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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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**Brownian Motion in AdS/CFT**Masaki Shigemori University of Amsterdam Tenth Workshop on Non-Perturbative QCDl’Institutd’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. • A. Atmaja, J. de Boer, K. Schalm, M.S., work in progress.**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**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**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**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**Preview: BM in AdS/CFT**endpoint =Brownian particle Brownian motion AdS boundary at infinity fundamentalstring horizon black hole**Outline**• Intro/motivation • BM • BM in AdS/CFT • Time scales • BM on stretched horizon**Brownian motion**Paul Langevin (1872-1946)**Langevin dynamics**Generalized Langevin eq: delayed friction random force**General properties of BM**Displacement: ballistic regime(init. velocity ) diffusive regime(random walk) diffusion constant**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**Bulk BM**• AdS Schwarzschild BH endpoint =Brownian particle Brownian motion AdS boundary at infinity fundamentalstring horizon r black hole**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**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**Brownian string**• Quadratic action • Mode expansion d=3: can be solved exactly d>3: can be solved in low frequency limit**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!**Semiclassical analysis**• Semiclassically, NH modes are thermally excited: Can use dictionary to compute x(t), s2(t) (bulk boundary) ballistic diffusive Does exhibitBrownian motion**Time scales**information about plasma constituents R(t) t**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,**Time scales from R-correlators**Can determine μ, thus tmfp • 2-pt func • Low-freq. 4-pt func tilde = Fourier transform**Sketch of derivation**k pulses … 0 Probability that there are k pulses in period [0,τ]: (Poisson dist.) 2-pt func:**Sketch of derivation**“disconnected part” Similarly, for 4-pt func, “connected part”**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**Times scales from AdS/CFT**• Resulting timescales: • weak coupling • conventional kinetictheory is good**Times scales from AdS/CFT**• Resulting timescales: • strong coupling . is also possible. • Multiple collisions occur simultaneously. • Cf. “fast scrambler”**BM on stretched horizon**(Jorge’s talk)**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

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