Derivation of the Boltzmann equation from hard-sphere dynamics (after Y. Deng, Z. Hani, and X. Ma)
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866911421729603584 |
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| author | Bodineau, Thierry Gallagher, Isabelle Saint-Raymond, Laure Simonella, Sergio |
| author_facet | Bodineau, Thierry Gallagher, Isabelle Saint-Raymond, Laure Simonella, Sergio |
| contents | Consider a microscopic system of $N$ hard spheres that are initially independent (modulo the exclusion condition on particle positions) and identically distributed in $\mathbb{R}^3$. When the number $N$ of particles goes to infinity and the diameter $\varepsilon$ of the particles goes to zero, and under the weak density assumption $N\varepsilon^2=1$, it has been known since the work of Lanford that the empirical measure for the particles converges to the solution of the Boltzmann equation in a short time interval. In particular, the particles remain dynamically independent, in this limit and in the short time interval where the correlations induced by their collisions remain under control. In a recent work, Y. Deng, Z. Hani and X. Ma successfully obtained the same convergence result in arbitrary large time; more precisely, the convergence result holds for any time interval on which the Boltzmann equation has a regular solution. In this note, we explain a few elements of their proof. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_04407 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Derivation of the Boltzmann equation from hard-sphere dynamics (after Y. Deng, Z. Hani, and X. Ma) Bodineau, Thierry Gallagher, Isabelle Saint-Raymond, Laure Simonella, Sergio Analysis of PDEs Mathematical Physics Probability Consider a microscopic system of $N$ hard spheres that are initially independent (modulo the exclusion condition on particle positions) and identically distributed in $\mathbb{R}^3$. When the number $N$ of particles goes to infinity and the diameter $\varepsilon$ of the particles goes to zero, and under the weak density assumption $N\varepsilon^2=1$, it has been known since the work of Lanford that the empirical measure for the particles converges to the solution of the Boltzmann equation in a short time interval. In particular, the particles remain dynamically independent, in this limit and in the short time interval where the correlations induced by their collisions remain under control. In a recent work, Y. Deng, Z. Hani and X. Ma successfully obtained the same convergence result in arbitrary large time; more precisely, the convergence result holds for any time interval on which the Boltzmann equation has a regular solution. In this note, we explain a few elements of their proof. |
| title | Derivation of the Boltzmann equation from hard-sphere dynamics (after Y. Deng, Z. Hani, and X. Ma) |
| topic | Analysis of PDEs Mathematical Physics Probability |
| url | https://arxiv.org/abs/2602.04407 |