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Brownian Motion in AdS/CFT

Brownian Motion in AdS/CFT. Masaki Shigemori University of Amsterdam Tenth Workshop on Non-Perturbative QCD l’Institut d’Astrophysique de Paris Paris, 11 June 2009. This talk is based on:. J. de Boer, V. E. Hubeny, M. Rangamani, M.S., “Brownian motion in AdS/CFT,” arXiv:0812.5112.

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Brownian Motion in AdS/CFT

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  1. Brownian Motion in AdS/CFT Masaki Shigemori University of Amsterdam Tenth Workshop on Non-Perturbative QCDl’Institutd’Astrophysique de Paris Paris, 11 June 2009

  2. This talk is based on: • 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.

  3. Intro / Motivation

  4. AdS/CFT and fluid-gravity • AdS • CFT black hole inquantum gravity plasma in stronglycoupled QFT difficult ? Long-wavelength approximation easier;better-understood horizon dynamicsin classical GR hydrodynamicsNavier-Stokes eq. Bhattacharyya+Minwalla+ Rangamani+Hubeny 0712.2456

  5. Macrophysics vs. microphysics • Hydro: coarse-grained •  BH in classical GR is also macro, approx. descriptionof underlying microphysics of QG BH! coarse grain • Can’t study microphysics within hydro framework(by definition) •  want to go beyond hydro approx

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

  7. Brownian motion in AdS/CFT  Do the same for hydro. in AdS/CFT! • Learn about QG from BM on boundary • How does Langevin dynamics come aboutfrom bulk viewpoint? • Fluctuation-dissipation theorem • 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 • BM • BM in AdS/CFT • Time scales • BM on stretched horizon

  10. Brownian motion Paul Langevin (1872-1946)

  11. Langevin dynamics Generalized Langevin eq: delayed friction random force

  12. General properties of BM Displacement: ballistic regime(init. velocity ) diffusive regime(random walk) diffusion constant

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

  14. BM in AdS/CFT

  15. Bulk BM • AdS Schwarzschild BH endpoint =Brownian particle Brownian motion AdS boundary at infinity fundamentalstring horizon r black hole

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

  17. BM in AdS/CFT • Probe approximation • Small gs • No interaction with bulk • The only interaction is at horizon • Small fluctuation • Expand Nambu-Goto actionto quadratic order • Transverse positionsare similar to Klein-Gordon scalars boundary horizon r

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

  19. 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!

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

  21. Time scales

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

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

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

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

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

  27. R-correlators from BM in AdS/CFT • Expansion of NG action to higher order: • Can compute tmfp from correction to 4-pt func. Can compute and thus tmfp

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

  29. Times scales from AdS/CFT • Resulting timescales: • strong coupling . is also possible. • Multiple collisions occur simultaneously. • Cf. “fast scrambler”

  30. BM on stretched horizon (Jorge’s talk)

  31. Conclusions

  32. Conclusions • 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

  33. Thanks!

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