  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.
