On thermal fluctuations and the generating functional in relativistic hydrodynamics

A bstract We discuss a real-time generating functional for correlation functions in dis-sipative relativistic hydrodynamics which takes into account thermal fluctuations of thehydrodynamic variables. Starting from the known form of these correlation functions in the linearized regime, we integrate to find a generating functional which we can interpret within the CTP formalism, provided the space-time and internal global symmetries are realized in a specific manner in the ( r, a ) sectors. We then verify that this symmetry real-ization, when implemented in an effective action for hydrodynamic fields in the ( r, a ) basis, leads to a consistent derivative expansion for the constitutive relations at the nonlinear level, modulo constraints associated with the existence of an equilibrium state.