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Understanding Transport Coefficients in Thermal Gauge Theories

Explore the regime characterized by fluid dynamics in gauge theories, focusing on transport coefficients like viscosity. Learn about string theory computations for gauge theories and their gravity duals.

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Understanding Transport Coefficients in Thermal Gauge Theories

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  1. Transport coefficientsfromstring theory: an update Andrei Starinets Perimeter Institute Wien 2005 workshop

  2. Collaboration: Dam Son Giuseppe Policastro Chris Herzog Alvaro Nunez Pavel Kovtun Alex Buchel Jim Liu Andrei Parnachev Paolo Benincasa References: hep-th/0205051 hep-th/0205052 hep-th/0302026 hep-th/0309213 hep-th/0405231 hep-th/0406124 hep-th/0506144 hep-th/0506184 hep-th/0507026

  3. Prologue • Our goal is to understand thermal gauge theories, e.g. thermal QCD • Of particular interest is the regime described by fluid dynamics, e.g. quark-gluon plasma • This near-equilibrium regime is completely characterized by values of transport coefficients, e.g. shear and bulk viscosity • Transport coefficients are hard to compute from “first principles”, even in perturbation theory. For example, no perturbative calculation of bulk viscosity in gauge theory is available.

  4. Prologue (continued) • Transport coefficients of some gauge theories can be computed in the regime described by string (gravity) duals – usually at large N and large ‘t Hooft coupling • Corrections can in principle be computed • Shear viscosity result is universal. Model-independent results may be of relevance for RHIC physics • Certain results are predicted by hydrodynamics. Finding them in gravity provides a check of the AdS/CFT conjecture

  5. Timeline and status report • 2001:shear viscosity for N=4 SYM computed • 2002: prescription to compute thermal correlators from gravity formulated and applied to N=4 SYM; shear and sound poles in correlators are found • 2002-03: other poles in N=4 SYM correlators identified with quasinormal spectrum in gravity • 2003-04: universality of shear viscosity; general formula for diffusion coefficient from “membrane paradigm”; correction to shear viscosity • 2004-05: general prescription for computing transport coefficients from gravity duals formulated; bulk viscosity and the speed of sound computed in two non-conformal theories; equivalence between AdS/CFT and the “membrane paradigm” formulas established; spectral density computed /preliminary/ • 2005-??Nonzero chemical potential (with D.Son).

  6. What is hydrodynamics? Hierarchy of times (example) 0 t | | | | Mechanical description Hydrodynamic approximation Equilibrium thermodynamics Kinetic theory Hierarchy of scales (L is a macroscopic size of a system)

  7. Holography and hydrodynamics Gravitational fluctuations Deviations from equilibrium Dispersion relations Quasinormal spectrum

  8. Gauge-gravity duality in string theory Perturbative string theory: open and closed strings (at low energy, gauge fields and gravity, correspondingly) Nonperturbativetheory: D-branes (“topological defects” in 10d) Complementary description of D-branes by open (closed) strings: perturbative gauge theory description OK perturbative gravity description OK

  9. Hydrodynamics as an effective theory Thermodynamic equilibrium: Near-equilibrium: Eigenmodes of the system of equations Shear mode (transverse fluctuations of ): Sound mode: For CFT we have and

  10. Computing transport coefficients from “first principles” Fluctuation-dissipation theory (Callen, Welton, Green, Kubo) Kubo formulae allows one to calculate transport coefficients from microscopic models In the regime described by a gravity dual the correlator can be computed using AdS/CFT

  11. Universality of Theorem: For any thermal gauge theory (with zero chemical potential), the ratio of shear viscosity to entropy density is equal to in the regime described by a corresponding dual gravity theory Remark: Gravity dual to QCD (if it exists at all) is currently unknown.

  12. Universality of shear viscosity in the regime described by gravity duals Graviton’s component obeys equation for a minimally coupled massless scalar. But then . Since theentropy (density) is we get

  13. Three roads to universality of • The absorption argument D. Son, P. Kovtun, A.S., hep-th/0405231 • Direct computation of the correlator in Kubo formula from AdS/CFTA.Buchel, hep-th/0408095 • “Membrane paradigm” general formula for diffusion coefficient + interpretation as lowest quasinormal frequency = pole of the shear mode correlator + Buchel-Liu theorem P. Kovtun, D.Son, A.S., hep-th/0309213, A.S., to appear, P.Kovtun, A.S., hep-th/0506184, A.Buchel, J.Liu, hep-th/0311175

  14. Shear viscosity in SYM P.Arnold, G.Moore, L.Yaffe, 2001 Correction to : A.Buchel, J.Liu, A.S., hep-th/0406264

  15. Viscosity of gases and liquids Gases (Maxwell, 1867): Viscosity of a gas is • independent of pressure • scales as square of temperature • inversely proportional to cross-section Liquids (Frenkel, 1926): • Wis the “activation energy” • In practice, A and W are chosen to fit data

  16. A viscosity bound conjecture P.Kovtun, D.Son, A.S., hep-th/0309213, hep-th/0405231

  17. Two-point correlation function of stress-energy tensor Field theory Zero temperature: Finite temperature: Dual gravity • Five gauge-invariant combinations • of and other fields determine • obey a system of coupled ODEs • Their (quasinormal) spectrum determines singularities of the correlator

  18. Classification of fluctuations and universality O(2) symmetry in x-y plane Shear channel: Sound channel: Scalar channel: Other fluctuations (e.g. ) may affect sound channel But not the shear channel universality of

  19. Bulk viscosity and the speed of sound in SYM is a “mass-deformed” (Pilch-Warner flow) • Finite-temperature version: A.Buchel, J.Liu, hep-th/0305064 • The metric is known explicitly for • Speed of sound and bulk viscosity:

  20. Relation to RHIC • IF quark-gluon plasma is indeed formed in heavy ion collisions • IF a hydrodynamic regime is unambiguously proven to exist • THEN hydrodynamic MODELS describe experimental results • for e.g. elliptic flows well, provided • Bulk viscosity and speed of sound results are • potentially interesting

  21. Epilogue • AdS/CFT gives insights into physics of thermal gauge theories in the nonperturbative regime • Generic hydrodynamic predictions can be used to check validity of AdS/CFT • General algorithm exists to compute transport coefficients and the speed of sound in any gravity dual • Model-independent statements can presumably be checked experimentally

  22. What is viscosity? Friction in Newton’s equation: Friction in Euler’s equations

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