Transport coefficients from string theory: an update

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Transport coefficients
from string theory
an update

• Andrei Starinets
• Perimeter Institute

Wien 2005 workshop
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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
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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.

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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

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Timeline and status report

• 2001 shear viscosity for N4 SYM computed
• 2002 prescription to compute thermal correlators
  from gravity formulated and applied to N4 SYM
  shear and sound poles in correlators are found
• 2002-03 other poles in N4 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).

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What is hydrodynamics?
Hierarchy of times (example)
0
t




Mechanical description
Kinetic theory
Hydrodynamic approximation
Equilibrium thermodynamics
Hierarchy of scales
(L is a macroscopic size of a system)
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Holography and hydrodynamics
Gravitational fluctuations
Deviations from equilibrium
Dispersion relations
Quasinormal spectrum
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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
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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
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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
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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.
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Universality of shear viscosity in the regime
described by gravity duals
Gravitons component obeys equation for a
minimally coupled massless scalar. But then
.
Since the entropy (density) is
we get
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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

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

• W is the activation energy

• In practice, A and W are chosen to fit data

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

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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
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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

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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

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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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What is viscosity?
Friction in Newtons equation
Friction in Eulers equations