Brownian Motion in AdS/CFT YITP Kyoto, 24 Dec 2009

1. Brownian Motion in AdS/CFTYITP Kyoto, 24 Dec 2009 Masaki Shigemori (Amsterdam)

2. 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.

3. Introduction / Motivation

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

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

6. 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) erratic motion Robert Brown (1773-1858) pollen particle

7. 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

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

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

10. Paul Langevin (1872-1946) Boundary BM - Langevin dynamics

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

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

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

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

15. How to determine γ, κ • Forced motion • No external force, admittance  read off μ measure known  read off κ

16. Bulk BM

17. Bulk setup • AdSd Schwarzschild BH(planar) endpoint =Brownian particle Brownian motion boundary fundamentalstring r horizon black hole

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

19. Assumptions • Probe approximation • Small gs • No interaction with bulk • Only interaction is at horizon • Small fluctuation • Expand Nambu-Goto actionto quadratic order • Transverse positions aresimilar to Klein-Gordon scalar boundary r horizon

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

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

22. Semiclassical analysis • Semiclassically, NH modes are thermally excited: • AdS3 Can use dictionary to compute x(t), s2(t) (bulk  boundary) : ballistic : diffusive Does exhibitBrownian motion

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

24. 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 b.c. changed from Neumann “flavor D-brane” freely infalling thermal

25. Forced motion: results (AdS3) • Admittance • Random force correlator • FD theorem (no λ) satisfied can be proven generally

26. Time scales

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

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

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

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

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

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

33. ⟨RRRR⟩from bulk BM • Holographic renormalization [Skenderis] • Similar to KG scalar, but not quite • Lorentzian AdS/CFT [Skenderis + van Rees] • IR divergence • Near the horizon, bulk integral diverges

34. IR divergence Reason: • Near the horizon, local temperature is high • String fluctuates wildly • Expansion of NG action breaks down Remedy: • Introduce an IR cut offwhere expansion breaks down • Expect that this givesright estimate of the full result cutoff

35. Times scales from AdS/CFT (weak) • Resulting timescales: • weak coupling • conventional kinetictheory is good

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

37. BM on stretched horizon

38. 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”?

39. Langevin eq. at stretched horizon • EOM for endpoint: • Postulate: r=rs r=rH r

40. Langevin eq. at stretched horizon • Can satisfy EOM if Ingoing (thermal) outgoing (any) Plug into EOM Friction precisely cancels outgoing modes Random force excites ingoing modes thermally • 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: qp’s are moving at speed of light (stretched horizon at )

43. Granular structure on stretched horizon • String endpoint collides with a quasiparticle once in time • Cf. Mean-free-path time read off Rs-correlator: • Scattering probability for string endpoint:

44. Conclusions

45. 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

46. 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

47. Thanks!