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, we determine the critical temperature to be Tc=Tfc(1+1.49ρ1/3a) to leading order. Here, Tfc is the critical temperature of the free Bose gas, ρ is the density of the gas and a is the scattering length of the pair-interaction potential V. We also prove asymptotic expansions for the free energy. In particular, we recover the Lee–Huang–Yang formula in the limit ∫V→8πa.