Heat flow in a periodically forced, unpinned thermostatted chain
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913796689231872 |
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| author | Komorowski, Tomasz Olla, Stefano Simon, Marielle |
| author_facet | Komorowski, Tomasz Olla, Stefano Simon, Marielle |
| contents | We prove the hydrodynamic limit for a one-dimensional harmonic chain of interacting atoms with a random flip of the momentum sign. The system is open: at the left boundary it is attached to a heat bath at temperature $T_-$, while at the right endpoint it is subject to an action of a force which reads as $\bar F + \frac 1{\sqrt n} \widetilde{\mathcal F} (n^2 t)$, where $\bar F \ge0$ and $\widetilde{\mathcal F}(t)$ is a periodic function. Here $n$ is the size of the microscopic system. Under a diffusive scaling of space-time, we prove that the empirical profiles of the two locally conserved quantities - the volume stretch and the energy - converge, as $n\to+\infty$, to the solution of a non-linear diffusive system of conservative partial differential equations with a Dirichlet type and Neumann boundary conditions on the left and the right endpoints, respectively. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_11408 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Heat flow in a periodically forced, unpinned thermostatted chain Komorowski, Tomasz Olla, Stefano Simon, Marielle Probability Mathematical Physics We prove the hydrodynamic limit for a one-dimensional harmonic chain of interacting atoms with a random flip of the momentum sign. The system is open: at the left boundary it is attached to a heat bath at temperature $T_-$, while at the right endpoint it is subject to an action of a force which reads as $\bar F + \frac 1{\sqrt n} \widetilde{\mathcal F} (n^2 t)$, where $\bar F \ge0$ and $\widetilde{\mathcal F}(t)$ is a periodic function. Here $n$ is the size of the microscopic system. Under a diffusive scaling of space-time, we prove that the empirical profiles of the two locally conserved quantities - the volume stretch and the energy - converge, as $n\to+\infty$, to the solution of a non-linear diffusive system of conservative partial differential equations with a Dirichlet type and Neumann boundary conditions on the left and the right endpoints, respectively. |
| title | Heat flow in a periodically forced, unpinned thermostatted chain |
| topic | Probability Mathematical Physics |
| url | https://arxiv.org/abs/2406.11408 |