1. PART 2 Formalism of relativistic ideal/viscous hydrodynamics

2. Advertisement: Lecture Notes Chapter 4

3. Hydrodynamics Framework to describe space-time evolution of thermodynamic variables Balance equations (equations of motion, conservation law) + equation of state (matter property) + constitutive equations (phenomenology)

4. 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

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

6. “ 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):

7. Intuitive Picture of Projection time like flow vector field

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

9. 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).

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

11. 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)

12. 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 can be relevant in H.I.C.

13. Relation btw. Landau and Eckart

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

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

16. 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.

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

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

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

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

22. Conserved Current Case

23. 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.)

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

25. Diffusion of Flow Heat equation (  : 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. 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

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)) Fourier’s law  : “relaxation time” Parabolic equation (heat equation) ACAUSAL! Finite  Hyperbolic equation (telegraph equation) Balance eq.: Constitutive eq.:

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

30. 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.

31. 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!?

32. 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

33. PART 3 Bjorken’s Scaling Solution with Viscosity

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

35. 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

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

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

38. 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

39. 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…

40. 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 where

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

42. Relaxation Equation?

43. 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 2008; A.Monnai and TH, in preparation. According to Rischke’s talk, constitutive equations with vanishing heat flow are

44. 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 3 rd terms for bulk.

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

46. Relativistic Ideal Gas Thermodynamic potential for relativistic ideal gases

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

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

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

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

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

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

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

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

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

56. 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

57. 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)

58. PART 4 Effect of Viscosity on Particle Spectra

59. 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.

60. 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.

61. 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

62. Relativistic Transport Theory Boltzmann equation Time evolution of phase space dist. for a rarefied gas: One-particle phase space distribution Collision term Lorentz invariant transition rate gain loss

63. Free Streaming, Gain and Loss x p Note: p is mass-on-shell. gain loss gain

64. H-theorem “Entropy current”:

65. H-theorem (contd.)

66. H-theorem (cond.2)

67. H-theorem (contd.3) =0

68. H-theorem (contd.4) 1 1 0 One can identify s  with entropy current

69. Collision Invariant Binary collisions satisfy energy-momentum consevation So where (collision invariant)

70. Collision Invariant (contd.) Q. Check this identity Conservation law 5 conserved currents (quantities)

71. Thus should be collision invariant Equilibrium Distribution  and   can depend on position x and are to be  /T and u  /T

72. Quantum Statistics Bose enhancement or Pauli blocking Collision invariant

73. Some Remarks Collision term vanishes if f 0 is plugged in. If  and  is constant globally, f 0 can be a solution of Boltzmann equation. Local equilibrium distribution is NOT a solution of Boltzmann equation. Deviation from local equilibrium distribution can be obtained from Boltzmann equation.

74. Microscopic Interpretation Single particle phase space density in local equilibrium (no entropy production in Boltzmann eq.): Kinetic definition of current and energy momentum tensor are

75. 1 st Moment u  is normalized, so we can always choose a  such that

76. 1 st Moment (contd.) Vanishing for = i due to odd function in integrant. Q. Go through all steps in the above derivation.

77. 2 nd Moment

78. 2 nd Moment (contd.) where

79. Deviation from Equilibrium Distribution trace part  scalar Unknowns: 4 + 10 = 14 Important in a multi-component gas (A.Monnai and TH, (2009))

80. Taylor Expansion around Equilibrium Distribution

81. Taylor Expansion around Equilibrium Distribution (contd.)

82. 14 Conditions Stability conditions (2) Viscosities (12) Epsilons can be expressed by dissipative currents

83. Stability Condition Q. Check the above derivation.

84. Stability Condition (contd.)

85. Relation btw. Coefficients and Dissipative Currents

86. Moments  Bosons do not contribute…

87. Solutions

88. Solutions (contd.) Epsilons are expressed using hydrodynamic variables  Able to calculate particle spectra

89. Cooper-Frye Formula in Viscous Case

90. Transverse Momentum Spectra for Pions Bulk pressure  Isotropic   = -  Shear stress tensor  Traceless   = 4  /3

91. Summary Effect of viscosity on particle spectra can be calculated using hydrodynamic variables. Important to constrain equation of state and transport coefficients from experimental data