Around the Quantum Lenard-Balescu equation
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866912810005430272 |
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| author | Bihan, Corentin Le |
| author_facet | Bihan, Corentin Le |
| contents | In the mean-field regime, a gas of quantum particles with Boltzmann statistics can be described by the Hartree-Fock equation. This dynamics becomes trivial if the initial distribution of particle is invariant by translation. However, the first correction is given on time of order $O(N)$ by the quantum Lenard--Balescu equation. In the first part of the present article, we justify this equation until time of order $O((\log N)^{1-δ})$ (for any $δ\in(0,1)$).
A similar phenomenon exists in the classical setting (with a similar validity time obtained by Duerinckx \cite{Duerinckx}). In a second time, we prove the convergence for dimension $d\geq 2$ of the solutions of the quantum Lenard--Balescu equation to the solutions of its classical counterpart in the semi-classical limit. This problem can be interpreted as a grazing collision limit: the quantum Lenard--Balescu equation looks like a cut-off Boltzmann equation, when the classical one looks like the Landau equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_06544 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Around the Quantum Lenard-Balescu equation Bihan, Corentin Le Analysis of PDEs In the mean-field regime, a gas of quantum particles with Boltzmann statistics can be described by the Hartree-Fock equation. This dynamics becomes trivial if the initial distribution of particle is invariant by translation. However, the first correction is given on time of order $O(N)$ by the quantum Lenard--Balescu equation. In the first part of the present article, we justify this equation until time of order $O((\log N)^{1-δ})$ (for any $δ\in(0,1)$). A similar phenomenon exists in the classical setting (with a similar validity time obtained by Duerinckx \cite{Duerinckx}). In a second time, we prove the convergence for dimension $d\geq 2$ of the solutions of the quantum Lenard--Balescu equation to the solutions of its classical counterpart in the semi-classical limit. This problem can be interpreted as a grazing collision limit: the quantum Lenard--Balescu equation looks like a cut-off Boltzmann equation, when the classical one looks like the Landau equation. |
| title | Around the Quantum Lenard-Balescu equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2501.06544 |