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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2511.05514 |
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| _version_ | 1866912940828917760 |
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| author | Barkman, Tristan |
| author_facet | Barkman, Tristan |
| contents | A new operator-level necessity result for the Chapman--Enskog expansion is established: in closed and unforced kinetic systems, the $O(\varepsilon)$ deviatoric stress arises if and only if the first Chapman--Enskog correction $f^{(1)}$ is nonzero. This resolves a gap in the classical kinetic-to-continuum literature, where the presence of first-order deviatoric stress is typically assumed or derived formally but not shown to be necessary under explicit functional-analytic hypotheses. Under precise nullspace structure, coercivity or quantitative hypocoercivity, and Fredholm solvability of the linearized collision operator--together with uniform $O(\varepsilon^2)$ remainder control--a sharp necessity theorem (Theorem 6.1) is proved: if $f^{(1)}\equiv 0$, then no $O(\varepsilon)$ deviatoric stress can appear in the hydrodynamic limit. The argument identifies the bounded mapping \[ f^{(0)} \mapsto f^{(1)} = -L^{-1}(\partial_t^{(0)} + v \cdot \nabla_x) f^{(0)}, \] and the induced moment-to-stress operator, and shows how remainder bounds preclude hidden $O(\varepsilon)$ contributions. A worked BGK example verifies the construction, transport coefficients, and operator constants. Detailed assumptions and analytic estimates are provided in Section 4 and Appendix A. The discussion concludes by describing how microscopic seeds (deterministic or finite-$N$) can project into macroscopic amplification channels relevant for transition and turbulence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_05514 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Molecular Seeds of Shear: An operator-level necessity result for first-order Chapman-Enskog deviatoric stress Barkman, Tristan Analysis of PDEs A new operator-level necessity result for the Chapman--Enskog expansion is established: in closed and unforced kinetic systems, the $O(\varepsilon)$ deviatoric stress arises if and only if the first Chapman--Enskog correction $f^{(1)}$ is nonzero. This resolves a gap in the classical kinetic-to-continuum literature, where the presence of first-order deviatoric stress is typically assumed or derived formally but not shown to be necessary under explicit functional-analytic hypotheses. Under precise nullspace structure, coercivity or quantitative hypocoercivity, and Fredholm solvability of the linearized collision operator--together with uniform $O(\varepsilon^2)$ remainder control--a sharp necessity theorem (Theorem 6.1) is proved: if $f^{(1)}\equiv 0$, then no $O(\varepsilon)$ deviatoric stress can appear in the hydrodynamic limit. The argument identifies the bounded mapping \[ f^{(0)} \mapsto f^{(1)} = -L^{-1}(\partial_t^{(0)} + v \cdot \nabla_x) f^{(0)}, \] and the induced moment-to-stress operator, and shows how remainder bounds preclude hidden $O(\varepsilon)$ contributions. A worked BGK example verifies the construction, transport coefficients, and operator constants. Detailed assumptions and analytic estimates are provided in Section 4 and Appendix A. The discussion concludes by describing how microscopic seeds (deterministic or finite-$N$) can project into macroscopic amplification channels relevant for transition and turbulence. |
| title | Molecular Seeds of Shear: An operator-level necessity result for first-order Chapman-Enskog deviatoric stress |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2511.05514 |