Download Brownian Motion in AdS/CFT

Masaki Shigemori
University of Amsterdam
Tenth Workshop on Non-Perturbative QCD
l’Institut d’Astrophysique de Paris
Paris, 11 June 2009
 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.
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3
AdS
CFT
black hole in
quantum gravity
plasma in strongly
coupled QFT
?
horizon dynamics
in classical GR
difficult
Long-wavelength
approximation
hydrodynamics
Navier-Stokes eq.
easier;
betterunderstood
Bhattacharyya+Minwalla+
Rangamani+Hubeny 0712.2456
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 Hydro: coarse-grained
 BH in classical GR is also macro, approx. description
of underlying microphysics of QG BH!
coarse
grain
 Can’t study microphysics within hydro framework
(by definition)
 want to go beyond hydro approx
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― 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 = 6 × 1023 < ∞
 Ubiquitous
 Langevin eq. (friction + random force)
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 Do the same for hydro. in AdS/CFT!
 Learn about QG from BM on boundary
 How does Langevin dynamics come about
from bulk viewpoint?
 Fluctuation-dissipation theorem
 Relation to RHIC physics?
Related work:
drag force: Herzog+Karch+Kovtun+Kozcaz+Yaffe, Gubser, Casalderrey-Solana+Teaney
transverse momentum broadening: Gubser, Casalderrey-Solana+Teaney
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endpoint =
Brownian particle
Brownian
motion
AdS boundary
at infinity
fundamental
string
horizon
black hole
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 Intro/motivation
 BM
 BM in AdS/CFT
 Time scales
 BM on stretched horizon
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Paul Langevin (1872-1946)
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Generalized Langevin eq:
𝑥, 𝑝 = 𝑚𝑥
𝑡
𝑑𝑡′ 𝛾 𝑡 − 𝑡 ′ 𝑝 𝑡′ + 𝑅(𝑡)
𝑝 𝑡 =−
−∞
delayed
friction
𝑅 𝑡
= 0,
𝑅 𝑡 𝑅 𝑡′
random
force
= 𝜅(𝑡 − 𝑡 ′ )
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Displacement:
𝑠2 𝑡
≡ 𝑥 𝑡 −𝑥 0
2
𝑇 2
𝑡
𝑚
≈
(𝑡 ≪ 𝑡𝑟𝑒𝑙𝑎𝑥 )
2𝐷𝑡
(𝑡 ≫ 𝑡𝑟𝑒𝑙𝑎𝑥 )
𝑇
𝐷
≡
, 𝛾 =
diffusion constant
𝛾0 𝑚 0
ballistic regime
(init. velocity 𝑥~ 𝑇/𝑚 )
diffusive regime
(random walk)
∞
𝑑𝑡 𝛾(𝑡)
0
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1
 Relaxation time 𝑡relax ≡
, 𝛾0 =
𝛾0
 Collision duration time 𝑡coll
𝑅 𝑡 𝑅 0
∼ 𝑒 −𝑡/𝑡 coll
 Mean-free-path time 𝑡mfp
∞
𝑑𝑡 𝛾(𝑡)
0
 time elapsed
in a single collision
 time between collisions
R(t)
Typically
𝑡coll
𝑡relax ≫ 𝑡mfp ≫ 𝑡coll
t
𝑡mfp
but not necessarily so
for strongly coupled plasma
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 AdS Schwarzschild BH
endpoint =
Brownian particle
2
𝑟
2
𝑑𝑠𝑑2 = 2 −ℎ 𝑟 𝑑𝑡 2 + 𝑑𝑋𝑑−2
𝑙
𝑙 2 𝑑𝑟 2
+ 2
𝑟 ℎ(𝑟)
AdS boundary
at infinity
𝑋𝑑−2
r
𝑟𝐻
ℎ 𝑟 =1−
𝑟
Brownian
motion
fundamental
string
horizon
𝑑−1
1
𝑑 − 1 𝑟𝐻
𝑇= =
𝛽
4𝜋𝑙 2
black hole
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endpoint =
Brownian particle
 Horizon kicks endpoint on horizon
(= Hawking radiation)
boundary
𝑋𝑑−2
 Fluctuation propagates to
AdS boundary
 Endpoint on boundary
(= Brownian particle) exhibits BM
Brownian
motion
transverse
fluctuation
r
horizon
kick
Whole process is dual to
quark hit by QGP particles
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𝑋𝑑−2
 Probe approximation
 Small gs
boundary
 No interaction with bulk
 The only interaction is at horizon
r
 Small fluctuation
 Expand Nambu-Goto action
𝑋 𝑡, 𝑟
horizon
to quadratic order
 Transverse positions
𝑋𝑑−2 (𝑡, 𝑟)
are similar to Klein-Gordon scalars
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 Quadratic action
𝑋 𝑡, 𝑟
𝑆NG = const + 𝑆2 + 𝑆4 + ⋯
𝑋2
𝑟4 ℎ 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
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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
in gauge theory
correlator of
radiation modes
Can learn about quantum gravity in principle!
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 Semiclassically, NH modes are thermally excited:
†
𝑎𝜔 𝑎𝜔
1
∝ 𝛽𝜔
𝑒 −1
Can use dictionary to compute x(t), s2(t) (bulk  boundary)
𝑠 2 (𝑡) ≡ : 𝑥 𝑡 − 𝑥 0
𝑡𝑟𝑒𝑙𝑎𝑥
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
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𝑡re lax
𝑡mfp
𝑡coll
information about
plasma constituents
R(t)
𝑡coll
t
𝑡mfp
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Simplifying assumptions:
 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
𝜖2 = 1
𝜖𝑖 = ±1 : random sign
−𝑓(𝑡 − 𝑡3 )
 Distribution of pulses = Poisson distribution
𝜇 : number of pulses per unit time, ∼ 1/𝑡mfp
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 2-pt func
𝑅 𝑡 𝑅 0
 Low-freq. 4-pt func
→ 𝑡coll
𝑅 𝜔1 𝑅 𝜔2 𝑅 𝜔3 𝑅 𝜔4
𝑅 0
2
= 𝜇𝑇𝑓 0
2
𝑅 0
4
= 3𝑅 0
2 2
𝑇 ≡ 2𝜋𝛿 0 ,
+ 𝜇𝑇𝑓 0
→ 𝑡mfp
4
tilde = Fourier transform
Can determine μ, thus tmfp
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k pulses
𝑘
𝑅 𝑡 =
…
0
𝑡1
𝜖𝑖 𝑓 𝑡 − 𝑡𝑖
𝑡𝑘 𝜏
𝑡2
𝑖=1
Probability that there are k pulses in period [0,τ]:
𝑃𝑘 𝜏 = 𝑒 −𝜇𝜏
2-pt func:
(Poisson dist.)
𝑘
∞
𝑅 𝑡 𝑅(𝑡 ′ ) =
𝜇𝜏 𝑘
𝑘!
𝑃𝑘 𝜏
𝑘=1
𝜖𝑖 𝜖𝑗 𝑓 𝑡 − 𝑡𝑖 𝑓(𝑡 − 𝑡𝑗 )
𝑘
𝑖,𝑗 =1
𝜖𝑖 = ±1 ∶ random signs → 𝜖𝑖 𝜖𝑗 = 𝛿𝑖𝑗
𝑓 𝑡 − 𝑡𝑖 𝑓 𝑡 ′ − 𝑡𝑖
𝑘
𝑘
=
𝜏
𝜏
𝑑𝑢 𝑓 𝑡 − 𝑢 𝑓(𝑡 ′ − 𝑢)
0
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∞
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 0
2
𝑅 0
4
= 𝜇𝑇𝑓 0
2
= 3𝑅 0
2 2
+ 𝜇𝑇𝑓 0
4
𝑇 ≡ 2𝜋𝛿 0
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 Can compute tmfp from correction 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
Can compute 𝑅𝑅𝑅𝑅
and thus tmfp
2
𝑋 𝑡, 𝑟
c
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Resulting timescales:
𝑡relax ~
𝑚
𝜆
𝑇2
 weak coupling
1
𝑡coll ~
𝑇
𝑡mfp ~
1
𝜆𝑇
𝑙4
𝜆 = ′2
𝛼
𝜆≪1
𝑡relax ≫ 𝑡mfp ≫ 𝑡coll
conventional kinetic theory is good
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Resulting timescales:
𝑡relax ~
𝑚
𝜆
𝑇2
1
𝑡coll ~
𝑇
𝑡mfp ~
𝜆≫1
≪ 𝑡coll . 𝑡relax ≪ 𝑡coll
1
𝜆𝑇
𝑙4
𝜆 = ′2
𝛼
 strong coupling
𝑡mfp
is also possible.
Multiple collisions occur simultaneously.
Cf. “fast scrambler”
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(Jorge’s talk)
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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
↔ absorption
 Time scales in strong coupling QGP:
𝑡relax , 𝑡mfp , 𝑡coll
 BM on stretched horizon (Jorge’s talk)
 Fluctuation-dissipation theorem
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