1. Masaki Shigemori University of Amsterdam Tenth Workshop on Non-Perturbative QCD l’Institut d’Astrophysique de Paris Paris, 11 June 2009

2. 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. 2

3. 3

4. 4 AdSCFT black hole in quantum gravity horizon dynamics in classical GR plasma in strongly coupled QFT hydrodynamics Navier-Stokes eq. Long-wavelength approximation difficult easier; better- understood Bhattacharyya+Minwalla+ Rangamani+Hubeny 0712.2456 ?

5. Hydro: coarse-grained 5  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 coarse grain

6. ― Historically, a crucial step toward microphysics of nature 1827 Brown Due to collisions with fluid particles Allowed to determine Avogadro #: Ubiquitous Langevin eq. (friction + random force) 6 Robert Brown (1773-1858) erratic motion pollen particle

7.  Do the same for hydro. in AdS/CFT! Learn about QG from BM on boundary How does Langevin dynamics come about from bulk viewpoint? Fluctuation-dissipation theorem Relation to RHIC physics? 7 Related work: drag force: Herzog+Karch+Kovtun+Kozcaz+Yaffe, Gubser, Casalderrey-Solana+Teaney transverse momentum broadening: Gubser, Casalderrey-Solana+Teaney

8. 8 horizon AdS boundary at infinity fundamental string black hole endpoint = Brownian particle Brownian motion

9. Intro/motivation BM BM in AdS/CFT Time scales BM on stretched horizon 9

10. 10 Paul Langevin (1872-1946)

11. 11 Generalized Langevin eq: delayed friction random force

12. 12 Displacement: diffusive regime (random walk) ballistic regime (init. velocity ) diffusion constant

13. 13 Relaxation time Collision duration time Mean-free-path time  time elapsed in a single collision Typically but not necessarily so for strongly coupled plasma R(t)R(t) t  time between collisions

14. 14

15. AdS Schwarzschild BH 15 horizon AdS boundary at infinity fundamental string black hole endpoint = Brownian particle Brownian motion r

16. Horizon kicks endpoint on horizon (= Hawking radiation) Fluctuation propagates to AdS boundary Endpoint on boundary (= Brownian particle) exhibits BM 16 horizon boundary endpoint = Brownian particle Brownian motion r transverse fluctuation kick Whole process is dual to quark hit by QGP particles

17. 17 horizon boundary r Probe approximation Small g s No interaction with bulk The only interaction is at horizon Small fluctuation Expand Nambu-Goto action to quadratic order Transverse positions are similar to Klein-Gordon scalars

18. Quadratic action 18 d=3: can be solved exactly d>3: can be solved in low frequency limit Mode expansion

19. Near horizon: 19 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

20. Semiclassically, NH modes are thermally excited: 20 Can use dictionary to compute x(t), s 2 (t) (bulk  boundary) ballistic diffusive Does exhibit Brownian motion

21. 21

22. 22 R(t)R(t) t information about plasma constituents

23. 23 Simplifying assumptions: : shape of a single pulse : random sign : number of pulses per unit time, R(t) : consists of many pulses randomly distributed Distribution of pulses = Poisson distribution

24. 24 Can determine μ, thus t mfp tilde = Fourier transform 2 -pt func Low-freq. 4 -pt func

25. 25 Probability that there are k pulses in period [0,τ]: 0 k pulses … (Poisson dist.) 2-pt func:

26. 26 Similarly, for 4-pt func, “disconnected part” “connected part”

27. 27 Expansion of NG action to higher order: Can compute t mfp from correction to 4-pt func. Can compute and thus t mfp

28. 28 conventional kinetic theory is good Resulting timescales: weak coupling

29. 29 Multiple collisions occur simultaneously. Resulting timescales: strong coupling is also possible.. Cf. “fast scrambler”

30. 30 (Jorge’s talk)

31. 31

32. 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 ↔ absorption Time scales in strong coupling QGP: BM on stretched horizon (Jorge’s talk) Fluctuation-dissipation theorem 32

33. 33