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Transport coefficients from string theory : an update

Transport coefficients from string theory : an update. Andrei Starinets Perimeter Institute. Wien 2005 workshop. Collaboration:. Dam Son Giuseppe Policastro Chris Herzog Alvaro Nunez Pavel Kovtun Alex Buchel Jim Liu Andrei Parnachev Paolo Benincasa. References:.

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Transport coefficients from string theory : an update

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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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