Transport coefficients from string theory: an update

Andrei Starinets

Perimeter Institute

Wien 2005 workshop

Collaboration: Dam SonGiuseppe PolicastroChris HerzogAlvaro NunezPavel KovtunAlex BuchelJim LiuAndrei ParnachevPaolo Benincasa

References:

hep-th/0205051 hep-th/0205052 hep-th/0302026 hep-th/0309213 hep-th/0405231 hep-th/0406124hep-th/0506144 hep-th/0506184 hep-th/0507026

PrologueOur 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 plasmaThis 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.

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 computedShear 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

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).

What is hydrodynamics?

0 t| | | |

Hierarchy of times (example)

Mechanical description

Kinetic theory

Hydrodynamic approximation

Equilibrium thermodynamics

Hierarchy of scales

(L is a macroscopic size of a system)

Holography and hydrodynamics

Gravitational fluctuations Deviations from equilibrium

Quasinormal spectrum Dispersion relations

Gauge-gravity duality in string theory

Perturbative string theory: open and closed strings(at low energy, gauge fields and gravity, correspondingly)

Nonperturbative theory: D-branes (“topological defects” in 10d)

Complementary description of D-branes by open (closed) strings:

perturbative gauge theory description OK

perturbative gravity description OK

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

Computing transport coefficients from “first principles”

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

Fluctuation-dissipation theory(Callen, Welton, Green, Kubo)

Universality of

Theorem:

For any thermal gauge theory (with zero chemicalpotential), the ratio of shear viscosity to entropy density is equal to in the regime describedby a corresponding dual gravity theory

Remark:

Gravity dual to QCD (if it exists at all) is currentlyunknown.

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 the entropy (density) is we get

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/CFT A.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

Shear viscosity in SYM

Correction to : A.Buchel, J.Liu, A.S., hep-th/0406264

P.Arnold, G.Moore, L.Yaffe, 2001

Viscosity of gases and liquids

Gases (Maxwell, 1867):

Viscosity of a gas is independent of pressure

inversely proportional to cross-section

scales as square of temperature

Liquids (Frenkel, 1926):

W is the “activation energy” In practice, A and W are chosen to fit data

A viscosity bound conjecture

P.Kovtun, D.Son, A.S., hep-th/0309213, hep-th/0405231

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

Classification of fluctuations and universality

O(2) symmetry in x-y plane

Scalar channel:

Shear channel:

Sound channel:

Other fluctuations (e.g. ) may affect sound channel

But not the shear channel universality of

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:

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

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

Friction in Newton’s equation:

Friction in Euler’s equations

What is viscosity?