The Bogoliubov Free Energy Functional II: The Dilute Limit.

We analyse the canonical Bogoliubov free energy functional in three dimensions at low temperatures in the dilute limit. We prove existence of a first-order phase transition and, in the limit ∫V→8πa<inline-graphic></inline-graphic>, we determine the critical temperature to be Tc=Tfc(1+1.49ρ1/3a)<inline-graphic></inline-graphic> to leading order. Here, Tfc<inline-graphic></inline-graphic> is the critical temperature of the free Bose gas, <italic>ρ</italic> is the density of the gas and <italic>a</italic> is the scattering length of the pair-interaction potential <italic>V</italic>. We also prove asymptotic expansions for the free energy. In particular, we recover the Lee-Huang-Yang formula in the limit ∫V→8πa<inline-graphic></inline-graphic>. [ABSTRACT FROM AUTHOR]