Abstract
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.
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References
K.-c. Chou, Z.-b. Su, B.-l. Hao and L. Yu, Equilibrium and Nonequilibrium Formalisms Made Unified, Phys. Rept. 118 (1985) 1 [INSPIRE].
E. Wang and U.W. Heinz, A Generalized fluctuation dissipation theorem for nonlinear response functions, Phys. Rev. D 66 (2002) 025008 [hep-th/9809016] [INSPIRE].
I.D. Schepper, H.V. Beyeren and M. Ernst, The nonexistence of the linear diffusion equation beyond Fick’s law, Physica 75 (1974) 1.
Y. Pomeau and P. Résibois, Time dependent correlation functions and mode-mode coupling theories, Phys. Rep. 19 (1975) 63.
P. Kovtun and L.G. Yaffe, Hydrodynamic fluctuations, long time tails and supersymmetry, Phys. Rev. D 68 (2003) 025007 [hep-th/0303010] [INSPIRE].
P. Kovtun, G.D. Moore and P. Romatschke, The stickiness of sound: An absolute lower limit on viscosity and the breakdown of second order relativistic hydrodynamics, Phys. Rev. D 84 (2011) 025006 [arXiv:1104.1586] [INSPIRE].
E.M. Lifshitz and L.P. Pitaevskii, Statistical Physics, Part 2, Pergamon (1980).
J.I. Kapusta, B. Müller and M. Stephanov, Relativistic Theory of Hydrodynamic Fluctuations with Applications to Heavy Ion Collisions, Phys. Rev. C 85 (2012) 054906 [arXiv:1112.6405] [INSPIRE].
P. Kovtun, G.D. Moore and P. Romatschke, Towards an effective action for relativistic dissipative hydrodynamics, JHEP 07 (2014) 123 [arXiv:1405.3967] [INSPIRE].
L.D. Landau and E.M. Lifshitz, Fluid Mechanics, Pergamon (1987).
P. Kovtun, Lectures on hydrodynamic fluctuations in relativistic theories, J. Phys. A 45 (2012) 473001 [arXiv:1205.5040] [INSPIRE].
S. Jeon, The Boltzmann equation in classical and quantum field theory, Phys. Rev. C 72 (2005) 014907 [hep-ph/0412121] [INSPIRE].
S. Jeon, Color Glass Condensate in Schwinger-Keldysh QCD, Annals Phys. 340 (2014) 119 [arXiv:1308.0263] [INSPIRE].
N. Banerjee et al., Constraints on Fluid Dynamics from Equilibrium Partition Functions, JHEP 09 (2012) 046 [arXiv:1203.3544] [INSPIRE].
K. Jensen et al., Towards hydrodynamics without an entropy current, Phys. Rev. Lett. 109 (2012) 101601 [arXiv:1203.3556] [INSPIRE].
J. Bhattacharya, S. Bhattacharyya, S. Minwalla and A. Yarom, A Theory of first order dissipative superfluid dynamics, JHEP 05 (2014) 147 [arXiv:1105.3733] [INSPIRE].
S. Endlich, A. Nicolis, R.A. Porto and J. Wang, Dissipation in the effective field theory for hydrodynamics: First order effects, Phys. Rev. D 88 (2013) 105001 [arXiv:1211.6461] [INSPIRE].
S. Grozdanov and J. Polonyi, Viscosity and dissipative hydrodynamics from effective field theory, Phys. Rev. D 91 (2015) 105031 [arXiv:1305.3670] [INSPIRE].
D.T. Son, Low-energy quantum effective action for relativistic superfluids, hep-ph/0204199 [INSPIRE].
C.P. Herzog, N. Lisker, P. Surowka and A. Yarom, Transport in holographic superfluids, JHEP 08 (2011) 052 [arXiv:1101.3330] [INSPIRE].
D.T. Son, Newton-Cartan Geometry and the Quantum Hall Effect, arXiv:1306.0638 [INSPIRE].
K. Jensen, Aspects of hot Galilean field theory, JHEP 04 (2015) 123 [arXiv:1411.7024] [INSPIRE].
K. Jensen and A. Karch, Revisiting non-relativistic limits, JHEP 04 (2015) 155 [arXiv:1412.2738] [INSPIRE].
F.M. Haehl, R. Loganayagam and M. Rangamani, Adiabatic hydrodynamics: The eightfold way to dissipation, JHEP 05 (2015) 060 [arXiv:1502.00636] [INSPIRE].
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ArXiv ePrint: 1502.03076
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Harder, M., Kovtun, P. & Ritz, A. On thermal fluctuations and the generating functional in relativistic hydrodynamics. J. High Energ. Phys. 2015, 25 (2015). https://doi.org/10.1007/JHEP07(2015)025
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DOI: https://doi.org/10.1007/JHEP07(2015)025