Relativistic Ideal and Viscous Hydrodynamics Tetsufumi Hirano Department of Physics The University of Tokyo The University of Tokyo Intensive Lecture YITP, December 11th, 2008 Intensive Lecture YITP, December 11th, 2008 TH, N. van der Kolk, A. Bilandzic, arXiv: [nucl-th]; to be published in “Springer Lecture Note in Physics”.

Plan of this Lecture 1 st Day Hydrodynamics in Heavy Ion Collisions Collective flow Dynamical Modeling of heavy ion collisions (seminar) 2 nd Day Formalism of relativistic ideal/viscous hydrodynamics Bjorken’s scaling solution with viscosity Effect of viscosity on particle spectra (discussion)

PART 3 Formalism of relativistic ideal/viscous hydrodynamics

Relativistic Hydrodynamics Energy-momentum conservation Current conservation Energy-Momentum tensor The i-th conserved current In H.I.C., N i  = N B  (net baryon current) Equations of motion in relativistic hydrodynamics

Tensor/Vector Decomposition Tensor decomposition with a given time-like and normalized four-vector u  where,

“ Projection ” Tensor/Vector u  is perpendicular to  . u  is local four flow velocity. More precise meaning will be given later. Naively speaking, u  (  ) picks up time- (space-)like component(s). Local rest frame (LRF):

Intuitive Picture of Projection time like flow vector field

Decomposition of T  :Energy density :(Hydrostatic+bulk) pressure P = P s +  :Energy (Heat) current :Shear stress tensor : Symmetric, traceless and transverse to u  & u

Decomposition of N  :charge density :charge current Q. Count the number of unknowns in the above decomposition and confirm that it is 10(T  )+4k(N i  ). Here k is the number of independent currents. Note: If you consider u  as independent variables, you need additional constraint for them. If you also consider P s as an independent variable, you need the equation of state P s =P s (e,n).

Ideal and Dissipative Parts Energy Momentum tensor Charge current Ideal part Dissipative part

Meaning of u  u  is four-velocity of “flow”. What kind of flow? Two major definitions of flow are 2. Flow of conserved charge (Eckart) 1. Flow of energy (Landau)

Meaning of u  (contd.) Landau (W  =0, u L  V  =0) Eckart (V  =0,u E  W  =0) uLuL VV uEuE WW Just a choice of local reference frame. Landau frame might be relevant in H.I.C.

Relation btw. Landau and Eckart

Relation btw. Landau and Eckart (contd.)

Entropy Conservation in Ideal Hydrodynamics Neglect “dissipative part” of energy momentum tensor to obtain “ideal hydrodynamics”. Q. Derive the above equation. Therefore,

Assumption (1 st order theory): Non-equilibrium entropy current vector has linear dissipative term(s) constructed from (V , ,  , (u  ). Entropy Current Thus,  = 0 since N  = 0, W  = 0 since considering the Landau frame, and  = 0 since u  S  should be maximum in equilibrium (stability condition). (Practical) Assumption: Landau frame (omitting subscript “L”). No charge in the system.

The 2 nd Law of Thermodynamics and Constitutive Equations The 2 nd thermodynamic law tells us Q. Check the above calculation.

Constitutive Equations (contd.) Thermodynamic force Transport coefficient “ Current ” tensorshear scalarbulk Newton Stokes

Equation of Motion : Expansion scalar (Divergence) : Lagrange (substantial) derivative

Equation of Motion (contd.) Q. Derive the above equations of motion from energy-momentum conservation. Note: We have not used the constitutive equations to obtain the equations of motion.

Intuitive Interpretation of EoM Work done by pressure Production of entropy Change of volume Dilution Compression

Conserved Current Case

Lessons from (Non-Relativistic) Navier-Stokes Equation Assuming incompressible fluids such that, Navier-Stokes eq. becomes Final flow velocity comes from interplay between these two effects. Source of flow (pressure gradient) Diffusion of flow (Kinematic viscosity, /, plays a role of diffusion constant.)

Generation of Flow x P Expand Pressure gradient Source of flow  Flow phenomena are important in H.I.C to understand EOS

Diffusion of Flow Heat equation (: heat conductivity ~diffusion constant) For illustrative purpose, one discretizes the equation in (2+1)D space:

Diffusion ~ Averaging ~ Smoothing R.H.S. of descretized heat/diffusion eq. i j i j x x y y subtract Suppose  i,j is larger (smaller) than an average value around the site, R.H.S. becomes negative (positive). 2 nd derivative w.r.t. coordinates  Smoothing

Shear Viscosity Reduces Flow Difference Shear flow (gradient of flow) Smoothing of flow Next time step Microscopic interpretation can be made. Net momentum flow in space-like direction.  Towards entropy maximum state.

Necessity of Relaxation Time Non-relativistic case (Cattaneo(1948)) Fourier’s law  : “relaxation time” Parabolic equation (heat equation) ACAUSAL! Finite  Hyperbolic equation (telegraph equation) Balance eq.: Constitutive eq.:

Heat Kernel x x causality perturbation on top of background Heat transportation

Instability The 1 st order equation is not only acausal but also unstable under small perturbation on a moving back-ground. W.A.Hiscock and L.Lindblom, PRD31,725(1985). For particle frame with new EoM, see K.Tsumura and T.Kunihiro, PLB668, 425(2008). For a possible relation btw. stability and causality, see G.S.Danicol et al., J.Phys.G35, (2008).

Entropy Current (2 nd ) Assumption (2 nd order theory): Non-equilibrium entropy current vector has linear + quadratic dissipative term(s) constructed from (V , ,  , (u  )). Stability condition O.K.

The 2 nd Law of Thermodynamics: 2 nd order case Same equation, but different definition of  and . Sometimes omitted, but needed.  Generalization of thermodynamic force!?

Summary: Constitutive Equations Relaxation terms appear (  and   are relaxation time). No longer algebraic equations! Dissipative currents become dynamical quantities like thermodynamic variables. Employed in recent viscous fluid simulations. (Sometimes the last term is neglected.) : vorticity

Plan of this Lecture 1 st Day Hydrodynamics in Heavy Ion Collisions Collective flow Dynamical Modeling of heavy ion collisions (seminar) 2 nd Day Formalism of relativistic ideal/viscous hydrodynamics Bjorken’s scaling solution with viscosity Effect of viscosity on particle spectra (discussion)

PART 4 Bjorken’s Scaling Solution with Viscosity

“ Bjorken ” Coordinate 0 z t Boost  parallel shift Boost invariant  Independent of  s

Bjorken ’ s Scaling Solution Hydrodynamic equation for perfect fluids with a simple EoS, Assuming boost invariance for thermodynamic variables P=P() and 1D Hubble-like flow

Conserved and Non-Conserved Quantity in Scaling Solution expansion pdV work

Bjorken ’ s Equation in the 1 st Order Theory (Bjorken’s solution) = (1D Hubble flow) Q. Derive the above equation.

Viscous Correction Correction from shear viscosity (in compressible fluids) Correction from bulk viscosity  If these corrections vanish, the above equation reduces to the famous Bjorken equation. Expansion scalar = theta = 1/tau in scaling solution

is obtained from Super Yang-Mills theory. is obtained from lattice. Bulk viscosity has a prominent peak around T c. Recent Topics on Transport Coefficients Need microscopic theory (e.g., Boltzmann eq.) to obtain transport coefficients. Kovtun, Son, Starinet,… Nakamura, Sakai,… Kharzeev, Tuchin, Karsch, Meyer…

Bjorken ’ s Equation in the 2 nd Order Theory New terms appear in the 2 nd order theory.  Coupled differential equations Sometimes, the last terms are neglected. Importance of these terms  see Natsuume and Okamura, [hep-th]. where

Why only  00 -  zz ? In EoM of energy density, appears in spite of constitutive equations. According to the Bjorken solution,

Relaxation Equation?

Digression: Full 2 nd order equation? Beyond I-S equation, see R.Baier et al., JHEP 0804,100 (2008); Tsumura-Kunihiro?; D. Rischke, talk at SQM According to Rischke’s talk, constitutive equations with vanishing heat flow are

Digression (contd.): Bjorken ’ s Equation in the “ full ” 2 nd order theory See also, R.Fries et al.,PRC78,034913(2008). Note that the equation for shear is valid only for conformal EOS and that no 2 nd and 3er terms for bulk.

Model EoS (crossover) Crossover EoS: T c = 0.17GeV  = T c /50 d H = 3, d Q = 37

Energy-Momentum Tensor at  0 in Comoving Frame In what follows, bulk viscosity is neglected.

Numerical Results (Temperature) Same initial condition (Energy momentum tensor is isotropic) T 0 = 0.22 GeV t 0 = 1 fm/c /s = 1/4   = 3/4p Numerical code (C++) is available upon request.

Numerical Results (Temperature) Same initial condition (Energy momentum tensor is anisotropic) T 0 = 0.22 GeV t 0 = 1 fm/c /s = 1/4   = 3/4p Numerical code (C++) is available upon request.

Numerical Results (Temperature) Numerical code (C++) is available upon request.

Numerical Results (Entropy) T 0 = 0.22 GeV t 0 = 1 fm/c /s = 1/4   = 3/4p Same initial condition (Energy momentum tensor is isotropic) Numerical code (C++) is available upon request.

Numerical Results (Entropy) T 0 = 0.22 GeV t 0 = 1 fm/c /s = 1/4   = 3/4p Same initial condition (Energy momentum tensor is anisotropic) Numerical code (C++) is available upon request.

Numerical Results (Entropy) Numerical code (C++) is available upon request.

Numerical Results (Shear Viscosity) Numerical code (C++) is available upon request.

Numerical Results (Initial Condition Dependence in the 2 nd order theory) Numerical code (C++) is available upon request.

Numerical Results (Relaxation Time dependence) Saturated values non-trivial Relaxation time larger  Maximum  is smaller Relaxation time smaller  Suddenly relaxes to 1 st order theory

Remarks Sometimes results from ideal hydro are compared with the ones from 1 st order theory. But initial conditions must be different. Be careful what is attributed for the difference between two results. Sensitive to initial conditions and new parameters (relaxation time for stress tensor)

Plan of this Lecture 1 st Day Hydrodynamics in Heavy Ion Collisions Collective flow Dynamical Modeling of heavy ion collisions (seminar) 2 nd Day Formalism of relativistic ideal/viscous hydrodynamics Bjorken’s scaling solution with viscosity Effect of viscosity on particle spectra (discussion)

PART 5 Effect of Viscosity on Particle Spectra

Particle Spectra in Hydrodynamic Model How to compare with experimental data (particle spectra)? Free particles (/L>>1) eventually stream to detectors. Need prescription to convert hydrodynamic (thermodynamic) fields (/L<<1) into particle picture. Need kinetic (or microscopic) interpretation of hydrodynamic behavior.

Microscopic Interpretation Single particle phase space density in local thermal equilibrium: Kinetic definition of current and energy momentum tensor are

Matter in (Kinetic) Equilibrium uu Kinetically equilibrated matter at rest Kinetically equilibrated matter at finite velocity pxpx pypy pxpx pypy Isotropic distribution Lorentz-boosted distribution

Cooper-Frye Formula No dynamics of evaporation. Just counting the net number of particles (out-going particles) - (in-coming particles) through hypersurface  Negative contribution can appear at some space-like hyper surface elements.

1 st Moment u  is normalized, so we can always choose a  such that

1 st Moment (contd.) Vanishing for = i due to odd function in integrant. Q. Go through all steps in the above derivation.

2 nd Moment where,

Deviation from Equilibrium Distribution Grad’s 14 moments EoM for epsilons can be also obtained from BE. Neglecting anti-particles,

Taylor Expansion around Equilibrium Distribution

Taylor Expansion around Equilibrium Distribution (contd.)

14 Conditions “Landau” conditions (2) Viscosities (12) Epsilons can be expressed by dissipative currents

Relation btw. Coefficients and Dissipative Currents For details, see Sec.6 and Appendix C in Israel-Stewart paper. (Ann.Phys.118,341(1979)) Finally,

Remarks Boltzmann equation gives a microscopic interpretation of hydrodynamics. However, hydrodynamics may be applied for gas/liquid where the Boltzmann equation can not be applied. Deviation from equilibrium can be incorporated into phase space distribution.

References Far from Complete List … General –L.D.Landau, E.M.Lifshitz, Fluid Mechanics, Section –L.P.Csernai, Introduction to Relativistic Heavy Ion Collisions –D.H.Rischke, nucl-th/ –J.-Y.Ollitrault, [nucl-th]. Viscous hydro + transport coefficient –C.Eckart, Phys.Rev.15,919(1940). –M.Namiki, C.Iso,Prog.Theor.Phys.18,591(1957) –C.Iso, K.Mori,M.Namiki, Prog.Theor.Phys.22,403(1959) –I.Mueller, Z. Phys. 198, 329 (1967) –W.Israel, Ann.Phys.100,310(1976) –W.Israel. J.M.Stewart, Ann.Phys.118,341(1979) –A.Hosoya, K.Kajantie, Nucl.Phys.B250, 666(1985) –P.Danielewicz,M.Gyulassy, Phys.Rev.D31,53(1985) –I.Muller, Liv.Rev.Rel

References for Astrophysics/Cosmology N.Andersson, G.L.Comer, Liv.Rev.Rel., R.Maartens, astro-ph/ Disclaimer: I’m not familiar with these kinds of review…