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  • Print publication year: 2015
  • Online publication date: May 2015

12 - Linear response and hydrodynamics

from Part III - Applications

Summary

A very successful and important application of gauge/gravity duality has emerged in the context of hydrodynamics. In generalisation of the dynamics of fluids, the term hydrodynamics generically refers to an effective field theory describing long-range, low-energy fluctuations about equilibrium.

Recently, experimental evidence has accumulated that the quarkgluon plasma observed in heavy-ion collision experiments is best described by a strongly coupled relativistic fluid, rather than by a gas of weakly interacting particles. Strongly coupled fluids are intrinsically difficult to describe by standard methods. This explains the success of applying gauge/gravity duality to this area of physics. In particular, gauge/gravity duality has made predictions of universal values of certain transport coefficients in strongly coupled fluids. The most famous example of this is the ratio of shear viscosity over entropy density, which takes a very small value. Beyond these results, gauge/gravity duality has provided a fresh look at relativistic hydrodynamics, for which many new non-trivial properties have been uncovered using the fluid/gravity correspondence.

We will describe these results in some detail. The starting point is to introduce linear response theory and Green's functions which respect the causal structure. Then we move on to an introduction to relativistic hydrodynamics. We consider the energy-momentum tensor and a conserved current and their dissipative contributions in an expansion in derivatives of fluctuations. We define the associated first-order transport coefficients and subsequently relate them to the retarded Green's function by virtue of appropriate Green–Kubo relations. This provides a link between macroscopic hydrodynamic properties and microscopic physics as described by the Green's functions. Using gauge/gravity duality methods to evaluate the relevant Green's functions, we compute the charge diffusion constant and the shear viscosity.

[1] Kovtun, Pavel K., and Starinets, Andrei o. 2005. Quasinormal modes and holography. Phys. Rev., D72, 086009.
[2] de Boer, Jan, Verlinde, Erik P., and Verlinde, Herman L. 2000. on the holographic renormalization group.J. High Energy Phys., 0008, 003.
[3] Papadimitriou, Ioannis, and Skenderis, Kostas. 2004. Correlation functions in holographic RG flows.J. High Energy Phys., 0410, 075.
[4] McGreevy, John. 2010. Holographic duality with a view toward many-body physics.Adv. High Energy Phys., 2010, 723105.
[5] Myers, Robert C., Starinets, Andrei o., and Thomson, Rowan M. 2007. Holographic spectral functions and diffusion constants for fundamental matter.J. High Energy Phys., 0711, 091.
[6] Baier, Rudolf, Romatschke, Paul, Son, Dam Thanh, Starinets, Andrei o., and Stephanov, Mikhail A. 2008. Relativistic viscous hydrodynamics, conformal invari-ance, and holography. J. High Energy Phys., 0804, 100.
[7] Bhattacharyya, Sayantani, Hubeny, Veronika E., Minwalla, Shiraz, and Rangamani, Mukund. 2008. Nonlinear fluid dynamics from gravity.J. High Energy Phys., 0802, 045.
[8] Haack, Michael, and Yarom, Amos. 2008. Nonlinear viscous hydrodynamics in various dimensions using AdS/CFT.J. High Energy Phys., 0810, 063.
[9] Rangamani, Mukund. 2009. Gravity and hydrodynamics: lectures on the fluid-gravity correspondence. Class.Quantum Grav., 26, 224003.
[10] Son, Dam T., and Starinets, Andrei o. 2007. Viscosity, black holes, and quantum field theory. Ann. Rev. Nucl. Part. Sci., 57, 95–118.
[11] Kovtun, Pavel. 2012. Lectures on hydrodynamic fluctuations in relativistic theories.J. Phys., A45, 473001.
[12] Policastro, G., Son, D. T., and Starinets, A. O. 2001. The shear viscosity of strongly coupled N = 4 supersymmetric Yang-Mills plasma.Phys. Rev. Lett., 87, 081601.
[13] Kovtun, P., Son, D. T., and Starinets, A. o. 2005. Viscosity in strongly interacting quantum field theories from black hole physics.Phys. Rev. Lett., 94, 111601.
[14] Buchel, Alex, and Liu, James T. 2004. Universality of the shear viscosity in supergravity. Phys. Rev. Lett., 93, 090602.
[15] Buchel, Alex, Liu, James T., and Starinets, Andrei o. 2005. Coupling constant dependence of the shear viscosity in N = 4 supersymmetric Yang-Mills theory.Nucl. Phys., B707, 56–68.
[16] Kats, Yevgeny, and Petrov, Pavel. 2009. Effect of curvature squared corrections in AdS on the viscosity of the dual gauge theory.J. High Energy Phys., 0901, 044.
[17] Brigante, Mauro, Liu, Hong, Myers, Robert C., Shenker, Stephen, and Yaida, Sho. 2008. Viscosity bound violation in higher derivative gravity.Phys. Rev., D77, 126006.
[18] Buchel, Alex, Myers, Robert C., and Sinha, Aninda. 2009. Beyond eta/s = 1/4pi. J. High Energy Phys., 0903, 084.
[19] Rebhan, Anton, and Steineder, Dominik. 2012. Violation of the holographic viscosity bound in a strongly coupled anisotropic plasma.Phys. Rev. Lett., 108, 021601.
[20] Erdmenger, Johanna, Kerner, Patrick, and Zeller, Hansjorg. 2011. Non-universal shear viscosity from Einstein gravity. Phys. Lett., B699, 301–304.
[21] Erdmenger, Johanna, Kerner, Patrick, and Zeller, Hansjorg. 2012. Transport in anisotropic superfluids: a holographic description. J. High Energy Phys., 1201, 059.
[22] Horowitz, Gary T., and Hubeny, Veronika E. 2000. Quasinormal modes of AdS black holes and the approach to thermal equilibrium.Phys. Rev., D62, 024027.
[23] Berti, Emanuele, Cardoso, Vitor, and Starinets, Andrei 0. 2009. Quasinormal modes of black holes and black branes.Class.Quantum Grav., 26, 163001.
[24] Son, Dam T., and Starinets, Andrei O. 2006. Hydrodynamics of R-charged black holes. J. High Energy Phys., 0603, 052.
[25] Balasubramanian, Vijay, and Kraus, Per. 1999. A stress tensor for anti-de Sitter gravity.Commun. Math. Phys., 208, 413–128.
[26] de Haro, Sebastian, Solodukhin, Sergey N., and Skenderis, Kostas. 2001. Holographic reconstruction of space-time and renormalization in the AdS/CFT correspondence.Commun. Math. Phys., 217, 595–622.
[27] Erdmenger, Johanna, Haack, Michael, Kaminski, Matthias, and Yarom, Amos. 2009. Fluid dynamics of R-charged black holes.J. High Energy Phys., 0901, 055.
[28] Banerjee, Nabamita, Bhattacharya, Jyotirmoy, Bhattacharyya, Sayantani, Dutta, Suvankar, Loganayagam, R., and Surowka, Piotr. 2011. Hydrodynamics from charged black branes.J. High Energy Phys., 1101, 094.
[29] Son, Dam T., and Surowka, Piotr. 2009. Hydrodynamics with triangle anomalies.Phys. Rev. Lett., 103, 191601.
[30] Fukushima, Kenji, Kharzeev, Dmitri E., and Warringa, Harmen J. 2008. The chiral magnetic effect.Phys. Rev., D78, 074033.
[31] Kalaydzhyan, Tigran, and Kirsch, Ingo. 2011. Fluid/gravity model for the chiral magnetic effect.Phys. Rev. Lett., 106, 211601.