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arXiv · 2604.21583

Derivation of Gibbs measure from Gibbs state with the fractional Bessel interaction in Two Dimensions

Phan Thành Nam·Rongchan Zhu·Xiangchan Zhu
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Abstract

We derive the classical Gibbs measure on $\mathbb{T}^2$ associated with the fractional Bessel interaction potential $\widehat{v}_\beta(k)=\langle k\rangle^{-\beta}$ from a renormalized grand-canonical quantum Bose gas with the same interaction. Our result covers the whole range $\frac32<\beta\leq2$, where $\widehat{v}_\beta(k)$ 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.

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Phan Thành Nam, Rongchan Zhu, Xiangchan Zhu. 2026-04-23. Derivation of Gibbs measure from Gibbs state with the fractional Bessel interaction in Two Dimensions. https://arxiv.org/abs/2604.21583

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