Viscosity, quark gluon plasma, and string theory

Viscosity,quark gluon plasma,

and string theory

D. T. Son

Institute for Nuclear Theory, University of Washington

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PlanViscosity

Heavy ion collisions and the quark gluon plasma

Gauge/gravity duality: a surpising by-product of string theory

Viscosity through gauge/gravity duality

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ViscosityViscosity: introduced by Claude Navier in 1822 into what would be later called theNavier-Stokes equation.

Friction force between two plates:

F = ηA∂ux

∂y

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Viscosity and kinetic theory of gas

Maxwell: in the kinetic theory ofgases, viscosity arises from col-lisions between gas molecules.

Maxwell’s estimate of the viscosity:

η ∼ ρvℓ = mass density × velocity × mean free path

Consequence: at fixed temperature viscosity is independent of pressure (density).Contradicts expectation at the time: the denser the gas, the larger the viscosity

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Viscosity and kinetic theory of gas

Maxwell: in the kinetic theory ofgases, viscosity arises from col-lisions between gas molecules.

Maxwell’s estimate of the viscosity:

η ∼ ρvℓ = mass density × velocity × mean free path

Consequence: at fixed temperature viscosity is independent of pressure (density).Contradicts expectation at the time: the denser the gas, the larger the viscosity

“Such a consequence of a mathematical theory is very startling, and the onlyexperiment I have met with on the subject does not seem to confirm it” (1860)

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Maxwell’s own experiment

During the next few years Maxwell, with the help of hiswife, designed and carried out his own expriment.

Reported result in 1865: viscosity of air is independentof pressure when the latter varies from about 1/60 to oneatmosphere.

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A counter-intuitive behaviorImagine one can turn off the interaction between molecules of a gas

According to Maxwell’s formula: viscosity → ∞ (large mean free path)

Shouldn’t one expect no disspation, η → 0?

Reason: two limits do not commute:

The limit of infinitely weak interaction

The hydrodynamic limit (lengths ≫ mean free path)

Viscosity is well defined only when size of experiment ≫ mean free path

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Viscosity of liquids: a huge rangeViscosity of fluids is much poorer undertsood ! " # $ % & ' ( ) * + , - , . / , 0 ) ( % 0 , 1 2 ' + 3 ' 4 ( 5 6 + / 78 2 0 9 : ; < = = 9 :> ) ( , 0 = ? < 9 :> ) ( , 0 @ = A < 9> ) ( , 0 9 = = ? < = @ :B C 5 + , 0 4 * @ = ; < 9 D EF , 0 + % 0 5 @ = ; < 9 GH I J , * ( ) * , @ = K < = @ L8 0 M 3 * : D N = @ :O , P Q @ N = = L L$ % / , 0 R % S T O , P U @ 9 N =B C ) ' ' V 9 = 4 WX 3 ( , ( Y ) ( Z & 5 / 3 / % C ) 0 + 3 * [ , * ( 4 \ 3 * ] ( Y , T , ' 4 M * ) ( S 3 *^ _ M C ) ' ' ` a 4 ' ) / / C S , T ( 3 ) * 5 T b ' 3 0 T , 0 , T . ) ( , 0 b ) C 3 * + ,4 ( ' [ S ' + 3 ' 4 ( 5 , c + , , T ' 9 = 4 W + /

d

efgh

(from Chaikin & Lubensky, Principles of Condensed Matter Physics)

Viscosity spans many orders of magnitudes, despite the fact that the density doesnot change much.

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Viscosity of liquid glycerinT (C) η (mPa s)

0 12070

10 3900

20 1410

30 612

40 284

50 142

60 81.3

70 50.6

80 31.9

100 14.8

120 7.8

140 4.7

167 2.8

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Liquid viscosity can be very largeThe pitch-drop experiment, University of Queensland

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8 drops since 1930

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8 drops since 1930

No one has witnessed a drop fall

Viscosity ∼ 1011 times of water

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8 drops since 1930

No one has witnessed a drop fall

Viscosity ∼ 1011 times of water

2005 Ig Nobel Prize in Physics

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Purcell and WeisskopfE. Purcell was very puzzled by the fact that liquid viscosity spans such ahuge range

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He thought about challenging Weisskopf to explain this fact

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Am. J. Phys. 45, 3 (1977).

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Am. J. Phys. 45, 3 (1977).

but he anticipated that Weisskopf would say

η ∼ exp(#)

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Am. J. Phys. 45, 3 (1977).

but he anticipated that Weisskopf would say

η ∼ exp(#)

So he added

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Purcell’s question

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Purcell’s question

Viscosity of liquids is unbounded from above, but seems to be bounded from below

A version of Purcell’s question: By playing with the interaction strength, can onemake η → 0 ?

Modern context: the quark gluon plasma

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Heavy ion collisionsRelativistic Heavy Ion Collider (RHIC) at Brookhaven National Lab;

Heavy ion experiments at LHC

RHIC: 200 GeV/pair of nucleons; LHC: 2.76 TeV/pair

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A typical heavy ion collision

Goal: to create and study the quark gluon plasma

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The quark gluon plasmaHadrons consists of quarks (3 quarks, or 1 quark and 1 antiquark)

mesons (pions etc)protons, neutrons

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The quark gluon plasmaHadrons consists of quarks (3 quarks, or 1 quark and 1 antiquark)

mesons (pions etc)protons, neutrons

hadron gas quark−gluon plasma

Quark-gluon plasma: quarks and gluons move relatively freely

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Phases of QCDOne version of the QCD phase diagram:

250 500 750 1000 1250 1500 1750 2000Baryon chemical potential @MeVD

25

50

75

100

125

150

175

200T

empe

ratu

re@M

eVD

Quark-gluon plasma

Hadron phase 2SC

NQCFL

(Ruester et al., hep-ph/0503184)

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Strongly interacting QGPUnambiguous identification of QGP difficult

However, RHIC has created some strongly interacting mediumSupression of back-to-back jet correlations:

“Elliptic flow”: signals that particles push each other (more later)

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Can the viscosity be calculated?Good: fundamental theory known: Quantum Chromodynamics (QCD)

Bad: perturbation theory does not work

!

!"#

!"$

!"%

# #! #!$

µ&'()

α*+µ,

Summary of the values of ( ) at the values of where they are

0.2

We are here!

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Convergence of perturbation series

1.5 2 2.5 3 3.5 4 4.5 5

0.6

0.8

1

1.2

1.4

1.5 2 2.5 3 3.5 4 4.5 5

0.6

0.8

1

1.2

1.4

T=T

P=P0g2

g3 g4g5

P/P0|T=2Tc= 1 − 0.2 + 0.4 + 0.2 − 0.8 + · · ·

Kinetic coefficents: worse convergence expected

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So what to do?Measure the viscosity experimentally, or

Try a different (simpler theory)

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The Gauge/Gravity DualityMaldacena 1997: stack of N D3-branes in type IIB string theory can be describedin two different pictures:

As a quantum field theorydescribing fluctuations of thebranes: N = 4 super-Yang-Millstheory

As string theory on a a curvedspacetime called AdS5×S5

ds2 =r2

R2(−dt2+dx2)+

R2

r2dr2+R2dΩ2

5

N

=

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HolographyGauge/gravity duality: most concrete relalization of the idea of holography (’tHooft, Susskind): a theory without gravity in (3+1)D is equivalent a theory withgravity in higher dimensions

(from Maldacena, Nature 2003)Toronto 2011 – p.21/39

Mapping of parameters

Rl s

g2Nc =R4

ℓ4s

g2Nc ≫ 1: string theory → Einstein’s gravity

Difficult regime in field theory = easy regime in string theory

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Gauge/gravity duality at finite temperature

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Gauge/gravity duality at finite temperature

=

“Quark gluon plasma” = black hole (in anti de-Sitter space)

Gives entropy of plasma at strong coupling (Gubser, Klebanov, Peet 1996)

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A theorist’s way of measuring viscosityHow do we measure the viscosity of a system?

Viscosity = response of the fluid under shear

Theorist: send gravitational wave through the system

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A theorist’s way of measuring viscosityHow do we measure the viscosity of a system?

Viscosity = response of the fluid under shear

Theorist: send gravitational wave through the system

This is the essence of Kubo’s formula

η = limω→0

1

2ω

Z

dt dxeiωt〈[T xy(t,x), T xy(0,0)]〉

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Viscosity on the light of dualityConsider a graviton that falls on this stack of N D3-branesWill be absorbed by the D3 branes.The process of absorption can be looked at from two different perspectives:

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Viscosity on the light of dualityConsider a graviton that falls on this stack of N D3-branesWill be absorbed by the D3 branes.The process of absorption can be looked at from two different perspectives:

=

Absorption by D3 branes (∼ viscosity) = absorption by black hole

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Viscosity/entropy density ratioCollecting facts so far:

Viscosity ∼ absorption cross section for low energy gravitons

Absorption cross section = area of horizon (a consequence of Einsteinequation)

Entropy: also ∼ area of horizon (Bekenstein-Hawking formula)

η

s=

1

4π

where η is the shear viscosity, s is the entropy per unit volume.

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Universality of η/sRestoring ~ and c in the formula: a surprise

η

s=

~

4π

No velocity of light c

Finite viscosity at infinitely strong coupling!

The same value in all theories with gravity duals Kovtun, DTS, Starinets 2003

Universality of η/s is related to properties of black hole horizons

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Why doesη/s have the dimensionality of~In kinetic theory

η ∼ ρvℓ, s ∼ n =ρ

m

η

s∼ mvℓ ∼ ~

mean free path

de Broglie wavelength

In kinetic theory, mean free path cannot be ≪ de Broglie wavelength.

Uncertainty principle: η/s bounded from below by #~, unknown coefficient

Theories with gravity dual reach ~/4π

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Why doesη/s have the dimensionality of~In kinetic theory

η ∼ ρvℓ, s ∼ n =ρ

m

η

s∼ mvℓ ∼ ~

mean free path

de Broglie wavelength

In kinetic theory, mean free path cannot be ≪ de Broglie wavelength.

Uncertainty principle: η/s bounded from below by #~, unknown coefficient

Theories with gravity dual reach ~/4π

Conjecture Kovtun, DTS, Starinets 2003

η

s&

~

4π

Reminds Purcell’s question to WeisskopfNo c: ~/(4π) can be compared with ordinary, nonrelativistic liquids

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Ordinary liquidsη/s of liquid water in unit of ~/(4π), as function of temperature (K), along thesaturation curve

10

100

1000

250 300 350 400 450 500 550 600 650

Liquid water

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General observation on liquidsη/s reaches minimum near the critical point (liquid=gas) Kapusta, McLerran

but not exactly at the critical point: η diverges there according to theory ofdynamic critical phenomena

The minimal value of η/s varies from substance to substance

Substance`

η

s

´

minSubstance

`

η

s

´

minSubstance

`

η

s

´

min

H, He 8.8Ne 17 H2O 25 CO 35Ar 37 H2S 35 CO2 32Kr 57 N2 23 SO2 39Xe 84 O2 28

(η/s is measured in unit of ~/(4π))

Minimum among substances is reached by the most quantum liquids: hydrogenand helium

Superfluids: normal component finite shear viscosity (Andronikashviliexperiments)

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determining viscosity of QGP

Collisions with nonzero impactparameter

Distribution of particles overmomentum is not axially symmetric:characterized by “elliptic flow”parameter v2

explanation: pressure gradientdepends on angle

Viscosity reduces v2

Hydrodynamic simulations can give estimates for η

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Recent numerical simulations

0 1 2 3 4p

T [GeV]

0

5

10

15

20

25

v 2 (p

erce

nt)

STAR non-flow corrected (est.)STAR event-plane

Glauber

η/s=10-4

η/s=0.08

η/s=0.16

0 1 2 3 4p

T [GeV]

0

5

10

15

20

25

v 2 (p

erce

nt)

STAR non-flow corrected (est).STAR event-plane

CGCη/s=10

-4

η/s=0.08

η/s=0.16

η/s=0.24

(from Luzum and Romatschke, arXiv:0804. LHC data seem to prefer Glauberinitial conditions.)

η

s= 0.1 ± 0.1(th) ± 0.08(exp)

Not too far way from 1/4π

QGP is strongly coupled (sQGP)

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High or low viscosity?Brookhaven National Lab press release 2005: “the degree of collective interaction,rapid thermalization, and extremely low viscosity of the matter being formed atRHIC make this the most nearly perfect liquid ever observed.

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Estimating the viscosity of the QGP:

η ∼~

4πs ∼

10−27erg · s

(10−13cm)3∼ 1014

cp

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η ∼~

4πs ∼

10−27erg · s

(10−13cm)3∼ 1014

cp

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ÎÏÐÑ

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η ∼~

4πs ∼

10−27erg · s

(10−13cm)3∼ 1014

cp

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789:In absolute terms, QGP is almost as viscous as glass!

Low viscosity: in the sense of small η/s, suggested by gauge/gravity duality

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Condensed matter analogyMott’s minimal metallic conductivity hypothesis: conductivity is bounded frombelow by uncertainty principle

(Phillips, Advanced Solid State Physics)

Mott’s minimal metallic conductivity is a useful concept, but not a real, absoluteminimum.

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Is there a viscosity bound?η

s≥ ~

4πfor all fluids? Kovtun, DTS, Starinets 2005

There exist theories where

η

s=

~

4π

“

1 −c

N

”

, N ≫ 1

Kats, Petrov; Buchel, Myers, Sinha

No reliable way to go to small N .

Model studies: bad things happen before η/s reaches 0 Brigante etc. PRL 2008

Within a model: causality is violated when η/s > 16

25

~

4π

What is the real minimum of η/s ?

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Further applicationsChiral separation in quark matter:

Rotating volume of quark matter with nonzero chemical potential

Left and right handed quarks separate along the axis of rotation

Seen in gauge-gravity duality, and understood to be general feature of chiralrelativistic fluids

Originates from quantum anomalies

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ConclusionSurprising applications of string theory to real-world problems

New perspective on old problemsQGP = black holeviscosity = gravity absorptionetc

Suggested η/s as a relevant ratio

Suggested the value ~/(4π) as particularly interesting.

Most serious challenge: connect to the correct theory of strong interactions, QCD

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The unity of physicsWe have also learned a lesson that physics is one single subject:

Hydrodynamics

Quantum field theory

Statistical mechanics

Gravity

String theory

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The End

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Viscosity, quark gluon plasma, and string theory

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