Derivation of kinetic and diffusion equations from a hard-sphere Rayleigh gas using collision trees and semigroups
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866907893212643328 |
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| author | Matthies, Karsten Syntaka, Theodora |
| author_facet | Matthies, Karsten Syntaka, Theodora |
| contents | We will revisit the classical questions of understanding the statistics of various deterministic dynamics of $N$ hard spheres of diameter $\varepsilon$ with random initial data in the Boltzmann-Grad scaling as $\varepsilon$ tends to zero and $N$ tends to infinity. The convergence of the empiric particle dynamics to the Boltzmann-type dynamics is shown using semigroup methods to describe probability measures on collision trees associated to physical trajectories in the case of a Rayleigh gas. As an application we derive the diffusion equation by a further rescaling. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_04449 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Derivation of kinetic and diffusion equations from a hard-sphere Rayleigh gas using collision trees and semigroups Matthies, Karsten Syntaka, Theodora Analysis of PDEs Dynamical Systems We will revisit the classical questions of understanding the statistics of various deterministic dynamics of $N$ hard spheres of diameter $\varepsilon$ with random initial data in the Boltzmann-Grad scaling as $\varepsilon$ tends to zero and $N$ tends to infinity. The convergence of the empiric particle dynamics to the Boltzmann-type dynamics is shown using semigroup methods to describe probability measures on collision trees associated to physical trajectories in the case of a Rayleigh gas. As an application we derive the diffusion equation by a further rescaling. |
| title | Derivation of kinetic and diffusion equations from a hard-sphere Rayleigh gas using collision trees and semigroups |
| topic | Analysis of PDEs Dynamical Systems |
| url | https://arxiv.org/abs/2405.04449 |