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									  Viscous hydrodynamics for
relativistic heavy-ion collisions
               Huichao Song

             Department of Physics
             The Ohio State University
             191 West Woodruff Avenue
             Columbus, OH 43210


     2007 National Nuclear Summer School
       FSU, FL. July 8 – July 21, 2007

      Advisor: U. Heinz, Supported by DOE
The QCD Phase Diagram
          The QGP Shear Viscosity
The QGP shear viscosity of is in hot theoretical debates recently:
-Weakly coupled QCD prediction:   / s  0.15 ~ 0.5 P.Arnold,G.Moore,&Y.Yaffe, 00,03
-Strongly coupled AdS/CFT prediction:       / s  1 / 4  0.08  D.T. Son, et,al. 03


  The first estimation of the QGP shear viscosity from the experimental data (RHIC):
 --blast wave model + the first order theory corrections
                                                    D. Teaney, PRC 68 (2003) 034913

                                           Blast wave model analysis:
                                           -only counts the viscous corrections to the
                                            spectra during freeze-out procedure
                                           -neglects shear viscosity effects on hydro
                                              dynamic evolution before freeze-out


                                           Viscous hydrodynamics is
                                           needed!
                       Ideal hydrodynamics



                                                                            S.Bass
Hydrodynamics:
-A macroscopic tool to describe the expansion of QGP or hadronic matter

Conservation laws                       5 equ. 14 independent variables
    N  ( x)  0                      -reduce # of independent variables (ideal hydro)
     T  ( x)  0                   -Provide more equations? (viscous hydro)

Ideal hydrodynamics: reduce the independent variable from 14 to 6,   S   0
  N   nu                                     n   n (  u )
                                                
  T   (  p )u  u  pg                  e  (e  p )(   u )
                                                
   Input: “EOS”           ( p, n)            u     p /( e  p )
                                                
       Causal viscous hydrodynamics
Full tensor decomposition in frame of u  :
                                                                 Dissipative flows
 N  nu  V
                                                           V  charge flow W energy flow
                                                                            
T   (  p)u  u  pg 
                                                              bulk pressure        entropy flow
         (W  u  W  u  )       
                                                             shear   pressure tensor
S   su   
                            



 Besides these modified transport equations for n, e,           u , one need more equations
 for these dissipative flows:                                                S   0
 The first order theory formulism:
                               --- Eckart,    Landau & Lifishitz around 1950’s


                    
        q      T  Tu 
                                                Problems in the first order theory formulism:
                                                  -Acuasl, infinite speed of signal propagation
                      
                2                             -process instabilities
       Causal viscous hydrodynamics
Full tensor decomposition in frame of u  :
                                                                   Dissipative flows
 N  nu  V
                                                             V  charge flow W energy flow
                                                                              
T   (  p)u  u  pg 
                                                                bulk pressure        entropy flow
         (W  u  W  u  )       
                                                               shear   pressure tensor
S   su   
                         



 The 2nd order Israel-Stewart formulation:                          W. Israel, J.Stewart 79
--expansion the entropy current to the 2nd order of dissipative flows,  S   0
                                                                                    

         1 T    0 u     0  q 
                                
                           T      
                 2                 
                        
  q  q  q      T  Tu  
                                              
                                                2
                                                              
                                                              T
                                                                      
                                                  q  T 2   1 2 u    0     1  
                                                                      
                                        2          
        
                    
                              2   T  
                                  
                                                  u    2 q
                                              2T 
                                                   
     Causal viscous hydro in (2+1)-d
                                             U. Heinz, H. Song & A. Chaudhuri, PRC 73(2006) 034904

-Bjorken approximation:             v z  z / t ( , x, y, ) coordinates 3+1                        2+1
 -zero net baryon density, zero bulk pressure, zero heat conduction
     T mn , m  0             Equivalent to the modified transport equations for            e, u 
     D mn  
                        
                          1
                               ( mn  2
                                                      m
                                                          u
                                                              n
                                                                     
                                                                  )  u m k  u n k Du k
                                                                           n        m
                                                                                               
                                                                  1                    1
  T   (  p)u  u  pg                        u   [  u   u  ]    u
                                                                  2                    3

where, the transport equations for energy momentum tensor are explicit written as:

 1                x         y     p   2 
    (T )   x (T )   y (T )                                                         T x   x
                                                                                   v x  
                                                                                          T  p   
 1
    (T x )   x ((T x  x )vx )   y ((T x  x)v y )   x ( p  xx )   y xy        T y   y
                                                                                             v y  
                                                                                                   T  p   
 1                             y                       y
    (T y )   x ((T   )vx )   y ((T   )v y )   y ( p  yy)   x xy
                           y                        y

 
   Viscous vs. ideal hydro: temperature & entropy
Initial & final conditions: (viscous & ideal)   e0  30GeV/fm 3 , 0  0.6fm/c, Tdec  130MeV
Other viscous hydro parameter: Shear viscosity& relaxation time      / s  1 / 4 ,    3 / sT




-slowing down of cooling process due to decelerated longitudinal expansion initially ,
 but faster cooling in middle and late stages due to stronger transverse expansion
-viscous effects are larger in early and middle stages, but neglectable in late stage
  Viscous vs. ideal hydro: radial flow & spectra




 dN     p  d 3 ( x)                                   p  d 3 ( x)              1 p p    ( x) 
E 3                 [ f eq ( x, p)  f ( x, p)]                               2 T 2 ( x) (e  p)( x) 
                                                                      f eq ( x, p)1                      
 d p       2  3                                        (2 )  3
                                                                                                          

-More radial flow, flatter spectra
-the viscous effects to the hadron spectra could be absorbed by starting viscous
  hydro later with lower initial energy density
              Viscous vs. ideal hydro: elliptical flow                      v2
For non-central collisions:
          initial energy
          density distribution




    dN        dN
E        
    d 3 p dypT dpT d
     1   dN
               [1  2v2 ( pT , b) cos( 2 )  ...]
    2 dypT dpT

-Elliptical flow is very sensitive to even minimal shear viscosity.
-Both the evolution corrections (viscous corrections to f 0 and spectra corrections
                                                           )
 (viscous corrections to f) have significant effects to v2 for low pTregion evolution
                                                             ,
 correction dominant.
          Summery and Conclusions
- Shear viscosity results in slowing down of the cooling process in early stage due
   to decelerated longitudinal expansion, faster cooling in middle and late stages due
   to stronger transverse expansion.

-Shear viscosity leads to more radial flow generation and flatter spectra, the effects
 of which could be absorbed by starting viscous hydro later with lower initial energy
 density.

-elliptical flow v2is very sensitive to shear viscosity, even the minimum value from
 AdS/CFT could result in a great suppression of v2.

-in contrast with the Teaney’s blast wave model method, which only counts viscous
 spectra correctionsfduring freeze-out procedure, our results shows that evolution
 -corrections ( f 0 ) are the main factor to influence the low pT spectra elliptical flow.

 -Do the results indicate that the QGP viscosity smaller than the AdS/CFT lowest
  limit? Need more investigations in the at least flowing aspects:
      a)Different initial conditions
      b) more realistic descriptions of the hadronic stage
Thank You
            1st order theory vs. 2nd order theory
As the relaxation time reduces to zero, the 2nd order theory formulism returns to the
1st order one.              First order theory , instable ?

     1st theory vs. 2nd order theory               v2vs. different relaxation time
   a)                                         b)




a)   calculated with the same velocity profile from the 2nd theory by
     1st
                                                                             mn  2mun
                                                                              1st

   not the real 1st order Navier-Stokes hydrodynamics
 b)Smaller relaxation time reduces the viscous effects on suppression of v2
Blast wave model vs. real viscous hydrodynamics
       Blast wave model               vs.               viscous hydrodynamics
+ the1st order theory corrections           (the 2 order theory Israel-Stewart formulation)
--with only spectra corrections             --with both evolution and spectra corrections

Compare the spectra corrections from blast wave model and from real hydrodynamics:




 -Spectra corrections are sensitive to the details of flow pattern at freeze out, and are
  not easily captured with blast-wave model parametrizations.
viscous vs. ideal hydro: central Cu+Cu collisions:
viscous vs. ideal hydro: central Cu+Cu collisions:
Viscous vs. ideal hydro: momentum anisotropy




           x2  y2              T0xx  T0yy               T xx  T   yy

    x              ,   p                     '
                                                      
           x2  y2              T0xx  T0yy               T xx  T
                                                  p                  yy
           Where ideal hydro works
pT spectra for both central collision and noncentral collisions :

                                 Hydro: U Heinz & P.Kolb   Data: STAR PHENIX
               Where ideal hydro works
 Elliptic flow coefficient     v2 at noncentral collisions:
               STAR PRL04 PHENIX PRL03                             STAR, PHENIX, PHOBOS




     v2 as a function of centrality               v2 ( pT ) for different identified hadrons

-Ideal hydro describes the data well at b  7 fm, (nch / n max  0.5), pT  1.5 ~ 2GeV

-It also gives a correct mass splitting of     v2at low   pT region

								
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