Relativistic hydrodynamics at large gradients Michał P. Heller Perimeter Institute for Theoretical Physics, Canada National Centre for Nuclear Research, Poland based on 03.3452, 302.0697, 409.5087 & 503.0754 with R. Janik, M. Spaliński and P. Witaszczyk
Hydrodynamization M. P. Heller, R. A. Janik and P. Witaszczyk, Phys. Rev. Lett. 08, 20602 (202), 03.3452
Relativistic hydrodynamics hydrodynamics is an EFT of the slow evolution of conserved currents in collective media close to equilibrium DOFs: always local energy density and local flow velocity u µ ( u u = ) EOMs: conservation eqns r µ ht µ i =0for ht µ i systematically expanded in gradients ht T µ µ i = u µ u + P ( ){ g µ + u µ u } ( ) µ ( ){ g µ + u µ u }(r u)+... microscopic EoS input: ( P ( ) = 3 for CFTs) shear viscosity bulk viscosity (vanishes for CFTs) Dissipation: r µ + P ( ) T u µ +... = 2T + T (r u)2 +... 0 /3
Early initialization of hydrodynamics in HIC Hydrodynamic codes in HIC are initialized in < fm/c with temperature O(400MeV) large gradients: Longitudinal direction: p µ µ init = O(Tinit ) initτ=0.4 fm/c 0 600 500 5 0 300 200-5 y [fm] Einit @x Einit = O(Tinit ) -4 Einitε [fm ] Transverse plane: x3y [fm] 400-00 -0 0-0 -5 0 xx2 [fm] 5 0 - Schenke et al. 009.3244 FIG. gradients : (Color online) Energy density Can hydrodynamics work when are large? If yes, distribution why? (left), and after τ = 6 fm/c for the ideal case (midd 2/3
Boost-invariant flow [Bjorken 982] > 0 x 0 const x 0 slice: =0 x x q Boost-invariance: in ( x 2 0 x 2, y arctanhx,x 2,x 3 ) coords no y -dep x 0 In a CFT: ht µ i = diag E( ), E E, E + 2 E, E + 2 E and via scale-invariance Gradient expansion: series in. w P E ht 2 2i ht yyi is a function of 3/3 w T E( ) 3 8 2 N 2 c /4
Hydrodynamization Ab initio calculation in N=4 SYM at strong coupling: 03.3452 (see also Chesler & Yaffe 0906.4426, 0.3562) 0.6 0.5 P E 0.4 0.3 0.2 Hydrodynamics works despite huge anisotropy captured by µ 0. 0.0 0.3 0.4 0.5 0.6 0.7 0.8 0.9.0 w P E hydro = 0.2 w + 0.0069 w 2 0.0049 w 3 +... 4/3
Why hydrodynamization can occur? M. P. Heller, R. A. Janik and P. Witaszczyk, Phys. Rev. Lett. 0, 2602 (203), 302.0697
Excitations in strongly-coupled plasmas see, e.g. Kovtun & Starinets [hep-th/050684] see also Romatschke s week talk 3 2.5 Re!/2 T 3rd 2nd -0.5 0.5.5 2 k/2 T st 2.5 st - -.5 2nd @! @k k!0 = c sound 0.5 k/2 T 0.5.5 2 Figure 6: Real and imaginary parts of three lowest quasinormal frequencies as function of spatial momentum. The curves for which 0 as 0 correspond to hydrodynamic sound mode in the dual finite temperature N =4 SYM theory. -2-2.5-3 Im Im!/2 T 3rd N=4 SYM!(k)! 0 as k! 0 : slowly dissipating modes (hydrodynamic sound waves) behavior of the lowest (hydrodynamic) frequency which is absent for E α and Z 3.ForE z and Z,hydrodynamicfrequenciesarepurelyimaginary(givenbyEqs. (4.6) and (4.32) for small ω and q), and presumably move off to infinity as q becomes large. For Z 2,thehydrodynamic frequency has both real and imaginary parts (given by Eq. (4.44) for small ω and q), and eventually (for large q)becomesindistinguishableinthetowerofothereigenfrequencies. As an example, dispersion relations for the three lowest quasinormal frequencies in the soundchannel (including the one of the sound wave) are shown in Fig. 6. The tables below give numerical values of quasinormal frequencies for =. Onlynon-hydrodynamicfrequenciesareshown in the tables. The position of hydrodynamic frequencies at = is = 3.250637i for the 5/3 R-charge diffusive mode, = 0.598066i for the shear mode, and = ±0.74420 0.286280i all the rest: far from equilibrium (QNM) modes damped over t therm = O()/T
Hydrodynamic gradient expansion is divergent In 302.0697 we computed f(w) 2 up to O(w 240 ): 3 + P 6 E N=4 SYM f(w) = X n=0 f n w n = = 2 3 + 9 w +... (n!) /(n+) n! e n 6/3
Hydrodynamics and QNMs 302.0697 Analytic continuation of f B ( ) X240 n=0 n! f n n revealed the following singularities: N=4 SYM 3 Branch cut singularities start at! 2 i! QNM 7/3
EOMs for viscous hydro and quasinormal modes M. P. Heller, R. A. Janik, M. Spaliński and P. Witaszczyk, Phys. Rev. Lett. 3, 2660 (204), 409.5087
Evolution equations for relativistic viscous fluids is acausal. µ r = u µ u + P ( ){ g µ + u µ u } ( ) µ ( ){ g µ + u µ u µ =0 }(r u)+... Remedy: make µ = ht µ i ( u µ u + P ( ){g µ + u µ u }) ( D + ) µ = µ a new DOF, e.g. Small perturbations obey Maxwell-Cattaneo equation @ 2 t u z s Take it seriously:! = i st k2 +... hydrodynamics and Generalization that adds Re(! QNM ) : T @2 x u z + @ t u z =0! = i + i st k2 +... purely imaginary quasinormal mode ( T D)2 +2! I T D +! 2 µ = 8/3 D! 2 µ 409.5087
Resumming gradient expansion M. P. Heller, M. Spaliński, to appear in Phys. Rev. Lett. this Friday, 503.0754
The boost-invariant attractor ( D + ) µ = µ +... 503.0754 C = s and C = T f 0 = C + 2 3 C f + 6 3 w 6 9 wf + 4C 9 C wf 4 f w +... attractor different solutions f(w) =f 0 + f w f(w) =f 0 + f w + f 2 w 2 9/3
Gradient expansion f 0 = C + 2 3 C f + 6 3 w f = X n=0 6 9 wf + 4C 9 C wf f n w n + f + O( f 2 ) exp ( 3 2C w)... 4 f w +... 503.0754 sing = 3 2C f B ( ) X200 n=0 n! f n n = X200 (ξ) n=0 f n n - - - - (ξ) 0/3
sed by a factor of 3. Note that in the latter amic attractor is attained at larger values rom Eq. (2). Transseries Hydrodynamic gradient expansion is vergence. Note that in Eq. (6) large w correspond small. To invert the Borel transform it is necessa 503.0754 know the analytic continuation of series (6), whic B ( ). The inverse Borel transform intrinsically denote by fambiguous: Z Z fr (w) = d e f B ( /w) = w d e w f B ( ) (ξ) tial value of f at small w which is arbi q. (3), the attractor can be determined C C the result shown in Fig.. C f characterizing the attractor is to exwhere C denotes a contour in the complex plane con derivatives of f this is an analog of the ing 0 and, is interpreted as a resummation of the on in theories of inflation (see e.g. [9]). inal divergent series (5). To carry out the integrati B ( ). erates a kind of gradient expansion, but is essential to know the analytic structure of f - C2 sual hydrodynamical expansion, since One way to perform the analytic continuation is t proximations to f are not polynomials Pade approximants [20] which is the approach we a - osing the correct branch of the square - - throughout this study. We begin by examining the d rs at leading order one can ensure that (ξ) nal Pade approximant given by a ratio of two polynom th approximation in powers of /w one of order 00. This function has a dense sequence of p The ambiguity goes away upon including the quasinormal mode ( fn = a0,n ) C mx X 3 m C f= (camb + r) w exp w am,n w n 2C m=0 n=0 /3
Resummed hydrodynamics and the attractor 503.0754 attractor resummed transseries f(w) =f 0 + f w f(w) =f 0 + f w + f 2 w 2 Note that matching to the attractor required choosing r = 0.049 (not 0!). 2/3
Executive summary 03.3452, 302.0697, 409.5087 & 503.0754 with R. Janik, M. Spaliński and P. Witaszczyk
Ab initio calculations in holography show that early applicability of viscous hydrodynamics in the presence of large gradients / large anisotropies is not crazy. 3/3