transport coefficients from string theory: an update andrei starinets perimeter institute wien 2005...
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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
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
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