Dipole Formation in the Two-Component Plasma
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913526694543360 |
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| author | Boursier, Jeanne Serfaty, Sylvia |
| author_facet | Boursier, Jeanne Serfaty, Sylvia |
| contents | We consider the two-dimensional two-component plasma, or Coulomb gas, consisting of $N$ positive and $N$ negative charges with logarithmic interaction. We introduce a suitable regularization of the interaction by smearing the charges over a small length scale $λ$, which allows us to give meaning to the system in the continuum at any temperature. We provide an expansion of the free energy in terms of the inverse temperature $β$, as $N \to \infty$ and $λ\to 0$. Doing so allows us to show that, for $β\geq 2$, the charges (for the most part) pair into neutral dipoles of very small size as $λ\to 0$. This complements the prior work of Leblé, Zeitouni, and the second author, which proved that this does not happen for $β< 2$, thereby implying a transition at $β= 2$ from free charges to dipole pairs. Moreover, we obtain an estimate on the size of linear statistics.
The description in terms of dipoles is made via a decomposition into nearest-neighbor graphs of the point configurations, à la Gunson-Panta. This is combined with new energy estimates obtained via an electric reformulation of the interaction energy and a ball-growth method, which are expressed in terms of the nearest-neighbor graph distances only. In this way, the model is compared to a reduced nearest-neighbor interaction model, by showing the relative smallness of the dipole-dipole interactions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_01025 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Dipole Formation in the Two-Component Plasma Boursier, Jeanne Serfaty, Sylvia Mathematical Physics Probability We consider the two-dimensional two-component plasma, or Coulomb gas, consisting of $N$ positive and $N$ negative charges with logarithmic interaction. We introduce a suitable regularization of the interaction by smearing the charges over a small length scale $λ$, which allows us to give meaning to the system in the continuum at any temperature. We provide an expansion of the free energy in terms of the inverse temperature $β$, as $N \to \infty$ and $λ\to 0$. Doing so allows us to show that, for $β\geq 2$, the charges (for the most part) pair into neutral dipoles of very small size as $λ\to 0$. This complements the prior work of Leblé, Zeitouni, and the second author, which proved that this does not happen for $β< 2$, thereby implying a transition at $β= 2$ from free charges to dipole pairs. Moreover, we obtain an estimate on the size of linear statistics. The description in terms of dipoles is made via a decomposition into nearest-neighbor graphs of the point configurations, à la Gunson-Panta. This is combined with new energy estimates obtained via an electric reformulation of the interaction energy and a ball-growth method, which are expressed in terms of the nearest-neighbor graph distances only. In this way, the model is compared to a reduced nearest-neighbor interaction model, by showing the relative smallness of the dipole-dipole interactions. |
| title | Dipole Formation in the Two-Component Plasma |
| topic | Mathematical Physics Probability |
| url | https://arxiv.org/abs/2410.01025 |