I have some problems unifying, the two way I learned how a Bose condensate appears. The main problem is that the observables seem to be quite different.
In statistical physics lecture one starts with a free theory of bosons and calculates the particle number. One finds that at a critical temperature, in certain dimension, the ground states gets macroscopically occupied and if the fluctuations are not macroscopically, one calls that Bose condensation.
Including interaction one then talk about a condensate amplituds and anormalous Greens function of the condensate phase. Where explicite symmetry breaking must be included. These things are, to my current understand, not possible in non interacting theories as discussed in the context of naive statistical physics.
Essentially i would like to know if there is a distinction between these kinds of Bose condensation or if i can also talk about condensate amplitudes etc. in the free case.