Derivation of Gibbs measure from Gibbs state with the fractional Bessel interaction in Two Dimensions
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866910160432136192 |
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| author | Nam, Phan Thành Zhu, Rongchan Zhu, Xiangchan |
| author_facet | Nam, Phan Thành Zhu, Rongchan Zhu, Xiangchan |
| contents | We derive the classical Gibbs measure on $\mathbb{T}^2$ associated with the fractional Bessel interaction potential $\widehat{v}_β(k)=\langle k\rangle^{-β}$ from a renormalized grand-canonical quantum Bose gas with the same interaction. Our result covers the whole range $\frac32<β\leq2$, where $\widehat{v}_β(k)$ is not summable and the quantum model cannot be written in the usual density-square form, as the associated self-energy diverges. We therefore need to renormalize the zero mode by a centered number-fluctuation term and then develop a detailed analysis for the high-frequency remainders. All this allows us to implement a low-frequency localization and obtain the convergence of the quantum relative free energy to the classical fractional-Bessel free energy, as well as the convergence of the reduced density matrices to the limiting Gibbs measure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_21583 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Derivation of Gibbs measure from Gibbs state with the fractional Bessel interaction in Two Dimensions Nam, Phan Thành Zhu, Rongchan Zhu, Xiangchan Mathematical Physics Analysis of PDEs Probability We derive the classical Gibbs measure on $\mathbb{T}^2$ associated with the fractional Bessel interaction potential $\widehat{v}_β(k)=\langle k\rangle^{-β}$ from a renormalized grand-canonical quantum Bose gas with the same interaction. Our result covers the whole range $\frac32<β\leq2$, where $\widehat{v}_β(k)$ is not summable and the quantum model cannot be written in the usual density-square form, as the associated self-energy diverges. We therefore need to renormalize the zero mode by a centered number-fluctuation term and then develop a detailed analysis for the high-frequency remainders. All this allows us to implement a low-frequency localization and obtain the convergence of the quantum relative free energy to the classical fractional-Bessel free energy, as well as the convergence of the reduced density matrices to the limiting Gibbs measure. |
| title | Derivation of Gibbs measure from Gibbs state with the fractional Bessel interaction in Two Dimensions |
| topic | Mathematical Physics Analysis of PDEs Probability |
| url | https://arxiv.org/abs/2604.21583 |