Brownian Motion in AdS/CFT YITP Kyoto, 24 Dec 2009 Masaki Shigemori (Amsterdam)

This talk is based on: 2  J. de Boer, V. Hubeny, M. Rangamani, M.S., “Brownian motion in AdS/CFT,” arXiv:  A. Atmaja, J. de Boer, M.S., in preparation.

Introduction / Motivation 3

thermodynamics Hierarchy of scales in physics 4 hydrodynamics atomic theory subatomic theory large small Scale macrophysics microphysics coarse-grain Brownian motion

AdS BH in classical GR AdS BH in classical GR Hierarchy in AdS/CFT 5 AdSCFT [Bhattacharyya+Minwalla+ Rangamani+Hubeny] thermodynamics of plasma horizon dynamics in classical GR horizon dynamics in classical GR hydrodynamics Navier-Stokes eq. hydrodynamics Navier-Stokes eq. quantum BH strongly coupled quantum plasma large small Scale long-wavelength approximation macro- physics micro- physics ? This talk

Brownian motion 6 ― Historically, a crucial step toward microphysics of nature  1827 Brown  Due to collisions with fluid particles  Allowed to determine Avogadro #: N A = 6x10 23 < ∞  Ubiquitous  Langevin eq. (friction + random force) Robert Brown ( ) erratic motion pollen particle

Brownian motion in AdS/CFT 7  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

Preview: BM in AdS/CFT 8 horizon AdS boundary at infinity fundamental string black hole endpoint = Brownian particle Brownian motion

Outline 9  Intro/motivation  Boundary BM  Bulk BM  Time scales  BM on stretched horizon

Boundary BM - Langevin dynamics 10 Paul Langevin ( )

Simplest Langevin eq. 11 (instantaneous) friction random force white noise

Simplest Langevin eq. 12 diffusive regime (random walk) Diffusion constant: ballistic regime (init. velocity )  Displacement: Relaxation time: (S-E relation) FD theorem 

Generalized Langevin equation 13 delayed friction random force external force  Qualitatively similar to simple LE  ballistic regime  diffusive regime

Time scales 14  Relaxation time  time elapsed in a single collision Typically but not necessarily so for strongly coupled plasma R(t)R(t) t  time between collisions  Collision duration time  Mean-free-path time

How to determine γ, κ Forced motion 2. No external force, measureknown  read off μ  read off κ admittance

Bulk BM 16

Bulk setup 17  AdS d Schwarzschild BH (planar) r horizon boundary fundamental string black hole endpoint = Brownian particle Brownian motion

Physics of BM in AdS/CFT 18  Horizon kicks endpoint on horizon (= Hawking radiation)  Fluctuation propagates to AdS boundary  Endpoint on boundary (= Brownian particle) exhibits BM transverse fluctuation Whole process is dual to quark hit by plasma particles r boundary black hole endpoint = Brownian particle Brownian motion kick

Assumptions 19 r horizon boundary  Probe approximation  Small g s  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

Transverse fluctuations 20  Quadratic action d=3: can be solved exactly d>3: can be solved in low frequency limit  Mode expansion

Bulk-boundary dictionary 21 Near horizon: outgoing mode ingoing mode phase shift : tortoise coordinate : cutoff observe BM in gauge theory correlator of radiation modes Can learn about quantum gravity in principle! Near boundary:

Semiclassical analysis 22  Semiclassically, NH modes are thermally excited:  AdS 3 Can use dictionary to compute x(t), s 2 (t) (bulk  boundary) : ballistic : diffusive Does exhibit Brownian motion

Semiclassical analysis 23  Momentum distribution  Diffusion constant  Maxwell-Boltzmann  Agrees with drag force computation [Herzog+Karch+Kovtun+Kozcaz+Yaffe] [Liu+Rajagopal+Wiedemann] [Gubser] [Casalderrey-Solana+Teaney]

Forced motion 24  Want to determine μ(ω), κ(ω) ― follow the two-step procedure Turn on electric field E(t)=E 0 e -iωt on “flavor D-brane” and measure response ⟨p(t)⟩ E(t) p freely infalling thermal b.c. changed from Neumann “flavor D-brane”

Forced motion: results (AdS 3 ) 25  Admittance  Random force correlator  FD theorem (no λ ) satisfied can be proven generally

Time scales 26

Time scales 27 R(t)R(t) t information about plasma constituents

Time scales from R-correlators 28  R(t) : consists of many pulses randomly distributed A toy model: : shape of a single pulse : random sign : number of pulses per unit time,  Distribution of pulses = Poisson distribution

Time scales from R-correlators 29 Can determine μ, thus t mfp tilde = Fourier transform 2 -pt func Low-freq. 4 -pt func

Sketch of derivation (1/2) 30 Probability that there are k pulses in period [0,τ]: 0 k pulses … (Poisson dist.) 2-pt func:

Sketch of derivation (2/2) 31 Similarly, for 4-pt func, “disconnected part” “connected part”

⟨RRRR⟩ from bulk BM 32  Expansion of NG action to higher order:  Can compute t mfp from connected to 4-pt func. Can compute and thus t mfp 4-point vertex

33  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

Times scales from AdS/CFT (weak) 35 conventional kinetic theory is good Resulting timescales:  weak coupling

Times scales from AdS/CFT (strong) 36 Multiple collisions occur simultaneously. strong coupling Cf. “fast scrambler” [Hayden+Preskill] [Sekino+Susskind] Resulting timescales:

BM on stretched horizon 37

Membrane paradigm? 38  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”? Q. Bulk BM was due to b.c. at horizon (ingoing: thermal, outgoing: any). Can we reproduce this by interaction between string and “membrane”?

Langevin eq. at stretched horizon 39 r r=r s r=r H  EOM for endpoint:  Postulate:

Langevin eq. at stretched horizon Plug into EOM Ingoing (thermal)outgoing (any) 40 Friction precisely cancels outgoing modes Random force excites ingoing modes thermally  Can satisfy EOM if  Correlation function:

Granular structure on stretched horizon 41  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?

Granular structure on stretched horizon 42  AdS d BH  One quasiparticle / string bit per Planck area  Proper distance from actual horizon: 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 R s -correlator:  Scattering probability for string endpoint:

Conclusions 44

Conclusions 45  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: t relax, t coll, t mfp  FD theorem

Conclusions 46  BM on stretched horizon  Analogue of Avogadro # N A = 6x10 23 < ∞?  Boundary:  Bulk:  energy of a Hawking quantum is tiny as compared to BH mass, but finite

47 Thanks!