# The Bogoliubov free energy functional

# The Bogoliubov free energy functional

Einstein's classic treatment of Bose-Einstein condensation for non-interacting bosons predicts at what temperature the phase transition to BEC occurs, but how this critical temperature changes when the bosons start to interact weakly is an old problem, first posed by Feynman in 1953. In this talk, I will address this question in a variational model. We have called it the 'Bogoliubov free energy functional', as it is related to Bogoliubov's 1947 theory of weakly-interacting Bose gases. I will describe several aspects of the model: its derivation, the existence of minimizers, the phase diagram, and finally the expression for the critical temperature, which has the form predicted by Lee and Yang in 1958 and a constant that agrees with numerical simulations.