Masaki ShigemoriUniversity of Amsterdam
Tenth Workshop on Non-Perturbative QCDl’Institut d’Astrophysique de Paris
Paris, 11 June 2009
Brownian Motion in AdS/CFT
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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!