The Bogoliubov free energy functional II. The dilute limit

TL;DR

This paper analyzes the dilute limit of the Bogoliubov free energy functional at low temperatures, establishing a phase transition, determining the critical temperature correction, and deriving asymptotic free energy expansions including the Lee-Huang-Yang formula.

Contribution

It provides the first rigorous analysis of the dilute limit of the Bogoliubov free energy functional, including phase transition proof and precise critical temperature correction.

Findings

Proves existence of a first order phase transition.

Determines the critical temperature correction in the dilute limit.

Derives asymptotic expansions for the free energy, recovering the Lee-Huang-Yang formula.

Abstract

We analyse the canonical Bogoliubov free energy functional at low temperatures in the dilute limit. We prove existence of a first order phase transition and, in the limit $a_0\to a$, we determine the critical temperature to be $T_{\rm{c}}=T_{\rm{fc}}(1+1.49(\rho^{1/3}a))$ to leading order. Here, $T_{\rm{fc}}$ is the critical temperature of the free Bose gas, $\rho$ is the density of the gas, $a$ is the scattering length of the pair-interaction potential $V$, and $a_0=(8\pi)^{-1}\widehat{V}(0)$ its first order approximation. We also prove asymptotic expansions for the free energy. In particular, we recover the Lee-Huang-Yang formula in the limit $a_0\to a$.