Transcript Brownian Motion in AdS/CFT

Brownian Motion in AdS/CFT
YITP Kyoto, 24 Dec 2009
Masaki Shigemori (Amsterdam)
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.
2
Introduction / Motivation
3
Hierarchy of scales in physics
Scale
large
thermodynamics
macrophysics
hydrodynamics
coarse-grain
Brownian
motion
atomic theory
microphysics
small
4
subatomic theory
Hierarchy in AdS/CFT
Scale
large
AdS
CFT
AdS BH
in classical GR
thermodynamics
of plasma
[Bhattacharyya+Minwalla+
Rangamani+Hubeny]
horizon dynamics
in classical GR
hydrodynamics
Navier-Stokes eq.
long-wavelength approximation
small
5
?
quantum BH
macrophysics
This
talk
strongly coupled
quantum plasma
microphysics
Brownian motion
― Historically, a crucial step
toward microphysics of nature
1827 Brown

erratic motion
Robert Brown (1773-1858)
pollen particle
Due to collisions with fluid particles
Allowed to determine Avogadro #: NA= 6x1023 < ∞
Ubiquitous
Langevin eq. (friction + random force)




6
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+Teaney
Transverse momentum broadening: Gubser, Casalderrey-Solana+Teaney
7
Preview: BM in AdS/CFT
endpoint =
Brownian particle
Brownian
motion
AdS boundary
at infinity
fundamental
string
horizon
black hole
8
Outline
9

Intro/motivation

Boundary BM

Bulk BM

Time scales

BM on stretched horizon
Paul Langevin (1872-1946)
Boundary BM
- Langevin dynamics
10
Simplest Langevin eq.
𝑥, 𝑝 = 𝑚𝑥
𝑝 𝑡 = −𝛾0 𝑝 𝑡 + 𝑅 𝑡
(instantaneous)
friction
𝑅 𝑡
= 0,
𝑅 𝑡 𝑅 𝑡′
random
force
= 𝜅0 𝛿(𝑡 − 𝑡 ′ )
white noise
11
Simplest Langevin eq.

Displacement:
𝑠2 𝑡
≡ 𝑥 𝑡 −𝑥 0
𝑇 2
𝑡
≈ 𝑚
2𝐷𝑡
(𝑡 ≪ 𝑡𝑟𝑒𝑙𝑎𝑥 )
ballistic regime
(init. velocity 𝑥~ 𝑇/𝑚 )
(𝑡 ≫ 𝑡𝑟𝑒𝑙𝑎𝑥 )
diffusive regime
(random walk)
𝑇
• Diffusion constant: 𝐷 ≡
𝛾0 𝑚
1
• Relaxation time: 𝑡relax =
𝛾0
12
2
(S-E relation)
𝜅0
FD theorem  𝛾0 =
2𝑚𝑇
Generalized Langevin equation
𝑡
𝑑𝑡′ 𝛾 𝑡 − 𝑡 ′ 𝑝 𝑡′ + 𝑅 𝑡 + 𝐹(𝑡)
𝑝 𝑡 =−
−∞
delayed
friction
𝑅 𝑡

𝑅 𝑡 𝑅 𝑡′
= 𝜅(𝑡 − 𝑡 ′ )
Qualitatively similar to simple LE


13
= 0,
random
force
ballistic regime
diffusive regime
external
force
Time scales


Relaxation time
𝑡relax
∞
𝑑𝑡 𝛾(𝑡)
0
Collision duration time 𝑡coll
𝑅 𝑡 𝑅 0

1
≡ , 𝛾0 =
𝛾0
∼ 𝑒 −𝑡/𝑡 coll
Mean-free-path time 𝑡mfp
 time between collisions
R(t)
Typically
𝑡coll
𝑡relax ≫ 𝑡mfp ≫ 𝑡coll
t
𝑡mfp
14
 time elapsed
in a single collision
but not necessarily so
for strongly coupled plasma
How to determine γ, κ
𝑅 𝜔 +𝐹 𝜔
𝑝 𝜔 =
≡𝜇 𝜔 𝑅 𝜔 +𝐹 𝜔
𝛾 𝜔 − 𝑖𝜔
admittance
1.
Forced motion 𝐹 𝑡 = 𝐹0 𝑒 −𝑖𝜔𝑡
𝑝 𝑡
2.
= 𝜇 𝜔 𝐹0 𝑒 −𝑖𝜔𝑡  read off μ
No external force, 𝐹 = 0
𝑝𝑝 = 𝜇
measure
15
2
known
𝑅𝑅
 read off κ
Bulk BM
16
Bulk setup

AdSd Schwarzschild BH
(planar)
endpoint =
Brownian particle
2
2
2
𝑟
𝑙
𝑑𝑟
2
𝑑𝑠𝑑2 = 2 −ℎ 𝑟 𝑑𝑡 2 + 𝑑𝑋𝑑−2
+ 2
𝑙
𝑟 ℎ(𝑟)
𝑟𝐻
ℎ 𝑟 =1−
𝑟
Brownian
motion
boundary
𝑋𝑑−2
𝑑−1
r
fundamental
string
horizon
1
𝑑 − 1 𝑟𝐻
𝑇= =
𝛽
4𝜋𝑙 2
𝑙: AdS radius
17
black hole
Physics of BM in AdS/CFT

Horizon kicks endpoint on horizon
(= Hawking radiation)

Fluctuation propagates to
AdS boundary

Endpoint on boundary
(= Brownian particle) exhibits BM
Whole process is dual to
quark hit by plasma particles
endpoint =
Brownian particle
boundary
𝑋𝑑−2
r
transverse
fluctuation
kick
black hole
18
Brownian
motion
Assumptions

Probe approximation

Small gs


No interaction with bulk

Only interaction is at horizon
Small fluctuation


19
Expand Nambu-Goto action
to quadratic order
𝑋𝑑−2 (𝑡, 𝑟)
Transverse positions are
similar to Klein-Gordon scalar
𝑋𝑑−2
boundary
r
𝑋 𝑡, 𝑟
horizon
Transverse fluctuations

Quadratic action
𝑆NG = const + 𝑆2 + 𝑆4 + ⋯
𝑋2
𝑟4ℎ 𝑟 ′ 2
𝑑𝑡 𝑑𝑟
−
𝑋
4
ℎ 𝑟
𝑙
1
𝑆2 = −
4𝜋𝛼 ′

Mode expansion
∞
𝑑𝜔 𝑓𝜔 𝑟 𝑒 −𝑖𝜔𝑡 𝑎𝜔 + h. c.
𝑋 𝑡, 𝑟 =
0
ℎ 𝑟
𝜔 + 4 𝜕𝑟 𝑟 4 ℎ 𝑟 𝜕𝑟
𝑙
2
𝑓𝜔 𝑟 = 0
d=3: can be solved exactly
d>3: can be solved in low frequency limit
20
𝑋 𝑡, 𝑟
Bulk-boundary dictionary
Near horizon:
∞
𝑋 𝑡, 𝑟 ∼
0
outgoing
mode
𝑑𝜔
𝑒 −𝑖𝜔 (𝑡−𝑟∗ ) + 𝑒 𝑖𝜃𝜔 𝑒 −𝑖𝜔 (𝑡+𝑟∗ ) 𝑎𝜔 + h. c.
2𝜔
Near boundary:
ingoing
mode
phase shift
𝑟∗ : tortoise coordinate
∞
𝑑𝜔 𝑓𝜔 (𝑟𝑐 )𝑒 −𝑖𝜔𝑡 𝑎𝜔 + h. c.
𝑋 𝑡, 𝑟c ≡ 𝑥(𝑡) =
𝑟c : cutoff
0
†
𝑥 𝑡1 𝑥 𝑡2 ⋯ ↔ 𝑎𝜔 1 𝑎𝜔
⋯
2
observe BM
correlator of
in gauge theory
radiation modes
Can learn about quantum gravity in principle!
21
Semiclassical analysis

Semiclassically, NH modes are thermally excited:
†
𝑎𝜔 𝑎𝜔
1
∝ 𝛽𝜔
𝑒 −1
Can use dictionary to compute x(t), s2(t) (bulk  boundary)

AdS3
𝑠 2 (𝑡) ≡ : 𝑥 𝑡 − 𝑥 0
𝑡relax
22
𝛼′ 𝑚
∼ 2 2
𝑙 𝑇
2:
≈
𝑇 2
𝑡
𝑚
(𝑡 ≪ 𝑡relax ) : ballistic
𝛼′
𝑡
2
𝜋𝑙 𝑇
(𝑡 ≫ 𝑡relax ) : diffusive
Does exhibit
Brownian motion
Semiclassical analysis

Momentum distribution
Probability distribution for 𝑝 = 𝑚𝑥
𝑝2
𝑓 𝑝 ∝ exp −𝛽𝐸𝑝 ,
𝐸𝑝 =
2𝑚
 Maxwell-Boltzmann

Diffusion constant
𝑑 − 1 2 𝛼′
𝐷=
8𝜋𝑙2 𝑇
 Agrees with drag force computation
[Herzog+Karch+Kovtun+Kozcaz+Yaffe] [Liu+Rajagopal+Wiedemann]
[Gubser] [Casalderrey-Solana+Teaney]
23
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
“flavor D-brane”
freely
infalling
24
thermal
b.c. changed
from Neumann
Forced motion: results (AdS3)

Admittance
1
𝛼 ′ 𝛽2 𝑚 1 − 𝑖𝜔/𝜋𝛼′𝑚
𝜇 𝜔 =
=
𝛾 𝜔 − 𝑖𝜔
2𝜋 1 − 𝛼 ′ 𝑚𝛽 2 𝜔/2𝜋

Random force correlator
4𝜋 1 − 𝛽𝜔/2𝜋 2
𝜅 𝜔 = ′ 3
𝛼 𝛽 1 − 𝜔/2𝜋𝛼′𝑚 2
𝑡coll
1
∼
𝑇
(no λ)

FD theorem
2𝑚 Re(𝛾 𝜔 ) = 𝛽𝜅 𝜔
satisfied
can be proven generally
25
Time scales
26
Time scales
𝑡relax
𝑡mfp
𝑡coll
information about
plasma constituents
R(t)
𝑡coll
t
𝑡mfp
27
Time scales from R-correlators
A toy model:

R(t) : consists of many pulses randomly distributed
𝑘
𝑅 𝑡 =
𝜖𝑖 𝑓 𝑡 − 𝑡𝑖
𝑓(𝑡 − 𝑡1 ) 𝑓(𝑡 − 𝑡2 )
𝑖=1
𝑓(𝑡) : shape of a single pulse
𝜖3 = −1
𝜖1 = 1
𝜖𝑖 = ±1 : random sign

−𝑓(𝑡 − 𝑡3 )
Distribution of pulses = Poisson distribution
𝜇 : number of pulses per unit time,
28
𝜖2 = 1
∼ 1/𝑡mfp
Time scales from R-correlators
 2-pt func
𝑅 𝑡 𝑅 0
 Low-freq. 4-pt func
→ 𝑡coll
𝑅 𝜔1 𝑅 𝜔2 𝑅 𝜔3 𝑅 𝜔4
𝑅(𝜔1 )𝑅(𝜔2 ) ≈ 2𝜋𝜇𝛿(𝜔1 + 𝜔2 )𝑓 0
𝑅 𝜔1 𝑅 𝜔2 𝑅 𝜔3 𝑅 𝜔4
→ 𝑡mfp
2
conn
≈ 2𝜋𝜇𝛿(𝜔1 + 𝜔2 + 𝜔3 + 𝜔4 )𝑓 0
4
tilde = Fourier transform
Can determine μ, thus tmfp
29
Sketch of derivation (1/2)
k pulses
𝑘
𝑅 𝑡 =
𝜖𝑖 𝑓 𝑡 − 𝑡𝑖
0
…
𝑖=1
𝑡2
𝑡1
Probability that there are k pulses in period [0,τ]:
𝑃𝑘 𝜏 = 𝑒 −𝜇𝜏
2-pt func:
∞
𝑅 𝑡 𝑅(𝑡 ′ ) =
𝜇𝜏 𝑘
𝑘!
𝑘
𝑃𝑘 𝜏
𝑘=1
(Poisson dist.)
𝜖𝑖 𝜖𝑗 𝑓 𝑡 − 𝑡𝑖 𝑓(𝑡 − 𝑡𝑗 )
𝑘
𝑖,𝑗 =1
𝜖𝑖 = ±1 ∶ random signs → 𝜖𝑖 𝜖𝑗 = 𝛿𝑖𝑗
𝜏
𝑘
𝑓 𝑡 − 𝑡𝑖 𝑓 𝑡 ′ − 𝑡𝑖 𝑘 =
𝑑𝑢 𝑓 𝑡 − 𝑢 𝑓(𝑡 ′ − 𝑢)
𝜏 0
30
𝑡𝑘 𝜏
Sketch of derivation (2/2)
∞
𝑅 𝑡 𝑅(𝑡 ′ ) = 𝜇
𝑑𝑢 𝑓 𝑡 − 𝑢 𝑓(𝑡 ′ − 𝑢)
−∞
𝑅 𝜔 𝑅(𝜔′) = 2𝜋𝜇𝛿 𝜔 + 𝜔′ 𝑓 𝜔 𝑓(𝜔′)
Similarly, for 4-pt func,
“disconnected part”
“connected
part”
𝑅 𝜔 𝑅(𝜔′)𝑅(𝜔′′)𝑅(𝜔′′′)
= 𝑅 𝜔 𝑅 𝜔′ 𝑅(𝜔′′)𝑅(𝜔′′′) + 2 more terms
+2𝜋𝜇𝛿 𝜔 + 𝜔′ + 𝜔′′ + 𝜔′′′ 𝑓 𝜔 𝑓 𝜔′ 𝑓 𝜔′′ 𝑓 𝜔′′′
𝑅(𝜔1 )𝑅(𝜔2 ) ≈ 2𝜋𝜇𝛿(𝜔1 + 𝜔2 )𝑓 0
𝑅 𝜔1 𝑅 𝜔2 𝑅 𝜔3 𝑅 𝜔4
31
conn
2
≈ 2𝜋𝜇𝛿(𝜔1 + 𝜔2 + 𝜔3 + 𝜔4 )𝑓 0
4
⟨RRRR⟩ from bulk BM
 Can compute tmfp from connected to 4-pt func.

Expansion of NG action to higher order:
𝑆NG = const + 𝑆2 + 𝑆4 + ⋯
1
𝑆4 =
16𝜋𝛼 ′
2
4
𝑋
𝑟 ℎ 𝑟 ′2
𝑑𝑡 𝑑𝑟
−
𝑋
4
ℎ 𝑟
𝑙
2
4-point vertex
Can compute 𝑅𝑅𝑅𝑅
and thus tmfp
32
conn
𝑋 𝑡, 𝑟
⟨RRRR⟩ from bulk BM

Holographic renormalization

Similar to KG scalar, but not quite

Lorentzian AdS/CFT

IR divergence

33
[Skenderis]
[Skenderis + van Rees]
Near the horizon, bulk integral diverges
IR divergence
Reason:



Near the horizon, local temperature is high
String fluctuates wildly
Expansion of NG action breaks down
Remedy:


34
Introduce an IR cut off
where expansion breaks down
Expect that this gives
right estimate of the full result
cutoff
Times scales from AdS/CFT (weak)
Resulting timescales:
𝑡relax ~
𝑚
𝜆 𝑇2
 weak coupling
1
𝑡coll ~
𝑇
1
𝑙4
𝑡mfp ~
𝜆 ≡ ′2
𝑇 log λ
𝛼
𝜆≪1
𝑡relax ≫ 𝑡mfp ≫ 𝑡coll
conventional kinetic theory is good
35
Times scales from AdS/CFT (strong)
Resulting timescales:
𝑡relax ~
𝑚
𝜆 𝑇2
 strong coupling
1
𝑡coll ~
𝑇
1
𝑙4
𝑡mfp ~
𝜆 ≡ ′2
𝑇 log λ
𝛼
𝜆≫1
𝑡mfp ≪ 𝑡coll
Multiple collisions occur simultaneously.
Cf. “fast scrambler”
[Hayden+Preskill]
[Sekino+Susskind]
36
BM on stretched horizon
37
Membrane paradigm?

Attribute physical properties to “stretched horizon”
E.g. temperature, resistance, …
stretched horizon
actual horizon
Q. Bulk BM was due to b.c. at horizon
(ingoing: thermal, outgoing: any).
Can we reproduce this by interaction
between string and “membrane”?
38
Langevin eq. at stretched horizon
𝑋 𝑡, 𝑟
𝐹𝑠 𝑡
r
r=rs
r=rH

EOM for endpoint: −#𝜕𝑟 𝑋 𝑡, 𝑟𝑠 = 𝐹𝑠 𝑡

Postulate:
𝑡
𝑑𝑡 ′ 𝛾𝑠 𝑡 − 𝑡 ′ 𝜕𝑡 𝑋 𝑡 ′ , 𝑟𝑠 + 𝑅𝑠 (𝑡)
𝐹𝑠 (𝑡) = −
−∞
𝑅𝑠 𝑡
39
= 0,
𝑅𝑠 𝑡 𝑅𝑠 𝑡 ′
= 𝜅𝑠 (𝑡 − 𝑡 ′ )
Langevin eq. at stretched horizon
𝑋 𝑡, 𝑟~𝑟𝐻 =
𝑑𝜔
+
2𝜔
[𝑎𝜔 𝑒 −𝑖𝜔
𝑡−𝑟∗
−
+ 𝑎𝜔 𝑒 −𝑖𝜔
Ingoing (thermal)
𝑡+𝑟∗
outgoing (any)
Plug into EOM

Can satisfy EOM if
(+)
𝛾𝑠 𝑡 ∝ 𝛿 𝑡 ,
𝑅𝑠 𝜔 ∝ 𝜔 𝑎𝜔
Friction precisely cancels
outgoing modes

Random force excites
ingoing modes thermally
Correlation function:
†
𝑅𝑠 𝜔 𝑅𝑠 𝜔
40
∝
+† +
𝑎𝜔 𝑎𝜔
𝜔
∝ 𝛽𝜔
𝑒 −1
+ h. c. ]
Granular structure on stretched horizon

BH covered by “stringy gas”
[Susskind et al.]

Frictional / random forces
can be due to this gas

Can we use Brownian string
to probe this?
41
Granular structure on stretched horizon

AdSd BH
2
2
2
𝑟
𝑙
𝑑𝑟
2
𝑑𝑠𝑑2 = 2 −ℎ 𝑟 𝑑𝑡 2 + 𝑑𝑋𝑑−2
+ 2
𝑙
𝑟 ℎ(𝑟)
𝑟𝐻
ℎ 𝑟 =1−
𝑟

𝑑−1
𝑇=
1
𝑑 − 1 𝑟𝐻
=
𝛽
4𝜋𝑙 2
One quasiparticle / string bit per Planck area
𝑙
𝛥𝑋 ∼ ℓ𝑃
𝑟𝐻
qp’s are moving at speed of light
𝛥𝑡 ∼

42
Δ𝑋
𝜖
∼
𝑙ℓ𝑃
𝜖 𝑟𝐻
(stretched horizon at 𝑟𝑠 = 1 + 2𝜖 𝑟𝐻 )
Proper distance from actual horizon:
𝐿∼ 𝜖𝑙
Granular structure on stretched horizon

String endpoint collides with a quasiparticle once in time
ℓ𝑃
Δ𝑡 ∼
𝑇𝐿

Cf. Mean-free-path time read off Rs-correlator:
𝑡mfp

1
∼
𝑇
Scattering probability for string endpoint:
𝛥𝑡
ℓ𝑃
1
𝜎=
∼
∼ #
𝑔s
𝑡mfp
𝐿
43
(𝐿 ∼ ℓP )
(𝐿 ∼ ℓs )
Conclusions
44
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
45
Conclusions

BM on stretched horizon

Analogue of Avogadro # NA= 6x1023 < ∞?

Boundary:
𝐸/𝑇 ∼ 𝒪 𝑁 2 < ∞

Bulk:
𝑀/𝑇 ∼ 𝒪 𝐺𝑁−1 < ∞
 energy of a Hawking quantum is tiny
as compared to BH mass, but finite
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Thanks!
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