Holographic hydrodynamization

Transcription

1 Holographic hydrodynamization Michał P. Heller University Amsterdam, The Nerls & National Centre for Nuclear Research, Pol (on leave) based on [hep-th] MPH, R. A. Janik & P. Witaszczyk (PRL 0 (03) 60)

2 Introduction

3 Modern relativistic (uncharged) hydrodynamics hydrodynamics is an EFT slow evolution conserved currents in collective media close equilibrium As any EFT it is based on idea gradient expansion DOFs: always local energy density local flow velocity u µ ( u u = ) EOMs: conservation eqns r µ T µ =0 for T µ systematically exped in gradients terms carrying more gradients T µ = u µ u + P ( ){ g µ + u µ u } ( ) µ ( ){ g µ + u µ u }(r u)+... microscopic input: perfect fluid stress tensor EoS (famous) shear viscosity /6 bulk viscosity (vanishes for CFTs)

4 Applicability hydrodynamics terms carrying more gradients T µ = u µ u + P ( ){ g µ + u µ u } ( ) µ ( ){ g µ + u µ u }(r u)+... perfect fluid stress tensor microscopic input: EoS (famous) shear viscosity bulk viscosity (vanishes for CFTs) Naively one might be inclined associate hydrodynamic regime with small gradients. But this is not how we should think about effective field ories! The correct way is underst hydrodynamic modes as low energy DOFs. Of course, re are also or DOFs in fluid. The pic my talk is use holography elucidate ir imprint on hydro. /6

5 Holographic plasmas ir degrees freedom

6 Holography From applicational perspective AdS/CFT is a ol for computing correlation functions in certain strongly coupled gauge ories, such as N =4SYM at large N c. For simplicity I will consider AdS+4 / CFT+3 focus on pure gravity secr. R ab Rg ab 6 L g ab =0 Different solutions correspond states in a dual CFT with different ht µ i. Minkowski spacetime at boundary 0 bulk AdS UV ds = L z n dz + µ dx µ dx + N c ht µ i z o??? IR 3/6

7 Excitations strongly coupled plasmas Consider small amplitude perturbations ( T µ = 8 N c T 4 diag (3,,, ) µ + T µ ( e i!(k) t+i ~ k ~x ) Kovtun & Starinets [hep-th/050684] T µ /N c T 4 ) on p a holographic plasma Dissipation leads modes with complex!(k), which in sound channel look like 3.5 Re!/ T 3rd nd k/ T st.5 st - k!0 = c sound 0.5 k/ T Figure 6: Real imaginary parts three lowest quasinormal frequencies as function spatial momentum. The curves for which 0 as 0 correspond hydrodynamic sound mode in dual finite temperature N =4 SYM ory. There are two different kinds modes:!(k)! 0 as k! 0 behavior lowest (hydrodynamic) frequency which is absent for E α Z 3.ForE z Z,hydrodynamicfrequenciesarepurelyimaginary(givenbyEqs. (4.6) (4.3) for small ω q), presumably move f infinity as q becomes large. For Z,hydrodynamic frequency has both real imaginary parts (given by Eq. (4.44) for small ω q), eventually (for large q)becomesindistinguishableinweroreigenfrequencies. As an example, dispersion relations for three lowest quasinormal frequencies in soundchannel (including one sound wave) are shown in Fig. 6. The tables below give numerical Im Im!/ T : slowly evolving dissipating modes (hydrodynamic sound waves) all rest: far from equilibrium (QNM) modes dampened over 4/6 3rd t rm = O()/T

8 Lesson : for hydrodynamics work all or DOFs need relax. 3.5 Re!/ T 3rd nd k/ T st.5 st - k!0 = c sound 0.5 k/ T Figure 6: Real imaginary parts three lowest quasinormal frequencies as function spatial momentum. The curves for which 0 as 0 correspond hydrodynamic sound mode in dual finite temperature N =4 SYM ory. 5/6 behavior lowest (hydrodynamic) frequency which is absent for E Z.ForE Im Im!/ T 3rd

9 3.5 Re!/ T 3rd nd k/ T st.5 st - k!0 = c sound Observation: 0.5 k/ T Figure 6: Real imaginary parts three lowest quasinormal frequencies as function spatial momentum. The curves for which 0 as 0 correspond hydrodynamic sound mode in dual finite temperature N =4 SYM ory. behavior lowest (hydrodynamic) frequency which is absent for E α Z 3.ForE z Z,hydrodynamicfrequenciesarepurelyimaginary(givenbyEqs. (4.6) (4.3) for small ω q), presumably move f infinity as q becomes large. For Z,hydrodynamic frequency has both real imaginary parts (given by Eq. (4.44) for small ω q), eventually (for large q)becomesindistinguishableinweroreigenfrequencies. As an example, dispersion relations for three lowest quasinormal frequencies in soundchannel (including one sound wave) are shown in Fig. 6. The tables below give numerical values quasinormal frequencies for =. Onlynon-hydrodynamicfrequenciesareshown in tables. The position hydrodynamic frequencies at = is = i for R-charge diffusive mode, = i for shear mode, = ± i for sound mode. The numerical values lowest five (non-hydrodynamic) quasinormal frequencies for electromagnetic perturbations are: Transverse channel Diffusive channel n Re Im Re Im -3 Im Im!/ T No matter how long one waits, re will be always remnants n-eq DOFs Lesson : Hydrodynamic gradient expansion cannot converge 6/6 3rd

10 Dynamical model

11 Fantastic y-model [Bjorken 98] The simplest model in which one can test se ideas is boost-invariant flow with no transverse expansion. x In Bjorken scenario dynamics depends only on proper time = q (x 0 ) (x ) ds = d + dy + dx + dx 3 stress tensor (for a CFT) is entirely expressed in terms local energy density x 0 ht µ i = diag{ ( ),p L ( ),p T ( ),p T ( )} with p p T ( ) = ( )+ 0 L ( ) = ( ) 0 ( ) ( ) =0 hadronic gas mixed phase described by hydrodynamics QGP described by pre-equilibrium stage AdS/CFT in this scenario x 7/6

12 Hydrodynamization [hep-th] PRL 08 (0) 060: MPH, R. A. Janik & P. Witaszczyk For hydrodynamics work all or DOFs need relax. 3.5 Re!/ T 3rd nd k/ T st.5 st - k!0 = c sound 0.5 Surprising consequence k/ T Figure 6: Real imaginary parts three lowest quasinormal frequencies as function spatial momentum. The curves for which 0 as 0 correspond hydrodynamic sound mode in dual finite temperature N =4 SYM ory. behavior lowest (hydrodynamic) frequency which is absent for E α Z 3.ForE z e - 3 p Z,hydrodynamicfrequenciesarepurelyimaginary(givenbyEqs. (4.6) (4.3) for small L ω q), presumably move f infinity as q becomes large. For Z e,hydrodynamic.4 grey: frequency full evolution has both real imaginary parts (given by Eq. (4.44) for small ω q), eventually (for large q)becomesindistinguishableinweroreigenfrequencies. As an. red: example, st order dispersion hydrodynamics relations for three lowest quasinormal frequencies in soundchannel.0 (including one sound wave) are shown in Fig. 6. The tables below give numerical 0.8 values quasinormal frequencies for =. Onlynon-hydrodynamicfrequenciesareshown 0.6 in tables. The position hydrodynamic Thus frequencies at = is = i for 0.4 R-charge diffusive mode, = i for shear mode, = ± i hydrodynamization = for sound mode. The numerical values lowest five (non-hydrodynamic) quasinormal 0. frequencies for electromagnetic perturbations are: t THtL Transverse channel Diffusive channel n Re Im8/6 Re Im Im Im!/ T Large anisotropy at onset hydrodynamics 3rd 3 p L isotropization rmalization

13 Boost-invariant hydrodynamics In hydrodynamics, stress tensor is expressed in terms T, u µ ir r Key observation: in Bjorken flow u µ is fixed by symmetries takes form u µ Its gradients will come thus from Christfel symbols ( ) ds = d + dy + dx + dx 3 y y = Lesson: in Bjorken flow hydrodynamic gradient expansion = late time power series At very late times p L = 0 = p T = + 0 4/3 T /3 In holographic hydrodynamics gradient expansion parameter is T µ = u µ u µ + P ( ) { µ + u µ u } T r µu For Bjorken flow is. /3 = /3 9/6 ( ) s( ) + P ( ) T T r µu µ +...

14 High order hydrodynamics

15 Hydrodynamic series at high orders [hep-th] PRL 0 (03) 60: MPH, R. A. Janik & P. Witaszczyk = 3 8 N c T 00 = ( ) 4/3 X / /3 n= n ( /3 ) n (T r µ u /3 ) at low orders behavior is different at large orders facrial growth gradient contributions with order (n!) /n ( n)/n e n First evidence that hydrodynamic expansion has zero radius convergence! 0/6

16 Singularities in Borel plane [hep-th] PRL 0 (03) 60: MPH, R. A. Janik & P. Witaszczyk A stard method for asymptic series is Borel transform Borel summation (u) X n u n (u = /3 ), B (ũ) n= X n= n! nũ n, Borel sum : Bs (u) = Z 0 u B (t)exp( t/u)dt This makes a difference only if we can find analytic continuation B (ũ). P 0 Idea: use Pade approximant B (ũ) = reveal singularities. m=0 c mũ m P 0 n=0 d nũ n B (ũ) green dots: zeros numerar gray dots: zeros denominar Im z those are real singularities Re z 0 those zeros cancel almost perfectly (up 0-50 ) /6

17 Hydrodynamic instanns hydrodynamic gradient expansion

18 Singularities Borel transform QNMs MPH, R. A. Janik & P. Witaszczyk In Borel summation outcome depends on conur connect 0 with. Here re are two inequivalent conurs (blue orange). e 3/ i! Borel /3 Borel Im ué Re!/ T Re ué Z Bs (u) = u [hep-th] PRL 0 (03) 60: B (t)exp( t/u)dt Bs (u) = Z u B (t)exp( t/u)dt behavior lowest (hydrodynamic) frequency w Z,hydrodynamicfrequenciesarepurelyimaginary - in tables. The position hydrodynamic freque 0.5 ω-.5 q), presumably move f infinity 3rdas q k/ Tfrequency has both real imaginary parts (give Im Im!/ T eventually (for large q)becomesindistinguishablein Figure 6: Real imaginary parts three lowest example, quasinormal dispersion frequencies relations asfor function three spatial lowest qua momentum. The curves for which 0 as 0 correspond (including hydrodynamic one soundmode wave) inare shown dual in =. Only! Borel = i Borel = i finite temperature N =4 SYM ory. values quasinormal frequencies for is frequency lowest non-hydrodynamic metric QNM at! Re!/ T Im!/ T nd k/ T 3rd k/ T Figure 6: Real 0.5 imaginary parts.5 three lowest q nd momentum The curves for which 0 as 0 correspo st finite temperature N =4 SYM ory. - st nd! Borel k =0 /6 st

19 x = cosh y x = sinh y. () ble on hydrodynamic degrees only freedom. On boosted black brane is slowly evolving captures lished that, when exped around = 0, energy der dition, consider can, incorporation no x = sinh y. () hydrodynamic degrees freedom. One in adcontains only even powersfield proper [7],evolving) degrees freedom by l nnergy casedensity (3+)-dimensional conformal ory time(fast dition, consider (03) incorporation nonhydrodynamic [hep-th] PRL 0 60: MPH, R. A. Janik & P. Witaszczyk this information turns out be hard implement in an plasma, most general stress tensor obeying symas in conformal field ory ensional stein s equations on p Ein hydrody (fast evolving) degrees freedom by linearizing, in we adopted metries unambiguous obeying problem inway coordinates (,following, y, x, x ) reads in stress tensor symi.e. B = Bhydro +term B,solution, similarly for Can wesimplest underst 3/ in exponent pre-exponential in stein s equations on p hydrodynamic continuation. coordinates (,µy, xanalytic, x ) reads looking for Bfor corresponding (at very µb = Bhydro/3 i.e. + B, similarly A, T = diag( pl, position pt, pt )3/,i! ()symmetric Figure shows, poles Borel Borel inorexponentially decaying contribution th e +... looking for B corresponding (at very large time)? µ Borel transformed energy den, pl, pt Pade, pt ) approximant, () ergent depending only on stress. Fortensor static bac exponentially decaying contribution where sity. energy density is a function proper time Let us note that some care must be taken as s [5], gous calculation would lead spectru depending only on. For static background analoyes, QNMs here are fast evolving modes on p slowly evolving background. nly longitudinal p transverse p pressures T poles on negapade approximant exhibits apparent is proper Ltime wer se-a function dynamic quasinormal modes carrying ze gous calculation would lead spectrum nonhydrore fully expressed in terms energy density [9] local restreal frame (as here), at leading order y(up only care about T. tive axis which, however, cancel almost perfectly plin transverse pt pressures subsewhichcarrying is knownzero be same as sp dynamic quasinormal modes momentum, 00 0 density accuracy) our ms energy [9] with zeroes numerar. As momentum quasinormal modes for which is known be same as spectrum zero 0 0 knowledge pseries pl = about. a finite (3) number T = is+limited field [3].for x (t) =!(t) x(t) with momentum quasinormal modes massless scalar But temperature here is time-dependent. Imagine solving esidues, leads an inshould resummation. It is structure tensor at each order, in analogy 0 ambiguity terms (first 4), we only trust pt = +. (3) In leadingorder graphs gradient exp (8) R field [3]. empting speculate that such an ambiguity can be un number Feynman at lar in close vicinity origin. Note, (), however, that x(t) e±i!(t)dt Note that, in varying proper time - rapidity coordinates with slowly frequency. The leading order result is modes, sulting on gravity side, indeed In leading order gradient expansion, redersod as appearance new, nonhydrodynamic, turbative calculations. lack poles on stress positive real () axis seems inhere -israpidity no momentum flow (), in tensor time coordinates duce indeed scalar essentially quasinormal modes bu Z axis. dicate sulting modes, on gravity side, re degrees freedom in system. Indeed, performborel pointing possible takes instann-like dependence 3 directions, Regarding a thought o flowstress velocity is summability, trivial 3 wards form u/3= /3 facrfuture ow in tensor () are damped expo ditional i! d +... = i! idenduce scalar quasinormal modes but obtain an ad QN M QN M /3 ng conur integration around cut originating from existence Borel-resummed all-order hydrodynamics. /3 = on coupling nomenological spin o our Letter is th. Hydrodynamic constituent relations lead, n, ( ) 3 ivial takes form u = [4]. Upon including viscous correction, g 3 m3/i)!borel are damped exponentially in Y M ditional facr leads approximate density following contribution The poles some complicated structure resumming hydrodynamic description an radient exped energy form ituent relations lead, n, tain a furr nontrivial powerlike preexp density [4]. Upon including viscous correction, modes [hep-th/060649] Janik &obpeschanski nergy at large proper time branch cuts. The pole nearest origin, from which a hydrodynamic stress tensor past regim Z density form trans- 3 majorbranch cut starts, tain furr powerlike preexponential facr3 sets a radiusnontrivial convergence qnm sequent low order terms give comparabl /3 exp ( i! = N + 3 /33 + term? +..., (4) How about pre-exponential Schematically + / d +qnm log 4 4/3, y c 4/3 /3 Borel /3 Borel transform hydrodynamic series. Its nu3 lead/3 exp ( i!borel (0) ), 8 refined criterium for t qnm This might exp ( i! ). (6) ann qnm 3 /3 + merical 4 4/3 +value... (multiplied, (4) by facr 3/i) reads gravity calculation Explicit hydrodynamics used in [, for 3] goe low l find where scale, in parwhere choice sets an overall energy spirit [4, 5]. mode yield Explicit gravity calculation for lowest!quasinormal = i (7)(! = The 4.565)!qnm (9) = , qnm =.54 icular for frequencies Borel Borel9). an overall energy scale, in paracknowledgments. MPH acknowledge Indeed agrees with!!! finite prefacr was chosen match N = 4 super Yang! = , = i. (7) al frequencies (7) 9). The = i. () qnm qnm Borel The frequency Organization!qnm agrees for withscientifi fr Nerls is consistent with coefficient obtained from fitting que Mills ory at large-n strong coupling. In folc match N = 4 super Yanglowest nonhydrodynamic scalar quasinor der Veni scheme would like 4 series.! facrial-like behavior original ation. The frequency agrees with frequency owing, we choose units by setting =. qnm 3/6 The contribution (0) ger with (9) () nd strong coupling. In fol- Slow fast modes

20 Interpretation possible relevance

21 Why hydro series might be asymptic? [hep-th] PRL 0 (03) 60: Famous examples asymptic expansions arise in pqfts MPH, R. A. Janik & P. Witaszczyk ] [ There, number Feynman graphs grows ~order! at large orders* ] [ We suspect analogous mechanism might work also in case hydro series* T µ = u µ u + P ( ){ g µ + u µ u } ( ) µ ( ){ g µ + u µ u }(r u)+... [ [ Π µν = ησ µν τ Π DΠ µν + d ] ] [ d Πµν ( u) [ + κτ [R µν (d )u α R α µν β Π DΠ µν + d ] [ ] ] u d Πµν ( u) β + κ [R µν (d )u α R α µν β u β + λ η Π µ λπ ν λ λ + λ η Π µ λω ν λ + λ 3 Ω µ λω ν λ. η Π µ λπ ν λ λ η Π µ λω ν λ + λ 3 Ω µ λω ν λ /6 st order hydro ( transport coeff) nd order hydro (5 transport coeffs)...

22 completely explained dissipative hydrodynamics. one by h, a perfect agreement w In order study transition hydrodynami for w > 0.63, on or ha more detail, will adopt a numerical criterion for t d Fhydrow(w) = Tef f (3) we pressure anisotropy in that regime w d w=, (4) (0) [hep-th] PRL : MPH, R. A. Janik & P. Witaszczyk malization which is deviation w from w d w completely explained byd dissipative ordercan hydro expression (5) form namely hydrodynamics, hydro equations In boost-invariant be recast in In order study transition t where Fhydro (w) is completely determined in terms d we will adopt a numeric detail, 3 more transport coe cients ory N =(w) 4 plasma d. For Fhydro d w 4 is < w = explicitly, upwith w =deviation Tef f ( ) =(4) Nc malization T3ef rd order f ( )which at strong coupling Fhydro (w)/w is known w d w F (w) 8 hydro rd order hydro expression (5) terms corresponding 3 order hydrodynamics [3] perfect fluid where Fhydro (w) is completely determined in terms bewildering variety nonequilib Despite d w exist, how ory. For plasma we will show below that d transport log coe cients 5 45 log + 4 log N = 4evolution, re < w coupling Fhydro 3 w3 3rd dynamics. order (w) is known explicitly upsurprising some regularities inf 3 9 w at7strong 97 (w)/w hydro rd (5) 3 order st terms 3rd order hydrohydrodynamics [3] nd corresponding Initial final entropy. Apart from ene Despite bewildering variety t d w momentum tensor components, a very important p dw log 5 45 log + 4 log evolution, we will show below that t cal property evolving plasma system is its ent w 3w 3dt w surprising regularities in d 3 9 w 7 97 w d This is quite reminiscent [] where all-order hydrodynamics density S (persome different data set transverse area unit (spacetime (5) was postulated in terms linearized AdS dynamics work with M.case, Spaliński pidity). In general pr Initial time-dependent final entropy. Apar momentum tensor components, a ve 0.8 cal property evolving plasma s This is quite reminiscent [] where all-order hydrodynamics density S (per transverse area 0.80 was postulated in terms linearized AdS dynamics. pidity). In general time-depende tions for scale invariant quantity Resummed hydrodynamics? = namely convergence a single curve! Idea: is it possible obtain (part ) this curve from Borel resummation? 5/ w

23 Summary

24 Summary [hep-th] PRL 0 (03) 60. Hydrodynamics is an asymptic series Re!/ T k/ T 3rd nd st igure 6: Real imaginary parts three lowest quasinormal frequencies as function spatial omentum. The curves for which 0 as 0 correspondhydrodynamicsoundmodeindual Im Im!/ T 6/6 k/ T st nd 3rd because in any fluid re are DOFs not captured by hydrodynamic approx.