Comment on "Hilbert's Sixth Problem: Derivation of Fluid Equations via Boltzmann's Kinetic Theory" by Deng, Hani, and Ma
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| Format: | Preprint |
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2025
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| author | Gao, Shan |
| author_facet | Gao, Shan |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_06297 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Comment on "Hilbert's Sixth Problem: Derivation of Fluid Equations via Boltzmann's Kinetic Theory" by Deng, Hani, and Ma Gao, Shan Analysis of PDEs Fluid Dynamics History and Philosophy of Physics 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. |
| title | Comment on "Hilbert's Sixth Problem: Derivation of Fluid Equations via Boltzmann's Kinetic Theory" by Deng, Hani, and Ma |
| topic | Analysis of PDEs Fluid Dynamics History and Philosophy of Physics |
| url | https://arxiv.org/abs/2504.06297 |