Brownian Motion in AdS/CFTYITP Kyoto, 24 Dec 2009

Masaki Shigemori (Amsterdam)

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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 theory

subatomic theory

large

small

Scale

macrophysics

microphysics

coarse-grain

Brownian motion

5

AdS BHin classical GR

Hierarchy in AdS/CFT

AdS CFT

[Bhattacharyya+Minwalla+

Rangamani+Hubeny]

thermodynamics of plasma

horizon dynamics

in classical GR

hydrodynamics

Navier-Stokes eq.

quantum BH

strongly 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

8

Preview: BM in AdS/CFT

horizon

AdS boundaryat infinity

fundamentalstring

black hole

endpoint =Brownian particle Brownian

motion

9

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

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

12

Simplest Langevin eq.

diffusive regime(random walk)

≈൞

𝑇𝑚𝑡2 (𝑡 ≪𝑡𝑟𝑒𝑙𝑎𝑥)2𝐷𝑡 (𝑡 ≫𝑡𝑟𝑒𝑙𝑎𝑥)

• Diffusion constant:

𝐷≡ 𝑇𝛾0𝑚

𝑠2ሺ𝑡ሻۃ ≡ۄ −ሾ𝑥ሺ𝑡ሻۃ 𝑥ሺ0ሻሿ2 ۄ

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

𝑅ሺ𝑡ሻۃ =ۄ 0, 𝑅ሺ𝑡ሻ𝑅ሺ𝑡′ሻۃ =ۄ 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)

𝑅ሺ𝑡ሻ𝑅ሺ0ሻۃ ∽ۄ 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

𝑝ሺ𝑡ሻۃ =ۄ 𝜇ሺ𝜔ሻ𝐹0𝑒−𝑖𝜔𝑡

𝑝𝑝ۃ =ۄ a2𝜇a22 𝑅𝑅ۃ ۄmeasure

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

particle

Brownian 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

particle

Brownian 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 actionto quadratic order

Transverse positions aresimilar 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 dictionary

Near 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:⟩ ≈ە۔

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 procedure

Turn 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

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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-correlators

R(t) : consists of many pulses randomly distributed

A 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

𝑅෨(𝜔1)𝑅෨(𝜔2)ۃ ≈ۄ 2𝜋𝜇𝛿(𝜔1 + 𝜔2)𝑓ሚሺ0ሻ2

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

tilde = Fourier transform

2-pt func

Low-freq. 4-pt func

𝑅ሺ𝑡ሻ𝑅ሺ0ሻۃ 𝑡coll→ۄ

𝑅෨ሺ𝜔1ሻ𝑅෨ሺ𝜔2ሻ𝑅෨ሺ𝜔3ሻ𝑅෨ሺ𝜔4ሻۃ 𝑡mfp→ۄ

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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

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

𝑘𝑖,𝑗=1

𝜖𝑖 = ±1 ∶ random signs → 𝜖𝑖𝜖𝑗ۃ =ۄ 𝛿𝑖𝑗

0

k pulses

𝑡1 𝑡2 𝑡𝑘 … 𝜏

(Poisson dist.)

−𝑓ሺ𝑡ۃ 𝑡𝑖ሻ𝑓ሺ𝑡′ − 𝑡𝑖ሻ 𝑘ۄ = 𝑘𝜏න 𝑑𝑢𝜏0 𝑓ሺ𝑡− 𝑢ሻ𝑓(𝑡′ − 𝑢)

2-pt func:

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Sketch of derivation (2/2)

𝑅ሺ𝑡ሻ𝑅(𝑡′)ۃ =ۄ 𝜇න 𝑑𝑢∞−∞ 𝑓ሺ𝑡− 𝑢ሻ𝑓(𝑡′ − 𝑢)

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

Similarly, for 4-pt func,

+2𝜋𝜇𝛿ሺ𝜔+ 𝜔′ + 𝜔′′ + 𝜔′′′ ሻ𝑓ሚሺ𝜔ሻ𝑓ሚሺ𝜔′ሻ𝑓ሚሺ𝜔′′ሻ𝑓ሚሺ𝜔′′′ሻ 𝑅෨ሺ𝜔ሻ𝑅෨(𝜔′)𝑅෨(𝜔′′)𝑅෨(𝜔′′′)ۃ ۄ𝑅෨ሺ𝜔ሻ𝑅෨ሺ𝜔′ሻ⟩ ⟨𝑅෨(𝜔′′)𝑅෨(𝜔′′′)ۃ ሺ2 more termsሻ+ۄ

“disconnected part”

“connected

part”=

𝑅෨(𝜔1)𝑅෨(𝜔2)ۃ ≈ۄ 2𝜋𝜇𝛿(𝜔1 + 𝜔2)𝑓ሚሺ0ሻ2

𝑅෨ሺ𝜔1ሻ𝑅෨ሺ𝜔2ሻ𝑅෨ሺ𝜔3ሻ𝑅෨ሺ𝜔4ሻۃ connۄ ≈ 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

𝑅𝑅𝑅𝑅ۃ connۄ Can compute and thus tmfp

4-point vertex

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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

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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

−∞

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

𝐹𝑠ሺ𝑡ሻ

EOM for endpoint: Postulate:

Langevin eq. at stretched horizon

𝑋ሺ𝑡,𝑟~𝑟𝐻ሻ=න𝑑𝜔ξ2𝜔[𝑎𝜔ሺ+ሻ𝑒−𝑖𝜔ሺ𝑡−𝑟∗ሻ+ 𝑎𝜔ሺ−ሻ𝑒−𝑖𝜔ሺ𝑡+𝑟∗ሻ+ h.c.]

𝛾𝑠ሺ𝑡ሻ∝ 𝛿ሺ𝑡ሻ, 𝑅𝑠ሺ𝜔ሻ∝ ξ𝜔 𝑎𝜔(+)

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

Plug into EOM

Ingoing (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

45

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

46

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< ∞

47

Thanks!