Effective action for relativistic hydrodynamics: fluctuations, dissipation, and entropy inflow

Abstract

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 [1]. 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 [2]. 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 Parisi-Sourlas. 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. arXiv:1803.11155v2 [hep-th] 10 Sep 2018 J Fixing the worldvolume connection 103 K Superspace expansion of further fields in the MMO limit 108 -1 -Recently an answer to this question has emerged, mainly in the context of treating fluid dynamics as an effective theory [2,23]. It was conjectured the correct local principle to enforce is to demand an emergent 'entropic' gauge symmetry dubbed U (1) T , which in real time provides the correct analytic continuation of the Euclidean periodicity. Its gauge current is the entropy current (this statement can be thought of as a generalization of Wald's idea that equilibrium entropy is a Noether charge [24]). This emergent KMS gauge symmetry can be understood in terms of the topological structure of the Schwinger-Keldysh construction. In particular, it was argued in [5] that the additional constraints coming from KMS invariance lead to a quartet of operations that act on the Schwinger-Keldysh super-operator algebra. Two of these are Grassmann odd, thermal counterparts of the BRST charges, Q KM S , Q KM S and the two others are Grassmann even generators Q 0 KM S , L KM S . 2 One can intuitively understand the thermal generators in the following fashion. For a thermal system one can view real time dynamics as occurring on a background spacetime that admits a fibration by a thermal circle. Recall that we are used to analyzing equilibrium dynami...

Nonlinear Langevin dynamics via holography

“…This would give a way to check various features of such effective theories proposed in the literature [61]. An especially interesting problem is to study fluid-gravity correspondence within holographic SK formalism and see what it can say about the non-Gaussianities in fluid fluctuations [62][63][64][65][66][67]. Another direction would be to study field theories in the black-brane backgrounds and derive dual open quantum field theories for the corresponding single trace primary, by integrating out the effect of the black-brane.…”

“…Another question is to trace where the entropy of the particle is created from ? Is there an entropy inflow from the complex spacetime that replaces the horizon in holographic SK contour [62,63,69] ? We see in this system the glimmer of many ideas which have recently appeared in the context of fluid SK effective theories [61].…”

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