Bose gas to Bose-Einstein Condensate by the Phase Transition of the Klein-Gordon equation

We rewrite the complex Klein-Gordon (KG) equation with a mexican-hat scalar field potential in a thermal bath with one loop contribution as a new Gross-Pitaevskii (GP)-like equation. We interpret it as a charged and finite temperature generalization of the GP equation. We find its hydrodynamic version as well and using it, we derive the corresponding thermodynamics. We obtain a generalized first law for a charged Bose-Einstein Condensate (BEC). We translate the breaking of the U(1) local symmetry of the KG field into the new version of the GP equation and demonstrate that this symmetry breaking corresponds to a phase transition of the gas into a BEC, and show the conditions for which this system naturally becomes superfluid and/or superconductor.