  We study the hydrodynamics of superconductors within the framework of
Schwinger-Keldysh Effective Field Theory. We show that in the vicinity of the
superconducting phase transition the most general leading-order EFT satisfying
the local Kubo-Martin-Schwinger condition is described by a version of the
Time-Dependent Ginzburg-Landau (TDGL) equations augmented with stochastic
terms. This version of TDGL is applicable in the gapless regime independently
of any microscopic details. Within this approach, it is possible to include
systematically the effects of non-uniform temperature and heat conductivity, as
well as explicit or spontaneous breaking of time reversal. We also introduce a
thermal version of the Josephson relation and use it to construct an exotic
hydrodynamics describing a phase of matter where heat can flow without
dissipation.
