) Comment on "Hilbert's Sixth Problem: Derivation of Fluid Equations via Boltzmann's Kinetic Theory" by Deng, Hani, and Ma. [Preprint]
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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 (ε → 0, Nεd−1 = α fixed) yielding the Boltzmann equation, followed by a hydrodynamic limit (α → ∞) to obtain fluid dynamics. Though mathematically rigorous, their approach harbors two critical physical flaws. First, the vanishing volume fraction (Nεd → 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.
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| Item Type: | Preprint | ||||||
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| Keywords: | Hilbert's Sixth Problem; Newtonian dynamics; Boltzmann equation; Navier-Stokes-Fourier equations | ||||||
| Subjects: | Specific Sciences > Mathematics > Explanation Specific Sciences > Complex Systems Specific Sciences > Physics > Classical Physics Specific Sciences > Physics > Statistical Mechanics/Thermodynamics |
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| Depositing User: | Prof. Shan Gao | ||||||
| Date Deposited: | 08 Apr 2025 21:30 | ||||||
| Last Modified: | 08 Apr 2025 21:30 | ||||||
| Item ID: | 25009 | ||||||
| Subjects: | Specific Sciences > Mathematics > Explanation Specific Sciences > Complex Systems Specific Sciences > Physics > Classical Physics Specific Sciences > Physics > Statistical Mechanics/Thermodynamics |
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| Date: | 7 April 2025 | ||||||
| URI: | https://philsci-archive.pitt.edu/id/eprint/25009 |
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