Fractional kinetics equation from a Markovian system of interacting Bouchaud trap models
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866929570798632960 |
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| author | Chiarini, Alberto Floreani, Simone Sau, Federico |
| author_facet | Chiarini, Alberto Floreani, Simone Sau, Federico |
| contents | We consider a partial exclusion process evolving on $\mathbb Z^d$ in a random trapping environment. In dimension $d\ge 2$, we derive the fractional kinetics equation \begin{equation*}\frac{\partial^βρ_t}{\partial t^β} = Δρ_t
\end{equation*} as a hydrodynamic limit of the particle system. Here, $\frac{\partial^β}{\partial t^β}$, $β\in(0,1)$, denotes the fractional derivative in the Caputo sense. We thus exhibit a Markovian interacting particle system whose empirical density field rescales to a sub-diffusive equation corresponding to a non-Markovian process, the Fractional Kinetics process. In contrast, we show that, when $d=1$, the system rescales to the solution to \begin{equation*}
\frac{\partial ρ_t}{\partial t}= \mathcal L_βρ_t\ , \end{equation*} where $\mathcal L_β$ is the random generator of the singular quasi-diffusion known as FIN diffusion. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_10156 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Fractional kinetics equation from a Markovian system of interacting Bouchaud trap models Chiarini, Alberto Floreani, Simone Sau, Federico Probability Mathematical Physics 60K35, 60K37, 60G57, 60G22, 35B27 We consider a partial exclusion process evolving on $\mathbb Z^d$ in a random trapping environment. In dimension $d\ge 2$, we derive the fractional kinetics equation \begin{equation*}\frac{\partial^βρ_t}{\partial t^β} = Δρ_t \end{equation*} as a hydrodynamic limit of the particle system. Here, $\frac{\partial^β}{\partial t^β}$, $β\in(0,1)$, denotes the fractional derivative in the Caputo sense. We thus exhibit a Markovian interacting particle system whose empirical density field rescales to a sub-diffusive equation corresponding to a non-Markovian process, the Fractional Kinetics process. In contrast, we show that, when $d=1$, the system rescales to the solution to \begin{equation*} \frac{\partial ρ_t}{\partial t}= \mathcal L_βρ_t\ , \end{equation*} where $\mathcal L_β$ is the random generator of the singular quasi-diffusion known as FIN diffusion. |
| title | Fractional kinetics equation from a Markovian system of interacting Bouchaud trap models |
| topic | Probability Mathematical Physics 60K35, 60K37, 60G57, 60G22, 35B27 |
| url | https://arxiv.org/abs/2302.10156 |