* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Brownian Motion in AdS/CFT
Survey
Document related concepts
Old quantum theory wikipedia , lookup
Noether's theorem wikipedia , lookup
Scalar field theory wikipedia , lookup
Canonical quantization wikipedia , lookup
History of quantum field theory wikipedia , lookup
Quantum chaos wikipedia , lookup
Strangeness production wikipedia , lookup
Renormalization wikipedia , lookup
Elementary particle wikipedia , lookup
Hawking radiation wikipedia , lookup
Theory of everything wikipedia , lookup
Transcript
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 46 Thanks! 47