On thermal fluctuations and the generating functional in relativistic hydrodynamics
Michael Harder, Pavel Kovtun, Adam Ritz
TL;DR
The paper develops a real-time generating functional for correlation functions in dissipative relativistic hydrodynamics that incorporates thermal fluctuations via a Schwinger-Keldysh (CTP) framework in the (r,a) basis. A bottom-up analysis shows how linearized diffusive, shear, and sound modes are encoded in the generating functional and its fluctuation structure, while a top-down symmetry-based approach imposes a nonlinear realization of doubled CTP symmetries to build a derivative-expanded effective action. The resulting framework reproduces known linear-response functions and yields constitutive relations with first-order transport coefficients, illustrating how fluctuations and dissipation can be captured within a unified effective action. The discussion highlights open questions on basis translations, equilibrium constraints, and measure, and suggests extensions to Galilean systems and connections with related classifications of transport phenomena.
Abstract
We discuss a real-time generating functional for correlation functions in dissipative relativistic hydrodynamics which takes into account thermal fluctuations of the hydrodynamic 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 realization, 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.