1、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:,hep-th/0205051 hep-th/0205052 hep-th/0302026
2、 hep-th/0309213 hep-th/0405231 hep-th/0406124 hep-th/0506144 hep-th/0506184 hep-th/0507026,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
3、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 co
4、efficients 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 resul
5、ts 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 computed2002: prescription to compute thermal correlators from gravity formulated and applied to N=4 SYM; shear and sound poles in correl
6、ators are found 2002-03: other poles in N=4 SYM correlators identified with quasinormal spectrum in gravity2003-04: universality of shear viscosity; general formula for diffusion coefficient from “membrane paradigm”; correction to shear viscosity2004-05: general prescription for computing transport
7、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 hydro
8、dynamics?,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 spe
9、ctrum,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) stri
10、ngs:,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 coef
11、ficients from “first principles”,Kubo formulae allows one to calculate transportcoefficients from microscopic models,In the regime described by a gravity dual the correlator can be computed usingAdS/CFT,Fluctuation-dissipation theory (Callen, Welton, Green, Kubo),Universality of,Theorem:,For any the
12、rmal 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.,Universality of shear viscosity in the regime described
13、 by gravity duals,Gravitons 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/0405231Direct computation of the correlator in Kubo formula from AdS/C
14、FT 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 theoremP. Kovtun, D.Son, A.S., hep-th/0309213, A.S., to appear,P.Kovtun, A.S., hep-th/0506184, A.Buchel, J.Liu, h
15、ep-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,Liqu
16、ids (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
17、 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 fluctuation
18、s (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:,Relatio
19、n 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 arepotentially interesting,Epilog
20、ue,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 Newtons equation:,Friction in Eulers equations,What is viscosity?,
