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.
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Introduction / Motivation
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thermodynamics
Hierarchy of scales in physics
hydrodynamics
atomic theorysubatomic
theory
large
small
Scale
macrophysics
microphysics
coarse-grain
Brownian motion
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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
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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
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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
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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 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
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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
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
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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
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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!