Abstract
Uncharged relativistic fluids in 3+1 dimensions have three independent thermodynamic transport coefficients at second order in the derivative expansion. Fluids with a single global U(1) current have nine, out of which seven are parity preserving. We derive the Kubo formulas for all nine thermodynamic transport coefficients in terms of equilibrium correlation functions of the energy-momentum tensor and the current. All parity-preserving coefficients can be expressed in terms of two-point functions in flat space without external sources, while the parity-violating coefficients require three-point functions. We use the Kubo formulas to compute the thermodynamic coefficients in several examples of free field theories.
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C. Gale, S. Jeon and B. Schenke, Hydrodynamic modeling of heavy-ion collisions, Int. J. Mod. Phys. A 28 (2013) 1340011 [arXiv:1301.5893] [INSPIRE].
S. Jeon and U. Heinz, Introduction to hydrodynamics, Int. J. Mod. Phys. E 24 (2015) 1530010 [arXiv:1503.03931] [INSPIRE].
R. Baier, P. Romatschke, D.T. Son, A.O. Starinets and M.A. Stephanov, Relativistic viscous hydrodynamics, conformal invariance and holography, JHEP 04 (2008) 100 [arXiv:0712.2451] [INSPIRE].
S. Bhattacharyya, V.E. Hubeny, S. Minwalla and M. Rangamani, Nonlinear fluid dynamics from gravity, JHEP 02 (2008) 045 [arXiv:0712.2456] [INSPIRE].
K. Jensen, M. Kaminski, P. Kovtun, R. Meyer, A. Ritz and A. Yarom, Parity-violating hydrodynamics in 2 + 1 dimensions, JHEP 05 (2012) 102 [arXiv:1112.4498] [INSPIRE].
N. Banerjee, J. Bhattacharya, S. Bhattacharyya, S. Jain, S. Minwalla and T. Sharma, Constraints on fluid dynamics from equilibrium partition functions, JHEP 09 (2012) 046 [arXiv:1203.3544] [INSPIRE].
K. Jensen, M. Kaminski, P. Kovtun, R. Meyer, A. Ritz and A. Yarom, Towards hydrodynamics without an entropy current, Phys. Rev. Lett. 109 (2012) 101601 [arXiv:1203.3556] [INSPIRE].
S. Bhattacharyya, Constraints on the second order transport coefficients of an uncharged fluid, JHEP 07 (2012) 104 [arXiv:1201.4654] [INSPIRE].
F.M. Haehl, R. Loganayagam and M. Rangamani, Adiabatic hydrodynamics: the eightfold way to dissipation, JHEP 05 (2015) 060 [arXiv:1502.00636] [INSPIRE].
G.D. Moore and K.A. Sohrabi, Kubo formulae for second-order hydrodynamic coefficients, Phys. Rev. Lett. 106 (2011) 122302 [arXiv:1007.5333] [INSPIRE].
P. Arnold, D. Vaman, C. Wu and W. Xiao, Second order hydrodynamic coefficients from 3-point stress tensor correlators via AdS/CFT, JHEP 10 (2011) 033 [arXiv:1105.4645] [INSPIRE].
G.D. Moore and K.A. Sohrabi, Thermodynamical second-order hydrodynamic coefficients, JHEP 11 (2012) 148 [arXiv:1210.3340] [INSPIRE].
P. Romatschke and D.T. Son, Spectral sum rules for the quark-gluon plasma, Phys. Rev. D 80 (2009) 065021 [arXiv:0903.3946] [INSPIRE].
E. Megias and M. Valle, Second-order partition function of a non-interacting chiral fluid in 3+1 dimensions, JHEP 11 (2014) 005[arXiv:1408.0165] [INSPIRE].
M. Buzzegoli, E. Grossi and F. Becattini, General equilibrium second-order hydrodynamic coefficients for free quantum fields, JHEP 10 (2017) 091 [Erratum ibid. 07 (2018) 119] [arXiv:1704.02808] [INSPIRE].
S. Chapman, C. Hoyos and Y. Oz, Superfluid Kubo formulas from partition function, JHEP 04 (2014) 186 [arXiv:1310.2247] [INSPIRE].
O. Philipsen and C. Schäfer, The second order hydrodynamic transport coefficient κ for the gluon plasma from the lattice, JHEP 02 (2014) 003 [arXiv:1311.6618] [INSPIRE].
S.I. Finazzo, R. Rougemont, H. Marrochio and J. Noronha, Hydrodynamic transport coefficients for the non-conformal quark-gluon plasma from holography, JHEP 02 (2015) 051 [arXiv:1412.2968] [INSPIRE].
S. Bhattacharyya, Entropy current from partition function: one example, JHEP 07 (2014) 139 [arXiv:1403.7639] [INSPIRE].
K. Jensen, R. Loganayagam and A. Yarom, Anomaly inflow and thermal equilibrium, JHEP 05 (2014) 134 [arXiv:1310.7024] [INSPIRE].
P. Kovtun, Thermodynamics of polarized relativistic matter, JHEP 07 (2016) 028 [arXiv:1606.01226] [INSPIRE].
R. Tolman and P. Ehrenfest, Temperature equilibrium in a static gravitational field, Phys. Rev. 36 (1930) 1791 [INSPIRE].
M.J. Duff, Twenty years of the Weyl anomaly, Class. Quant. Grav. 11 (1994) 1387 [hep-th/9308075] [INSPIRE].
J.F. Fuini and L.G. Yaffe, Far-from-equilibrium dynamics of a strongly coupled non-Abelian plasma with non-zero charge density or external magnetic field, JHEP 07 (2015) 116 [arXiv:1503.07148] [INSPIRE].
C. Eling, Y. Oz, S. Theisen and S. Yankielowicz, Conformal anomalies in hydrodynamics, JHEP 05 (2013) 037 [arXiv:1301.3170] [INSPIRE].
M.L. Bellac, Thermal field theory, Cambridge University Press, Cambridge, U.K., (1996) [INSPIRE].
J.I. Kapusta and C. Gale, Finite-temperature field theory, Cambridge University Press, Cambridge, U.K., (2011) [INSPIRE].
L.E. Parker and D. Toms, Quantum field theory in curved spacetime, Cambridge University Press, Cambridge, U.K., (2009) [INSPIRE].
J. Hernandez and P. Kovtun, Relativistic magnetohydrodynamics, JHEP 05 (2017) 001 [arXiv:1703.08757] [INSPIRE].
P. Kovtun, Lectures on hydrodynamic fluctuations in relativistic theories, J. Phys. A 45 (2012) 473001 [arXiv:1205.5040] [INSPIRE].
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Kovtun, P., Shukla, A. Kubo formulas for thermodynamic transport coefficients. J. High Energ. Phys. 2018, 7 (2018). https://doi.org/10.1007/JHEP10(2018)007
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DOI: https://doi.org/10.1007/JHEP10(2018)007