Brownian Motion in AdS/CFTYITP Kyoto, 24 Dec 2009

Masaki Shigemori (Amsterdam)

2

This talk is based on:

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

A. Atmaja, J. de Boer, M.S., in preparation.

3

Introduction / Motivation

4

thermodynamics

Hierarchy of scales in physics

hydrodynamics

atomic theorysubatomic

theory

large

small

Scale

macrophysics

microphysics

coarse-grain

Brownian motion

5

AdS BHin classical GR

Hierarchy in AdS/CFTAdS CFT

[Bhattacharyya+Minwalla+

Rangamani+Hubeny]

thermodynamics of plasma

horizon dynamics

in classical GR

hydrodynamics

Navier-Stokes eq.

quantum BHstrongly coupled quantum plasma

large

small

Scale

long-wavelength approximation

macro-physics

micro-physics

?

This

talk

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Brownian motion― Historically, a crucial step toward microphysics of nature

1827 Brown

Due to collisions with fluid particles Allowed to determine Avogadro #: NA= 6x1023 < ∞ Ubiquitous Langevin eq. (friction + random force)

Robert Brown (1773-1858)

erratic motion

pollen particle

7

Brownian motion in AdS/CFT Do the same in AdS/CFT!

Brownian motion of an external quark in CFT plasma

Langevin dynamics from bulk viewpoint? Fluctuation-dissipation theorem Read off nature of constituents of strongly

coupled plasma Relation to RHIC physics?

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 particle Brownian motion

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Outline

Intro/motivation Boundary BM Bulk BM Time scales BM on stretched horizon

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Boundary BM

- Langevin dynamics

Paul Langevin (1872-1946)

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Simplest Langevin eq.

𝑥,𝑝= 𝑚𝑥ሶ 𝑝ሶሺ𝑡ሻ= −𝛾0𝑝ሺ𝑡ሻ+ 𝑅ሺ𝑡ሻ (instantaneo

us)friction

random force

=ۄ����������𝑅ሺ𝑡ሻۃ������� 0, =ۄ����������𝑅ሺ𝑡ሻ𝑅ሺ𝑡′ሻۃ������� 𝜅0𝛿(𝑡− 𝑡′) white noise

12

Simplest Langevin eq.

diffusive regime(random walk)

≈൞

𝑇𝑚𝑡2 (𝑡 ≪𝑡𝑟𝑒𝑙𝑎𝑥 )2𝐷𝑡 (𝑡 ≫𝑡𝑟𝑒𝑙𝑎𝑥 )

• Diffusion constant:

𝐷≡ 𝑇𝛾0𝑚

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

ballistic regime(init. velocity )

𝑥ሶ~ඥ𝑇/𝑚

Displacement:

𝑡relax = 1𝛾0 • Relaxation time:

(S-E relation) 𝛾0 = 𝜅02𝑚𝑇 FD theorem

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Generalized Langevin equation

𝑝ሶሺ𝑡ሻ= −න 𝑑𝑡′𝑡−∞ 𝛾ሺ𝑡− 𝑡′ሻ𝑝ሺ𝑡′ሻ+ 𝑅ሺ𝑡ሻ+ 𝐹(𝑡)

delayed

friction

random force

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

external force

Qualitatively similar to simple LE ballistic regime diffusive regime

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Time scales Relaxation time𝑡relax ≡ 1𝛾0 , 𝛾0 = න 𝑑𝑡∞

0 𝛾(𝑡)

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

Collision duration time

Mean-free-path time

How to determine γ, κ

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1. Forced motion

2. No external force,

𝑝ሺ𝜔ሻ= 𝑅ሺ𝜔ሻ+ 𝐹ሺ𝜔ሻ𝛾ሾ𝜔ሿ− 𝑖𝜔 ≡ 𝜇ሺ𝜔ሻሾ𝑅ሺ𝜔ሻ+ 𝐹ሺ𝜔ሻሿ

𝐹ሺ𝑡ሻ= 𝐹0𝑒−𝑖𝜔𝑡

=ۄ����������𝑝ሺ𝑡ሻۃ������� 𝜇ሺ𝜔ሻ𝐹0𝑒−𝑖𝜔𝑡

=ۄ����������𝑝𝑝ۃ������� a2𝜇a22ۃ�������𝑅𝑅ۄ���������� measure

known

read off μ

read off κ

admittance

𝐹= 0

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Bulk BM

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Bulk setup AdSd Schwarzschild BH

(planar)𝑑𝑠𝑑2 = 𝑟2𝑙2 ൫−ℎሺ𝑟ሻ𝑑𝑡2 + 𝑑𝑋Ԧ𝑑−22 ൯+ 𝑙2𝑑𝑟2𝑟2ℎ(𝑟)

ℎሺ𝑟ሻ= 1−ቀ𝑟𝐻𝑟ቁ𝑑−1

𝑇= 1𝛽 = ሺ𝑑− 1ሻ𝑟𝐻4𝜋𝑙2

r

𝑋Ԧ𝑑−2

horizon

boundary

fundamentalstring

black hole

endpoint =Brownian

particleBrownian motion

𝑙:AdS radius

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Physics of BM in AdS/CFT

Horizon kicks endpoint on horizon(= Hawking radiation)

Fluctuation propagates toAdS boundary

Endpoint on boundary(= Brownian particle) exhibits BM

transverse fluctuationWhole process is dual to

quark hit by plasma particles

r

𝑋Ԧ𝑑−2 boundary

black hole

endpoint =Brownian

particleBrownian motion

kick

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Assumptions

𝑋Ԧ𝑑−2(𝑡,𝑟)

𝑋ሺ𝑡,𝑟ሻ r

𝑋Ԧ𝑑−2

horizon

boundary

Probe approximation Small gs

No interaction with bulk Only interaction is at horizon

Small fluctuation Expand Nambu-Goto action

to quadratic order Transverse positions are

similar to Klein-Gordon scalar

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Transverse fluctuations Quadratic action

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

𝑆NG = const+ 𝑆2 + 𝑆4 + ⋯

Mode expansion

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Bulk-boundary dictionaryNear horizon:

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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Semiclassical analysis Semiclassically, NH modes are thermally excited:

AdS3

†𝑎𝜔𝑎𝜔ۃ������� ∝ۄ���������� 1𝑒𝛽𝜔 − 1

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

𝑠2(𝑡) ≡ ⟨:ሾ𝑥ሺ𝑡ሻ− 𝑥ሺ0ሻሿ2:⟩ ≈VVە۔����

ۓ�������������������

𝑇𝑚𝑡2 (𝑡 ≪𝑡relax )𝛼′𝜋𝑙2𝑇𝑡 (𝑡 ≫𝑡relax )

𝑡relax ∼ 𝛼′𝑚𝑙2𝑇2

: ballistic: diffusiveDoes exhibit

Brownian motion

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Semiclassical analysis Momentum

distribution

Diffusion constant

Probability distribution for 𝑝= 𝑚𝑥ሶ 𝑓ሺ𝑝ሻ∝ exp൫−𝛽𝐸𝑝൯, 𝐸𝑝 = 𝑝22𝑚

Maxwell-Boltzmann

𝐷= ሺ𝑑− 1ሻ2𝛼′8𝜋𝑙2𝑇

Agrees with drag force computation[Herzog+Karch+Kovtun+Kozcaz+Yaffe]

[Liu+Rajagopal+Wiedemann][Gubser] [Casalderrey-Solana+Teaney]

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Forced motion Want to determine μ(ω), κ(ω)

― follow the two-step procedureTurn on electric field E(t)=E0e-iωt

on “flavor D-brane” and measure response ⟨p(t)⟩ E(t)p

freely infalling therm

al

b.c. changed from Neumann

“flavor D-brane”

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Forced motion: results (AdS3) Admittance

Random force correlator

FD theorem

𝜇ሺ𝜔ሻ= 1𝛾ሾ𝜔ሿ− 𝑖𝜔= 𝛼′𝛽2𝑚2𝜋 1− 𝑖𝜔/𝜋𝛼′𝑚1− 𝛼′𝑚𝛽2𝜔/2𝜋

𝜅ሺ𝜔ሻ= 4𝜋𝛼′𝛽3 1−ሺ𝛽𝜔/2𝜋ሻ21−ሺ𝜔/2𝜋𝛼′𝑚ሻ2 𝑡coll ∼ 1𝑇 (no λ)

2𝑚 Re(𝛾ሾ𝜔ሿ) = 𝛽𝜅ሺ𝜔ሻ satisfiedcan be proven

generally

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Time scales

27

Time scales

R(t)

t𝑡mfp

𝑡coll

𝑡relax

𝑡mfp 𝑡coll

information about plasma

constituents

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Time scales from R-correlators

R(t) : consists of many pulses randomly distributedA toy model:

𝑅ሺ𝑡ሻ= 𝜖𝑖𝑓ሺ𝑡− 𝑡𝑖ሻ𝑘

𝑖=1 𝑓(𝑡) : shape of a single

pulse　　𝜖𝑖 = ±1 : random sign

𝜇 : number of pulses per unit time,∼ 1/𝑡mfp

𝜖1 = 1 𝜖2 = 1

𝜖3 = −1

𝑓(𝑡− 𝑡1) 𝑓(𝑡− 𝑡2)

−𝑓(𝑡− 𝑡3) Distribution of pulses = Poisson distribution

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Time scales from R-correlators

Can determine μ, thus tmfp

≈ۄ����������𝑅෨(𝜔1)𝑅෨(𝜔2)ۃ������� 2𝜋𝜇𝛿(𝜔1 + 𝜔2)𝑓ሚሺ0ሻ2

connۄ����������𝑅෨ሺ𝜔1ሻ𝑅෨ሺ𝜔2ሻ𝑅෨ሺ𝜔3ሻ𝑅෨ሺ𝜔4ሻۃ������� ≈ 2𝜋𝜇𝛿(𝜔1 + 𝜔2 + 𝜔3 + 𝜔4)𝑓ሚሺ0ሻ4

tilde = Fourier transform

2-pt func

Low-freq. 4-pt func

𝑡coll→ۄ����������𝑅ሺ𝑡ሻ𝑅ሺ0ሻۃ�������

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

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Sketch of derivation (1/2)

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 (2/2)=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������� “disconnected part”

“connected

part”=

≈ۄ����������𝑅෨(𝜔1)𝑅෨(𝜔2)ۃ������� 2𝜋𝜇𝛿(𝜔1 + 𝜔2)𝑓ሚሺ0ሻ2

connۄ����������𝑅෨ሺ𝜔1ሻ𝑅෨ሺ𝜔2ሻ𝑅෨ሺ𝜔3ሻ𝑅෨ሺ𝜔4ሻۃ������� ≈ 2𝜋𝜇𝛿(𝜔1 + 𝜔2 + 𝜔3 + 𝜔4)𝑓ሚሺ0ሻ4

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⟨RRRR⟩ from bulk BM

Expansion of NG action to higher order: 𝑋ሺ𝑡,𝑟ሻ 𝑆NG = const+ 𝑆2 + 𝑆4 + ⋯

Can compute tmfp from connected to 4-pt func.

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

connۄ����������𝑅𝑅𝑅𝑅ۃ������� Can compute and thus tmfp

4-point vertex

33

Holographic renormalization [Skenderis]

Similar to KG scalar, but not quite

Lorentzian AdS/CFT [Skenderis + van Rees]

IR divergence

Near the horizon, bulk integral diverges

⟨RRRR⟩ from bulk BM

34

Reason: Near the horizon, local

temperature is high String fluctuates wildly Expansion of NG action breaks

down

Remedy: Introduce an IR cut off

where expansion breaks down Expect that this gives

right estimate of the full result

IR divergence

cutoff

35

Times scales from AdS/CFT (weak)

conventional kinetic theory is good

𝑡relax ~ 𝑚ξ𝜆 𝑇2 𝑡coll ~1𝑇

Resulting timescales:

𝜆≡ 𝑙4𝛼′2

weak coupling𝜆≪1 𝑡relax ≫𝑡mfp ≫𝑡coll

𝑡mfp ~ 1 𝑇logλ

36

Times scales from AdS/CFT (strong)

Multiple collisions occur simultaneously.

strong coupling𝜆≫1 𝑡mfp ≪𝑡coll

Cf. “fast scrambler”

[Hayden+Preskill] [Sekino+Susskind]

𝑡relax ~ 𝑚ξ𝜆 𝑇2 𝑡coll ~1𝑇

Resulting timescales:

𝜆≡ 𝑙4𝛼′2 𝑡mfp ~ 1 𝑇logλ

37

BM on stretched horizon

38

Membrane paradigm? Attribute physical properties to “stretched

horizon”E.g. temperature, resistance, …

stretched horizon

actual horizonQ. Bulk BM was due to b.c. at horizon (ingoing: thermal, outgoing: any). Can we reproduce this by interaction between string and “membrane”?

39

Langevin eq. at stretched horizon

𝑋ሺ𝑡,𝑟ሻ r

r=rsr=rH

−#𝜕𝑟𝑋ሺ𝑡,𝑟𝑠ሻ= 𝐹𝑠ሺ𝑡ሻ 𝐹𝑠(𝑡) = −න 𝑑𝑡′𝛾𝑠ሺ𝑡− 𝑡′ሻ𝜕𝑡𝑋ሺ𝑡′,𝑟𝑠ሻ+ 𝑅𝑠(𝑡)𝑡

−∞

=ۄ����������𝑅𝑠ሺ𝑡ሻۃ������� 0, =ۄ����������𝑅𝑠ሺ𝑡ሻ𝑅𝑠ሺ𝑡′ሻۃ������� 𝜅𝑠(𝑡− 𝑡′)

𝐹𝑠ሺ𝑡ሻ

EOM for endpoint: Postulate:

Langevin eq. at stretched horizon𝑋ሺ𝑡,𝑟~𝑟𝐻ሻ=න

𝑑𝜔ξ2𝜔[𝑎𝜔ሺ+ሻ𝑒−𝑖𝜔ሺ𝑡−𝑟∗ሻ+ 𝑎𝜔ሺ−ሻ𝑒−𝑖𝜔ሺ𝑡+𝑟∗ሻ+ h.c.]

𝛾𝑠ሺ𝑡ሻ∝ 𝛿ሺ𝑡ሻ, 𝑅𝑠ሺ𝜔ሻ∝ ξ𝜔 𝑎𝜔(+)

∝ۄ����������𝑅𝑠ሺ𝜔ሻ†𝑅𝑠ሺ𝜔ሻۃ������� ∝ۄ����������𝑎𝜔ሺ+ሻ†𝑎𝜔ሺ+ሻۃ������� 𝜔𝑒𝛽𝜔 − 1

Plug into EOMIngoing (thermal)

outgoing (any)

40

Friction precisely cancels outgoing

modes

Random force excites ingoing

modes thermally

Can satisfy EOM if

Correlation function:

41

Granular structure on stretched horizon

BH covered by “stringy gas”[Susskind et al.]

Frictional / random forcescan be due to this gas

Can we use Brownian stringto probe this?

42

Granular structure on stretched horizon AdSd BH

One quasiparticle / string bit per Planck area

Proper distance from actual horizon:

𝑑𝑠𝑑2 = 𝑟2𝑙2 ൫−ℎሺ𝑟ሻ𝑑𝑡2 + 𝑑𝑋Ԧ𝑑−22 ൯+ 𝑙2𝑑𝑟2𝑟2ℎ(𝑟)

ℎሺ𝑟ሻ= 1−ቀ𝑟𝐻𝑟ቁ𝑑−1

𝑇= 1𝛽 = ሺ𝑑− 1ሻ𝑟𝐻4𝜋𝑙2

𝛥𝑋∼ 𝑙𝑟𝐻ℓ𝑃

𝛥𝑡 ∼ Δ𝑋ξ𝜖 ∼ 𝑙ℓ𝑃ξ𝜖 𝑟𝐻 𝑟𝑠 = ሺ1+ 2𝜖ሻ𝑟𝐻 𝐿∼ ξ𝜖 𝑙

qp’s are moving at speed of light

(stretched horizon at )

Granular structure on stretched horizon

43

String endpoint collides with a quasiparticle once in time

Cf. Mean-free-path time read off Rs-correlator:

Scattering probability for string endpoint:

Δ𝑡 ∼ ℓ𝑃𝑇𝐿

𝑡mfp ∼ 1𝑇

𝜎= 𝛥𝑡𝑡mfp ∼ ℓ𝑃𝐿 ∼൜1 (𝐿∼ ℓP)𝑔s# (𝐿∼ ℓs)

44

Conclusions

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Conclusions Boundary BM ↔ bulk “Brownian string”

Can study QG in principle Semiclassically, can reproduce Langevin dyn. from

bulk random force ↔ Hawking rad. (“kick” by horizon)friction ↔ absorption

Time scales in strong coupling QGP: trelax, tcoll, tmfp FD theorem

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Conclusions

BM on stretched horizon Analogue of Avogadro # NA= 6x1023 < ∞?

Boundary: Bulk: energy of a Hawking quantum is tiny

as compared to BH mass, but finite

𝐸/𝑇∼ 𝒪ሺ𝑁2ሻ< ∞ 𝑀/𝑇∼ 𝒪ሺ𝐺𝑁−1ሻ< ∞

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Thanks!