When we consider a non interacting or a weakly interacting dilute thermal gas containing in a vessel, there is a symmetry in the system. That means for an atom, there is not a preferred direction, all directions look the same and it has a random phase. Once the gas is cooled to some low temperature in a trap, it enters in a new phase called a condensate, in which all atoms lie in the same quantum state and have the same macroscopic quantum phase, which is a completely different situation in comparison to the thermal gas. Therefore, the phenomenon of Bose-Einstein condensation is a spontaneous symmetry breaking !
Note: This posting is in progress !
