The Bogoliubov free energy functional II. The dilute limit

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 $T_{\mathrm{c}}=T_{\mathrm{fc}}(1+1.49({\rho }^{1/3}a))$ to leading order. Here, $T_{\mathrm{fc}}$ 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 $a_0=(8\pi {)}^{-1}\hat{V}(0)$ 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 $a_0\to a$.

Published version, 58 pages, 2 figures

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