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

TL;DR

This paper establishes a rigorous connection between a quantum Bose gas with fractional Bessel interactions and the classical Gibbs measure on a two-dimensional torus, covering a non-summable interaction range.

Contribution

It derives the classical Gibbs measure from a quantum model with non-summable fractional Bessel interactions, using renormalization and localization techniques.

Findings

Convergence of quantum relative free energy to classical fractional-Bessel free energy.

Reduced density matrices converge to the Gibbs measure.

Analysis covers the entire range 3/2<β≤2 where interactions are non-summable.

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.