Masaki ShigemoriUniversity of Amsterdam

Tenth Workshop on Non-Perturbative QCDl’Institut d’Astrophysique de Paris

Paris, 11 June 2009

Brownian Motion in AdS/CFT

2

J. de Boer, V. E. Hubeny, M. Rangamani, M.S., “Brownian motion in AdS/CFT,” arXiv:0812.5112.

A. Atmaja, J. de Boer, K. Schalm, M.S., work in progress.

This talk is based on:

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Intro / Motivation

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AdS/CFT and fluid-gravityAdS CFT

black hole inquantum gravity

horizon dynamicsin classical GR

plasma in stronglycoupled QFT

hydrodynamicsNavier-Stokes eq.

Long-wavelength approximation

difficult

easier;better-understood

Bhattacharyya+Minwalla+Rangamani+Hubeny 0712.2456

?

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Hydro: coarse-grained

Macrophysics vs. microphysics

BH in classical GR is also macro, approx. description of underlying microphysics of QG BH!

Can’t study microphysics within hydro framework(by definition)

want to go beyond hydro approx

coarsegrain

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― Historically, a crucial step toward microphysics of nature

1827 Brown

Due to collisions with fluid particlesAllowed to determine Avogadro #:UbiquitousLangevin eq. (friction + random force)

Brownian motion

Robert Brown (1773-1858)

erratic motion

pollen particle

𝑁𝐴 = 6× 1023 < ∞

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Do the same for hydro. in AdS/CFT!Learn about QG from BM on boundaryHow does Langevin dynamics come about

from bulk viewpoint?Fluctuation-dissipation theoremRelation to RHIC physics?

Brownian motion in AdS/CFT

Related work:drag force: Herzog+Karch+Kovtun+Kozcaz+Yaffe, Gubser, Casalderrey-Solana+Teaneytransverse momentum broadening: Gubser, Casalderrey-Solana+Teaney

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Preview: BM in AdS/CFT

horizon

AdS boundaryat infinity

fundamentalstring

black hole

endpoint =Brownian

particleBrownian

motion

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Intro/motivation BM BM in AdS/CFT Time scales BM on stretched horizon

Outline

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Brownian motion Paul Langevin (1872-1946)

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Langevin dynamics

𝑝ሶሺ𝑡ሻ= −න 𝑑𝑡′𝑡−∞ 𝛾ሺ𝑡− 𝑡′ሻ𝑝ሺ𝑡′ሻ+ 𝑅(𝑡)

Generalized Langevin eq: 𝑥,𝑝= 𝑚𝑥ሶ

delayed

friction

random force

=ۄ����������𝑅ሺ𝑡ሻۃ������� 0, =ۄ����������𝑅ሺ𝑡ሻ𝑅ሺ𝑡′ሻۃ������� 𝜅(𝑡− 𝑡′)

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General properties of BMDisplacement:

≡ۄ����������𝑠2ሺ𝑡ሻۃ������� −ሾ𝑥ሺ𝑡ሻۃ������� 𝑥ሺ0ሻሿ2ۄ����������

diffusive regime(random walk)

ballistic regime(init. velocity )

𝑥ሶ~ඥ𝑇/𝑚 ≈൞

𝑇𝑚𝑡2 (𝑡 ≪𝑡𝑟𝑒𝑙𝑎𝑥)2𝐷𝑡 (𝑡 ≫𝑡𝑟𝑒𝑙𝑎𝑥)

𝐷≡ 𝑇𝛾0𝑚, 𝛾0 = න 𝑑𝑡∞0 𝛾(𝑡) diffusion

constant

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Time scales𝑡relax ≡ 1𝛾0 , 𝛾0 = න 𝑑𝑡∞0 𝛾(𝑡)

∽ۄ����������𝑅ሺ𝑡ሻ𝑅ሺ0ሻۃ������� 𝑒−𝑡/𝑡coll

Relaxation timeCollision duration time

Mean-free-path time

time elapsed in a single collision

𝑡coll

𝑡mfp

Typically𝑡relax ≫𝑡mfp ≫𝑡coll but not necessarily sofor strongly coupled plasma

R(t)

t𝑡mfp

𝑡coll

time between collisions

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BM in AdS/CFT

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AdS Schwarzschild BH

Bulk BM

𝑑𝑠𝑑2 = 𝑟2𝑙2 ൫−ℎሺ𝑟ሻ𝑑𝑡2 + 𝑑𝑋Ԧ𝑑−22 ൯

+ 𝑙2𝑑𝑟2𝑟2ℎ(𝑟)

ℎሺ𝑟ሻ= 1−ቀ𝑟𝐻𝑟ቁ𝑑−1

𝑇= 1𝛽 = ሺ𝑑− 1ሻ𝑟𝐻4𝜋𝑙2

horizon

AdS boundaryat infinity

fundamentalstring

black hole

endpoint =Brownian

particleBrownian

motion

r

𝑋Ԧ𝑑−2

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Horizon kicks endpoint on horizon(= Hawking radiation)

Fluctuation propagates toAdS boundary

Endpoint on boundary(= Brownian particle) exhibits BM

Physics of BM in AdS/CFT

horizon

boundary

endpoint =Brownian

particleBrownian

motion

r

𝑋Ԧ𝑑−2

transverse fluctuation

kick

Whole process is dual to quark hit by QGP particles

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BM in AdS/CFT

horizon

boundary

r

𝑋Ԧ𝑑−2 Probe approximationSmall gs

No interaction with bulk The only interaction is at

horizonSmall fluctuation

Expand Nambu-Goto actionto quadratic order

Transverse positionsare similar to Klein-Gordon scalars

𝑋Ԧ𝑑−2(𝑡,𝑟)

𝑋ሺ𝑡,𝑟ሻ

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Quadratic action

Brownian string

𝑆2 = − 14𝜋𝛼′ න𝑑𝑡 𝑑𝑟ቈ𝑋ሶ2ℎሺ𝑟ሻ− 𝑟4ℎሺ𝑟ሻ𝑙4 𝑋′2

ቈ𝜔2 + ℎሺ𝑟ሻ𝑙4 𝜕𝑟ሺ𝑟4ℎሺ𝑟ሻ𝜕𝑟ሻ 𝑓𝜔ሺ𝑟ሻ= 0

𝑋ሺ𝑡,𝑟ሻ= න 𝑑𝜔∞0 ൫𝑓𝜔ሺ𝑟ሻ𝑒−𝑖𝜔𝑡𝑎𝜔 + h.c.൯

d=3: can be solved exactlyd>3: can be solved in low frequency limit

𝑋ሺ𝑡,𝑟ሻ

Mode expansion

𝑆NG = const + 𝑆2 + 𝑆4 + ⋯

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Near horizon:

Bulk-boundary dictionary

𝑋ሺ𝑡,𝑟cሻ≡ 𝑥(𝑡) = න 𝑑𝜔∞0 �𝑓𝜔(𝑟𝑐)𝑒−𝑖𝜔𝑡𝑎𝜔 + h.c.൧

𝑋ሺ𝑡,𝑟ሻ∼න𝑑𝜔ξ2𝜔∞

0 �൫𝑒−𝑖𝜔(𝑡−𝑟∗) + 𝑒𝑖𝜃𝜔𝑒−𝑖𝜔(𝑡+𝑟∗)൯𝑎𝜔 + h.c.൧ outgoin

gmode

ingoingmode

phase shift

𝑟∗

: tortoise coordinate

𝑟c : cutoff

⋯𝑥ሺ𝑡1ሻ𝑥ሺ𝑡2ሻۃ������� ↔ۄ���������� †𝑎𝜔1𝑎𝜔2ۃ������� ⋯ ۄ����������

observe BMin gauge theory

correlator ofradiation

modesCan learn about quantum gravity in principle!

Near boundary

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Semiclassically, NH modes are thermally excited:

Semiclassical analysis

†𝑎𝜔𝑎𝜔ۃ������� ∝ۄ���������� 1𝑒𝛽𝜔 − 1

Can use dictionary to compute x(t), s2(t) (bulk boundary)

𝑠2(𝑡) ≡ ⟨:ሾ𝑥ሺ𝑡ሻ− 𝑥ሺ0ሻሿ2:⟩ ≈ە��۔����

ۓ������������������

𝑇𝑚𝑡2 (𝑡 ≪𝑡𝑟𝑒𝑙𝑎𝑥)𝛼′𝜋𝑙2𝑇𝑡 (𝑡 ≫𝑡𝑟𝑒𝑙𝑎𝑥)

𝑡𝑟𝑒𝑙𝑎𝑥 ∼ 𝛼′𝑚𝑙2𝑇2

ballistic

diffusive

Does exhibitBrownian motion

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Time scales

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Time scales

R(t)

t𝑡mfp

𝑡coll

𝑡relax 𝑡mfp 𝑡coll

information about plasma

constituents

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Time scales from R-correlatorsSimplifying assumptions:

𝑅ሺ𝑡ሻ= 𝜖𝑖𝑓ሺ𝑡− 𝑡𝑖ሻ𝑘

𝑖=1 𝑓(𝑡) : shape of a single

pulse　　𝜖𝑖 = ±1 : random sign

𝜇 : number of pulses per unit time,

R(t) : consists of many pulses randomly distributed

Distribution of pulses = Poisson distribution ∼ 1/𝑡mfp

𝜖1 = 1 𝜖2 = 1

𝜖3 = −1

𝑓(𝑡− 𝑡1) 𝑓(𝑡− 𝑡2)

−𝑓(𝑡− 𝑡3)

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Time scales from R-correlators

Can determine μ, thus tmfp

=ۄ����������𝑅෨ሺ0ሻ2ۃ������� 𝜇𝑇𝑓ሚሺ0ሻ2 =ۄ����������𝑅෨ሺ0ሻ4ۃ������� 2ۄ����������𝑅෨ሺ0ሻ2ۃ�������3 + 𝜇𝑇𝑓ሚሺ0ሻ4 𝑇≡ 2𝜋𝛿ሺ0ሻ, tilde = Fourier

transform

2-pt func

Low-freq. 4-pt func

𝑡coll→ۄ����������𝑅ሺ𝑡ሻ𝑅ሺ0ሻۃ�������

𝑡mfp→ۄ����������𝑅෨ሺ𝜔1ሻ𝑅෨ሺ𝜔2ሻ𝑅෨ሺ𝜔3ሻ𝑅෨ሺ𝜔4ሻۃ�������

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Sketch of derivation

𝑃𝑘ሺ𝜏ሻ= 𝑒−𝜇𝜏ሺ𝜇𝜏ሻ𝑘𝑘!

Probability that there are k pulses in period [0,τ]:

𝑅ሺ𝑡ሻ= 𝜖𝑖𝑓ሺ𝑡− 𝑡𝑖ሻ𝑘

𝑖=1

=ۄ����������𝑅ሺ𝑡ሻ𝑅(𝑡′)ۃ������� 𝑃𝑘ሺ𝜏ሻ∞𝑘=1 −𝜖𝑖𝜖𝑗𝑓ሺ𝑡ۃ������� 𝑡𝑖ሻ𝑓(𝑡− 𝑡𝑗)ۄ����������𝑘

𝑘𝑖,𝑗=1

𝜖𝑖 = ±1 ∶ random signs → =ۄ����������𝜖𝑖𝜖𝑗ۃ������� 𝛿𝑖𝑗

0

k pulses

𝑡1 𝑡2 𝑡𝑘 … 𝜏

(Poisson dist.)

−𝑓ሺ𝑡ۃ������� 𝑡𝑖ሻ𝑓ሺ𝑡′ − 𝑡𝑖ሻۄ����������𝑘 = 𝑘𝜏න 𝑑𝑢𝜏0 𝑓ሺ𝑡− 𝑢ሻ𝑓(𝑡′ − 𝑢)

2-pt func:

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Sketch of derivation=ۄ����������𝑅ሺ𝑡ሻ𝑅(𝑡′)ۃ������� 𝜇න 𝑑𝑢∞

−∞ 𝑓ሺ𝑡− 𝑢ሻ𝑓(𝑡′ − 𝑢)

=ۄ����������𝑅෨ሺ𝜔ሻ𝑅෨(𝜔′)ۃ������� 2𝜋𝜇𝛿ሺ𝜔+ 𝜔′ሻ𝑓ሚሺ𝜔ሻ𝑓ሚ(𝜔′)

Similarly, for 4-pt func,

+2𝜋𝜇𝛿ሺ𝜔+ 𝜔′ + 𝜔′′ + 𝜔′′′ ሻ𝑓ሚሺ𝜔ሻ𝑓ሚሺ𝜔′ሻ𝑓ሚሺ𝜔′′ሻ𝑓ሚሺ𝜔′′′ሻ ۄ����������𝑅෨ሺ𝜔ሻ𝑅෨(𝜔′)𝑅෨(𝜔′′)𝑅෨(𝜔′′′)ۃ�������

ሺ2 more termsሻ+ۄ����������𝑅෨ሺ𝜔ሻ𝑅෨ሺ𝜔′ሻ⟩ ⟨𝑅෨(𝜔′′)𝑅෨(𝜔′′′)ۃ������� =ۄ����������𝑅෨ሺ0ሻ2ۃ������� 𝜇𝑇𝑓ሚሺ0ሻ2

=ۄ����������𝑅෨ሺ0ሻ4ۃ������� 2ۄ����������𝑅෨ሺ0ሻ2ۃ�������3 + 𝜇𝑇𝑓ሚሺ0ሻ4

𝑇≡ 2𝜋𝛿ሺ0ሻ

“disconnected part”

“connected

part”=

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R-correlators from BM in AdS/CFT

𝑋ሺ𝑡,𝑟ሻ

Expansion of NG action to higher order:

𝑆NG = const + 𝑆2 + 𝑆4 + ⋯

Can compute tmfp from correction to 4-pt func.

𝑆4 = 116𝜋𝛼′ න𝑑𝑡 𝑑𝑟ቈ𝑋ሶ2ℎሺ𝑟ሻ− 𝑟4ℎሺ𝑟ሻ𝑙4 𝑋′22

c Can computeۄ����������𝑅𝑅𝑅𝑅ۃ�������and thus tmfp

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Times scales from AdS/CFT

conventional kinetic theory is good

𝑡relax ~ 𝑚ξ𝜆 𝑇2 𝑡coll ~1𝑇

Resulting timescales:

𝜆= 𝑙4𝛼′2

weak coupling𝜆≪1 𝑡relax ≫𝑡mfp ≫𝑡coll

𝑡mfp ~ 1ξ𝜆 𝑇

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Times scales from AdS/CFT

Multiple collisions occur simultaneously.

𝑡relax ~ 𝑚ξ𝜆 𝑇2 𝑡coll ~1𝑇

Resulting timescales:

𝜆= 𝑙4𝛼′2

strong coupling𝜆≫1

𝑡mfp ~ 1ξ𝜆 𝑇

𝑡mfp ≪𝑡coll 𝑡relax ≪𝑡coll is also possible.

.

Cf. “fast scrambler”

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BM on stretched horizon

(Jorge’s talk)

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Conclusions

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Boundary BM ↔ bulk “Brownian string”can study QG in principle

Semiclassically, can reproduce Langevin dyn. from bulk

random force ↔ Hawking rad. (kicking by horizon)

friction ↔ absorptionTime scales in strong coupling QGP:BM on stretched horizon (Jorge’s talk)Fluctuation-dissipation theorem

Conclusions

𝑡relax ,𝑡mfp ,𝑡coll

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Thanks!