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

Abstract

We derive the classical Gibbs measure on ${\mathbb{T}}^2$ associated with the fractional Bessel interaction potential ${\hat{v}}_{\beta }(k)=⟨k{⟩}^{-\beta }$ from a renormalized grand-canonical quantum Bose gas with the same interaction. Our result covers the whole range $\frac{3}{2}<\beta \le 2$, where ${\hat{v}}_{\beta }(k)$ 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.

66 pages