Diffusion Limit of the Low-Density Magnetic Lorentz Gas
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909429856731136 |
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| author | Nota, Alessia Nowak, Dominik Saffirio, Chiara |
| author_facet | Nota, Alessia Nowak, Dominik Saffirio, Chiara |
| contents | We consider the magnetic Lorentz gas proposed by Bobylev et al. [4], which describes a point particle moving in a random distribution of hard-disk obstacles in $\mathbb{R}^2$ under the influence of a constant magnetic field perpendicular to the plane. We show that, in the coupled low-density and diffusion limit, when the intensity of the magnetic field is smaller than $\frac{8π}{3}$, the non-Markovian effects induced by the magnetic field become sufficiently weak. Consequently, the particle's probability distribution converges to the solution of the heat equation with a diffusion coefficient dependent on the magnetic field and given by the Green-Kubo formula. This formula is derived from the generator of the generalized Boltzmann process associated with the generalized Boltzmann equation, as predicted in [4]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_11134 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Diffusion Limit of the Low-Density Magnetic Lorentz Gas Nota, Alessia Nowak, Dominik Saffirio, Chiara Mathematical Physics Analysis of PDEs Probability We consider the magnetic Lorentz gas proposed by Bobylev et al. [4], which describes a point particle moving in a random distribution of hard-disk obstacles in $\mathbb{R}^2$ under the influence of a constant magnetic field perpendicular to the plane. We show that, in the coupled low-density and diffusion limit, when the intensity of the magnetic field is smaller than $\frac{8π}{3}$, the non-Markovian effects induced by the magnetic field become sufficiently weak. Consequently, the particle's probability distribution converges to the solution of the heat equation with a diffusion coefficient dependent on the magnetic field and given by the Green-Kubo formula. This formula is derived from the generator of the generalized Boltzmann process associated with the generalized Boltzmann equation, as predicted in [4]. |
| title | Diffusion Limit of the Low-Density Magnetic Lorentz Gas |
| topic | Mathematical Physics Analysis of PDEs Probability |
| url | https://arxiv.org/abs/2412.11134 |