Relativistic Ideal and Viscous Hydrodynamics

1. Intensive Lecture YITP, December 11th, 2008 Relativistic Ideal and Viscous Hydrodynamics Tetsufumi Hirano Department of Physics The University of Tokyo TH, N. van der Kolk, A. Bilandzic, arXiv:0808.2684[nucl-th]; to be published in “Springer Lecture Note in Physics”.

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

3. PART 3 Formalism of relativistic ideal/viscous hydrodynamics

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

5. Tensor/Vector Decomposition Tensor decomposition with a given time-like and normalized four-vector um where,

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

7. Intuitive Picture of Projection time like flow vector field

8. Decomposition of Tmu :Energy density :Energy (Heat) current :Shear stress tensor :(Hydrostatic+bulk) pressure P = Ps + P <…>: Symmetric, traceless and transverse to um & un

9. Decomposition of Nm :charge density :charge current Q. Count the number of unknowns in the above decomposition and confirm that it is 10(Tmn)+4k(Nim). Here k is the number of independent currents. Note: If you consider um as independent variables, you need additional constraint for them. If you also consider Ps as an independent variable, you need the equation of state Ps=Ps(e,n).

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

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

12. Meaning of um (contd.) Landau (Wm=0, uLmVm=0) Eckart (Vm=0,uEmWm=0) Wm Vm uEm uLm Just a choice of local reference frame. Landau frame might be relevant in H.I.C.

13. Relation btw. Landau and Eckart

14. Relation btw. Landau and Eckart (contd.)

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

16. Entropy Current Assumption (1st order theory): Non-equilibrium entropy current vector has linear dissipative term(s) constructed from (Vm, P, pmn, (um)). • (Practical) Assumption: • Landau frame (omitting subscript “L”). • No charge in the system. Thus, a = 0 since Nm = 0, Wm = 0 since considering the Landau frame, and g = 0 since um Sm should be maximum in equilibrium (stability condition).

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

18. Constitutive Equations (contd.) Newton Stokes

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

20. 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.

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

22. Conserved Current Case

23. Diffusion of flow (Kinematic viscosity, h/r, plays a role of diffusion constant.) Source of flow (pressure gradient) 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.

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

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

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

27. 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.

28. Necessity of Relaxation Time Non-relativistic case (Cattaneo(1948)) Balance eq.: Constitutive eq.: Fourier’s law t : “relaxation time” Parabolic equation (heat equation) ACAUSAL! Finite t Hyperbolic equation (telegraph equation)

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

30. Instability • The 1st 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, 115102(2008).

31. Entropy Current (2nd) Assumption (2nd order theory): Non-equilibrium entropy current vector has linear + quadratic dissipative term(s) constructed from (Vm, P, pmn, (um)). Stability condition O.K.

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

33. Summary:Constitutive Equations w: vorticity • Relaxation terms appear (tp and tP 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.)

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

35. PART 4 Bjorken’s Scaling Solution with Viscosity

36. “Bjorken” Coordinate t Boost  parallel shift Boost invariant  Independent of hs z 0

37. Bjorken’s Scaling Solution Assuming boost invariance for thermodynamic variables P=P(t) and 1D Hubble-like flow Hydrodynamic equation for perfect fluids with a simple EoS,

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

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

40. 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

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

42. Bjorken’s Equationin the 2nd Order Theory where New terms appear in the 2nd order theory.  Coupled differential equations Sometimes, the last terms are neglected. Importance of these terms  see Natsuume and Okamura, 0712.2917[hep-th].

43. Why only p00-pzz? In EoM of energy density, appears in spite of constitutive equations. According to the Bjorken solution,

44. Relaxation Equation?

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

46. Digression (contd.): Bjorken’s Equationin the “full” 2nd 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 2nd and 3er terms for bulk.

47. Model EoS (crossover) Crossover EoS: Tc = 0.17GeV D = Tc/50 dH = 3, dQ = 37

48. Energy-Momentum Tensorat t0 in Comoving Frame In what follows, bulk viscosity is neglected.

49. Numerical Results (Temperature) T0 = 0.22 GeV t0 = 1 fm/c h/s = 1/4p tp= 3h/4p Same initial condition (Energy momentum tensor is isotropic) Numerical code (C++) is available upon request.

50. Numerical Results (Temperature) T0 = 0.22 GeV t0 = 1 fm/c h/s = 1/4p tp= 3h/4p Same initial condition (Energy momentum tensor is anisotropic) Numerical code (C++) is available upon request.