# Viscosity, quark gluon plasma, and string theory

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

8 drops since 1930

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

8 drops since 1930

No one has witnessed a drop fall

Viscosity ∼ 1011 times of water

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

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

He thought about challenging Weisskopf to explain this fact

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

He thought about challenging Weisskopf to explain this fact

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

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He thought about challenging Weisskopf to explain this fact

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

but he anticipated that Weisskopf would say

η ∼ exp(#)

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He thought about challenging Weisskopf to explain this fact

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

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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

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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

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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:

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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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=

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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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.

Estimating the viscosity of the QGP:

η ∼~

4πs ∼

10−27erg · s

(10−13cm)3∼ 1014

cp

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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.

Estimating the viscosity of the QGP:

η ∼~

4πs ∼

10−27erg · s

(10−13cm)3∼ 1014

cp

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ÎÏÐÑ

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Estimating the viscosity of the QGP:

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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