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Effective action for relativistic hydrodynamics: fluctuations, dissipation, and entropy inflow. (English) Zbl 1402.83021
Summary: We present a detailed and self-contained analysis of the universal Schwinger-Keldysh effective field theory which describes macroscopic thermal fluctuations of a relativistic field theory, elaborating on our earlier construction [ibid. 2016, No. 4, Paper No. 39, 21 p. (2016; Zbl 1388.83351)]. We write an effective action for appropriate hydrodynamic Goldstone modes and fluctuation fields, and discuss the symmetries to be imposed. The constraints imposed by fluctuation-dissipation theorem are manifest in our formalism. Consequently, the action reproduces hydrodynamic constitutive relations consistent with the local second law at all orders in the derivative expansion, and captures the essential elements of the eightfold classification of hydrodynamic transport of [the authors, ibid. 2015, No. 5, Paper No. 60, 214 p. (2015; Zbl 1388.81456)]. We demonstrate how to recover the hydrodynamic entropy and give predictions for the non-Gaussian hydrodynamic fluctuations. The basic ingredients of our construction involve (i) doubling of degrees of freedom a la Schwinger-Keldysh, (ii) an emergent gauge $$U(1)_{T}$$ symmetry associated with entropy which is encapsulated in a Noether current a la Wald, and (iii) a BRST/topological supersymmetry imposing the fluctuation-dissipation theorem a la G. Parisi and N. Sourlas [Nucl. Phys., B 206, No. 2, 321–332 (1982; Zbl 0968.81547)]. The overarching mathematical framework for our construction is provided by the balanced equivariant cohomology of thermal translations, which captures the basic constraints arising from the Schwinger-Keldysh doubling, and the thermal Kubo-Martin-Schwinger relations. All these features are conveniently implemented in a covariant superspace formalism. An added benefit is that the second law can be understood as being due to entropy inflow from the Grassmann-odd directions of superspace.

##### MSC:
 83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems) 76Y05 Quantum hydrodynamics and relativistic hydrodynamics 81T45 Topological field theories in quantum mechanics 82B05 Classical equilibrium statistical mechanics (general) 83C55 Macroscopic interaction of the gravitational field with matter (hydrodynamics, etc.) 81T60 Supersymmetric field theories in quantum mechanics
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