Fractional diffusion as the limit of a short range potential Rayleigh gas
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915592637775872 |
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| author | Matthies, Karsten Syntaka, Theodora |
| author_facet | Matthies, Karsten Syntaka, Theodora |
| contents | The fractional diffusion equation is rigorously derived as a scaling limit from a deterministic Rayleigh gas, where
particles interact via short range potentials with support of size $\varepsilon$ and the background is distributed in space $\mathbb{R}^3$ according to a Poisson process with intensity $N$ and in velocity according to some fat-tailed distribution. As an intermediate step a linear Boltzmann equation is obtained in the Boltzmann-Grad limit as $\varepsilon$ tends to zero and $N$ tends to infinity with $N \varepsilon^2 =c$. 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. The fractional diffusion equation is a hydrodynamic limit for times $t \in [0,T]$, where $T$ and inverse mean free path $c$ can both be chosen as some negative rational power $\varepsilon^{-k}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_19025 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fractional diffusion as the limit of a short range potential Rayleigh gas Matthies, Karsten Syntaka, Theodora Analysis of PDEs Mathematical Physics Probability The fractional diffusion equation is rigorously derived as a scaling limit from a deterministic Rayleigh gas, where particles interact via short range potentials with support of size $\varepsilon$ and the background is distributed in space $\mathbb{R}^3$ according to a Poisson process with intensity $N$ and in velocity according to some fat-tailed distribution. As an intermediate step a linear Boltzmann equation is obtained in the Boltzmann-Grad limit as $\varepsilon$ tends to zero and $N$ tends to infinity with $N \varepsilon^2 =c$. 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. The fractional diffusion equation is a hydrodynamic limit for times $t \in [0,T]$, where $T$ and inverse mean free path $c$ can both be chosen as some negative rational power $\varepsilon^{-k}$. |
| title | Fractional diffusion as the limit of a short range potential Rayleigh gas |
| topic | Analysis of PDEs Mathematical Physics Probability |
| url | https://arxiv.org/abs/2405.19025 |