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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2412.04396 |
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| _version_ | 1866910728848408576 |
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| author | Erhard, Dirk Franco, Tertuliano Xu, Tiecheng |
| author_facet | Erhard, Dirk Franco, Tertuliano Xu, Tiecheng |
| contents | In \cite{fgn1}, the hydrodynamic limit in the diffusive scaling of the symmetric simple exclusion process with a finite number of slow bonds of strength $n^{-β}$ has been studied. Here $n$ is the scaling parameter and $β>0$ is fixed. As shown in \cite{fgn1}, when $β>1$, such a limit is given by the heat equation with Neumann boundary conditions. In this work, we find more non-trivial super-diffusive scaling limits for this dynamics. Assume that there are $k$ equally spaced slow bonds in the system.
If $k$ is fixed and the time scale is $k^2n^θ$, with $θ\in (2,1+β)$, the density is asymptotically constant in each of the $k$ boxes, and equal to the initial expected mass in that box, i.e., there is no time evolution. If $k$ is fixed and the time scale is $k^2n^{1+β}$, then the density is also spatially constant in each box, but evolves in time according to the discrete heat equation. Finally, if the time scale is $k^2n^{1+β}$ and, additionally, the number of boxes $k$ increases to infinity, then the system converges to the continuous heat equation on the torus, with no boundary conditions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_04396 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Superdiffusive Scaling Limits for the Symmetric Exclusion Process with Slow Bonds Erhard, Dirk Franco, Tertuliano Xu, Tiecheng Probability 60K35 In \cite{fgn1}, the hydrodynamic limit in the diffusive scaling of the symmetric simple exclusion process with a finite number of slow bonds of strength $n^{-β}$ has been studied. Here $n$ is the scaling parameter and $β>0$ is fixed. As shown in \cite{fgn1}, when $β>1$, such a limit is given by the heat equation with Neumann boundary conditions. In this work, we find more non-trivial super-diffusive scaling limits for this dynamics. Assume that there are $k$ equally spaced slow bonds in the system. If $k$ is fixed and the time scale is $k^2n^θ$, with $θ\in (2,1+β)$, the density is asymptotically constant in each of the $k$ boxes, and equal to the initial expected mass in that box, i.e., there is no time evolution. If $k$ is fixed and the time scale is $k^2n^{1+β}$, then the density is also spatially constant in each box, but evolves in time according to the discrete heat equation. Finally, if the time scale is $k^2n^{1+β}$ and, additionally, the number of boxes $k$ increases to infinity, then the system converges to the continuous heat equation on the torus, with no boundary conditions. |
| title | Superdiffusive Scaling Limits for the Symmetric Exclusion Process with Slow Bonds |
| topic | Probability 60K35 |
| url | https://arxiv.org/abs/2412.04396 |