Relativistic magnetohydrodynamics

Open Access
Regular Article - Theoretical Physics

Abstract

We present the equations of relativistic hydrodynamics coupled to dynamical electromagnetic fields, including the effects of polarization, electric fields, and the derivative expansion. We enumerate the transport coefficients at leading order in derivatives, including electrical conductivities, viscosities, and thermodynamic coefficients. We find the constraints on transport coefficients due to the positivity of entropy production, and derive the corresponding Kubo formulas. For the neutral state in a magnetic field, small fluctuations include Alfvén waves, magnetosonic waves, and the dissipative modes. For the state with a non-zero dynamical charge density in a magnetic field, plasma oscillations gap out all propagating modes, except for Alfvén-like waves with a quadratic dispersion relation. We relate the transport coefficients in the “conventional” magnetohydrodynamics (formulated using Maxwell’s equations in matter) to those in the “dual” version of magnetohydrodynamics (formulated using the conserved magnetic flux).

Keywords

Holography and quark-gluon plasmas Thermal Field Theory 

Notes

Open Access

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References

  1. [1]
    L.D. Landau and E.M. Lifshitz, Fluid Mechanics, Pergamon (1987).Google Scholar
  2. [2]
    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].ADSMathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    S. Bhattacharyya, V.E. Hubeny, S. Minwalla and M. Rangamani, Nonlinear Fluid Dynamics from Gravity, JHEP 02 (2008) 045 [arXiv:0712.2456] [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    D.T. Son and P. Surowka, Hydrodynamics with Triangle Anomalies, Phys. Rev. Lett. 103 (2009) 191601 [arXiv:0906.5044] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    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].ADSMathSciNetCrossRefGoogle Scholar
  6. [6]
    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].ADSCrossRefGoogle Scholar
  7. [7]
    S. Bhattacharyya, Entropy current and equilibrium partition function in fluid dynamics, JHEP 08 (2014) 165 [arXiv:1312.0220] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    S. Bhattacharyya, Entropy Current from Partition Function: One Example, JHEP 07 (2014) 139 [arXiv:1403.7639] [INSPIRE].ADSCrossRefGoogle Scholar
  9. [9]
    I. Fouxon and Y. Oz, Exact Scaling Relations In Relativistic Hydrodynamic Turbulence, Phys. Lett. B 694 (2010) 261 [arXiv:0909.3574] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    M.P. Heller and M. Spalinski, Hydrodynamics Beyond the Gradient Expansion: Resurgence and Resummation, Phys. Rev. Lett. 115 (2015) 072501 [arXiv:1503.07514] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    F.M. Haehl, R. Loganayagam and M. Rangamani, Adiabatic hydrodynamics: The eightfold way to dissipation, JHEP 05 (2015) 060 [arXiv:1502.00636] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  12. [12]
    S.A. Hartnoll, P.K. Kovtun, M. Muller and S. Sachdev, Theory of the Nernst effect near quantum phase transitions in condensed matter and in dyonic black holes, Phys. Rev. B 76 (2007) 144502 [arXiv:0706.3215] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    J. Crossno et al., Observation of the Dirac fluid and the breakdown of the Wiedemann-Franz law in graphene, Science 351 (2016) 1058 [arXiv:1509.04713].ADSCrossRefGoogle Scholar
  14. [14]
    A. Lucas, J. Crossno, K.C. Fong, P. Kim and S. Sachdev, Transport in inhomogeneous quantum critical fluids and in the Dirac fluid in graphene, Phys. Rev. B 93 (2016) 075426 [arXiv:1510.01738] [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    A. Lucas, R.A. Davison and S. Sachdev, Hydrodynamic theory of thermoelectric transport and negative magnetoresistance in Weyl semimetals, Proc. Nat. Acad. Sci. 113 (2016) 9463 [arXiv:1604.08598] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    K. Jensen, R. Loganayagam and A. Yarom, Anomaly inflow and thermal equilibrium, JHEP 05 (2014) 134 [arXiv:1310.7024] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    P. Kovtun, Thermodynamics of polarized relativistic matter, JHEP 07 (2016) 028 [arXiv:1606.01226] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    P. Kovtun, Lectures on hydrodynamic fluctuations in relativistic theories, J. Phys. A 45 (2012) 473001 [arXiv:1205.5040] [INSPIRE].ADSMathSciNetMATHGoogle Scholar
  19. [19]
    C. Eling, Y. Oz, S. Theisen and S. Yankielowicz, Conformal Anomalies in Hydrodynamics, JHEP 05 (2013) 037 [arXiv:1301.3170] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    K. Jensen, P. Kovtun and A. Ritz, Chiral conductivities and effective field theory, JHEP 10 (2013) 186 [arXiv:1307.3234] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  21. [21]
    E.G. Harris, Relativistic magnetohydrodynamics, Phys. Rev. 108 (1957) 1357.ADSMathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    S. Grozdanov, D.M. Hofman and N. Iqbal, Generalized global symmetries and dissipative magnetohydrodynamics, arXiv:1610.07392 [INSPIRE].
  23. [23]
    D. Schubring, Dissipative String Fluids, Phys. Rev. D 91 (2015) 043518 [arXiv:1412.3135] [INSPIRE].ADSMathSciNetGoogle Scholar
  24. [24]
    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].ADSCrossRefGoogle Scholar
  25. [25]
    X.-G. Huang, A. Sedrakian and D.H. Rischke, Kubo formulae for relativistic fluids in strong magnetic fields, Annals Phys. 326 (2011) 3075 [arXiv:1108.0602] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  26. [26]
    S.I. Finazzo, R. Critelli, R. Rougemont and J. Noronha, Momentum transport in strongly coupled anisotropic plasmas in the presence of strong magnetic fields, Phys. Rev. D 94 (2016) 054020 [arXiv:1605.06061] [INSPIRE].ADSGoogle Scholar
  27. [27]
    R. Critelli, S.I. Finazzo, M. Zaniboni and J. Noronha, Anisotropic shear viscosity of a strongly coupled non-Abelian plasma from magnetic branes, Phys. Rev. D 90 (2014) 066006 [arXiv:1406.6019] [INSPIRE].ADSGoogle Scholar
  28. [28]
    W. Israel and J.M. Stewart, Transient relativistic thermodynamics and kinetic theory, Annals Phys. 118 (1979) 341 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  29. [29]
    S. Pu, T. Koide and D.H. Rischke, Does stability of relativistic dissipative fluid dynamics imply causality?, Phys. Rev. D 81 (2010) 114039 [arXiv:0907.3906] [INSPIRE].ADSGoogle Scholar
  30. [30]
    E.M. Lifshitz and L.P. Pitaevskii, Physical Kinetics, Pergamon (1981).Google Scholar

Copyright information

© The Author(s) 2017

Authors and Affiliations

  1. 1.Department of Physics and AstronomyUniversity of VictoriaVictoriaCanada

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