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Numerics, Initial
Conditions, EOS, Freeze-out and
the Role of Dissipation in
Relativistic Hydro.



Udo Ornik
SoulTek GbR
Homberg/Ohm
Content
   Numerical Issues of HYLANDER
       Solution of PDE-System
       Grid
       Numerical PT as LHC
       Stabilty, Quality, Tests
   Initial Conditions
       Partonic or Collective
       Collision of an Ideal Fluid
       Taking into account Uncertainty
       Baryon-Model
       Navier-Stokes Solutions
   EOS & Freeze-Out Problems
       Inconsistencies between Fluid and Free Flow Stage
       Absorption Problem
       Resonances and Pion chemical Potential
       Leakage in Hydro
       How to deal with Hyper surfaces
Solution of the PDE-System I

  M i                                    1 
                     ( M iv1  P )              ( rM i v 2 )
   t             x                       r r
    E                                     1 
                     ( v1 ( E  P ))            ( rv 2 ( E  P ))
    t            x                        r r
    B                             1 
                     ( Bv 1 )           ( rBv 2 )
    t            x                r r
     E   (   Pv )
              2               2



    M i   (  P ) v i
              2



     B  b
     P  c ( , b )
              2




Requirement: spatial and time derivatives have to be separable!
1d Navier-Stokes




                           
D q  v t (       ) z(       )
               T            T
                                            In 1d Viscosity leads to an
    t    z ( v )                     effective Pressure
Solutions require implicit algorithms or iteration procedure
e.g.
                                    ~ n 1
                                    Dq  Dq
                                           n

   Approximation:                    n 1            ~ n 1
                                             0 .5 ( D q  D q )
                                                             n
                                    Dq
Solution of the PDE-System II
               Numerical Operations                                        f        g
                                                                      :         
               for a simple example                                        t         x
                                   k 1
               f ( x, t )                fi
                                              k
                                  fi
                                                                                First Iteration
                   t                        t
                                                                                 k 1             t
                                  f   f  1
                                       k          k
               f ( x, t )                                                      fi      fi
                                                                                            k 1
                                                                                                     ( f i  f i 1 )
                                                                                                           k      k
                             
                  x                       x                                                      x
                 mod( k , 2 )  i
                                                                   k 1
, p :         f ( x, t )       fi
                                      k 1
                                              0 .5 ( f i
                                                            k
                                                                 fi )
                                                                                    Second Iteration
                   t                                 t
                                      k 1        k 1
               f ( x, t )       fp           f p 1                                                                k 1  t  k 1  k 1
                                                                               fi
                                                                                     k 1
                                                                                             0 .5 ( f i
                                                                                                           k 1
                                                                                                                   fi )     ( f p  f p 1 )
                  x                         x                                                                            x
alternating
forward/       p  mod(   i  1, 2 )  i
backward
differences

              Algorithm generalized from nonrelativistic Flow-Problems
Solution of the PDE-System III

        tk+1




        tk+1
                                  k 1  k 1                         k 1
                                                                                         k 1
                                                                               0 . 5 ( E ij  E ij )
                                                                                                  k
t   r                  F        Fi  1 j  Fi , j       E       E
                                                              
                                                                      ij
                             
                       z                z              t                         t
                        Forward Derivative
    z
        tk
                                                                               k 1
                                 F F                                         E ij  E ij
                                     k          k                                       k
                       F                     i 1 , j        E
                             
                                   ij
                                                                      
                        z               z                   t                  t
                       Backward Derivative
Grid Growth for high density problems

                       x i 1  c  x i
                      c 1




                          Very slow cooling of
                          edges (relativistic
                          effect)
Numerical
                               „Phase
                               Transition“
                               at LHC



Real*4  Real *8

v cm  0.9999999
 cm  3000


To reduce computing time:
grid-Size has to be enlarged
during collision
Stability, Quality & Tests
                        v   c0
Stability Condition:                t  x
                       1  v 


Tests:1d Khalatnikov, Shock, 1d Weiner/Masuda (asymmetric)
      1d rarefaction wave
      Global E,P,S conservation
                                               At least 10
                                               points for
 Accuracy: 1d < 0.1%                           profile
            3d 0.5-5%



 Duration: 1minute/[fm/c] on a 2 GH Celeron
            650x350 Grid cells
Initial Conditions I
   PDE-System needs initial condition
   Long History of Thermal/Hydro-Models
    by Fermi/Landau (1950+X).
   Yet unsolved question if initial state dominated
    by parton interactions ( as<1 ) or collective (as>1 )
   If parton dominated then initial collision cannot
    described by hydro, shocks
   But what happens before local equilibrium?
   However Landau was invented for cosmic rays
    and also works on earth until Tevatron energies
   Shows an extended 1d stage
   Hydro Collisions violates Quantum Mechanics at
    SPS
1st Hydro Models Landau/Khalatnikov

   3d
   radial

            Extended
            1d




Reason for success of Landau up to cosmic ray is extended
1d stage. 3d from beginning fails with too narrow distributions
Space-time of Collision
        Freeze-out Isotherm
                                                      f

                          Tf
                                                   0
                                                           sudden freeze-out


                              Quantum Limit
                                              1
                                      xi 
                                              Ti
Local equilibrium
Hydro
                                                           t0   Hydro
Pre-equilibrium                          c2
Shock
2-Fluid
Transport                                                  t0   Collision
RQMD
…
Uncertainty-model

           pi          p  p                f (T i ) for a system in local eq.
                                2            2



                                 1
            xi   i 
                                 p i Fluid Cell
                                                                                     1
Inelasticity                                                               c 
                                                                            2


                   K        s                                                        3
           i 
                  R  i                i
                        2



                    B part
           bi 
                  R  i
                       2
                                                                                i

           Ti ( i , bi )

                                                                      Ideal gas:
          Self consistent set of equations
          to determine  i                                            i 
                                                                               0 .1
                                                                           T [ GeV ]
Results UC-Model
Initial Conditions III

            Initial longitudinal Extension [fm]

1,4




                                                                   px  1
1,2
            EOS: Lattice
 1


0,8
                                                                            0 .1
                                                                    x 
                                                        Landau
                                                        UC-Model
0,6
                    Hydro Collisions violate
0,4
                    Quantum Limits                                            T
0,2


 0
      AGS   SPS                RHIC               LHC



At Energies > SPS Landau and all Hydro
Codes with collision violates QM
Baryonmodel
Parametrization
for SPS Pb+Pb Data
                     Initial Parametrization taking
                     into account:
                     Landau, QM, Leading Particle/
                     Inelasticity 2 Fluid effects
Shock Solution of RNS-Equation
Collision of 2 infinite extended nuclei




                                          Ornik,Mornas (1994)
Dissipation and Transparency

                            RNS-shock possible




             no RNS-shock
Width of Shock
1d Navier Stokes vs. 1d Euler
Navier-Stokes in the Expansion-Phase



 analytical solution:




 Remarkable:

 •Thermal conductivity terms disappear in 1d

 •Terms containing derivatives with viscosity
  disappear
1d Navier-Stokes for Bjorken-like
Expansion




          Strong effect   Weak effect for
          for collision   expansion
EOS
  Nucleonic Interaction inspired -> extendet to small baryon densities
  Hadron Gas/QGP inspired -> extended to large baryon densities
  Best results with Lattice and Hagedorn Gas




Ornik, Mornas 1994
EOS II
Newer results, invented by Blaschke, Bugaev show a deeper
relation between old Hagedorn and Lattice QCD


                                  Actual calculation by
                                  D. Blaschke and V. Yudichev.


                                           However:
                                           Problem for
                                           freeze-out

                                            HG  7  

 Programm ist to extend this
 to finite Baryon no. and recalculate up
 to LHC
Freeze-out I
   Simplest Ansatz: Final Particle Spektra
    at T(tf,x),b(tf,x) , sudden freeze-out 
    violates causality
   Better: Freeze-out Isotherm T = m
    Integration on isotherms.
   More accurate:Dynamical criterions
                                             1
     coll  n   i v   exp  (   u )
    chemical freeze-out for different
    particle species
   Final State interactions-decay of
    resoances  very time consuming (e.g.
    Parsifal Program)
Some results
EOS at freeze-out

      EOS -Fluid                  Free Particles

                                      ~ ~
                                  ~ , P ,b
        , P, b                   f f f

In general:
Fluid EOS contains Interaction, mean Fields, etc
free Particle State: all final particles one likes to take into
account
                 ~ ~
             ~ , P ,b
  , P, b   f  f   f

 No energy conservation or unphysical shocks instead
     of smooth transition
Resonances
             •Soft pt behavior was successfully explained by resonance
              decays

             •The decays can be effectively described by pion chemical
              potential

             •It also helps to overcome the smooth freeze-out
             problem:
                 ~
                f    ( u  , T )   Strange ( u s , T )   Baryons ( u s , T )  ....
Resonances
                   ~
             P  P f  P ( u  , T )  PStrange ( u s , T )  PBaryons ( u s , T )  ....
                 ~
             b  b  b (u b , u b , T )
Pion         s~ 0
                 s
ground
state         … and enormously speeds up the calculation!
Cooper, Fry Schoenberg Formula

         i
    dN                          
E    3
    d p
                   fi ( x, p ) p d 

i-particle species
Problem: Not coupled to Hydro-Expansion
unphysical particles, crossing the fluid
without absorption.
Freeze-out III
 Usually Hydro-Expansion and Freeze-Out treated independently
         Absorption Processes in the space-like area are not
        correctly treated!
                                          Small
                           
                    p d  0             (surface) effect –
                                          only in the
                                          rarefaction region




              




                                     [From PHD Thesis 1990]
Leakage Term

               dN                           
                               f ( x, p ) p d   ( p  d )
                                                               u
         E      3
              d p
                                                               
          T                 S          d   ( p  d )
             Astrophysical Pendant: Neutrino Cooling
             Faster Cooling by Leakage
             Numerical Advantage. Leaked part don’t
               have to be calculated any more
                                                             1
          coll  n   i v   exp  (   u )
t

     Fluid                                         Free

                                  Freeze-Out Surface                   x
Freeze-Out
                 How to connect and cover the Hyper surface
                                              t

                                                                                 15
                                                                            14 13
                                                                        12 11
                                                                     10     9
                                                          8               7
                                                 6
dx 2  ( t i  1  t i , z i  1  z i )                                5
                                                                      4
i  1,..,15                                               dx 2
                                                                    3


                                                                 2




                  3                                       1              d 2
             
        dx 2                                                                          z

            2
        1        d     2
                              ( z 2  z 1 , t1  t 2 )
Freeze-out
Conclusions I
   Initial State for E>AGS only as input for
    Eulerian-Hydro (Preequilibrium
    models, Relativistic Navier-Stokes, 2 Fluid
    Hydro or appropriate Parameterizations:
    e.g. Baryon-Model)
   Influence of Transport Terms low during
    the Expansion
    Good Approximation to use 1d -Rel.
    Navier-Stokes during collision
Conclusions II

   Freeze-Out on Hyper surface (T
    isotherm) requires consistency of
    fluid and particle densities
   Absorption effects on large volumes
    might be important
   Description by a Leakage Term
    ->faster cooling
Conclusions III
   At LHC „Phase Transition“ to double
    precision for the initial state
   To avoid time consuming Calculations:
      1d Approximation (and RNS) for the
       first fermi
      Grid-space adaptation steps to reduce
       Data
      Pion spectra by Pion chemical Potential
       instead of resonance production and
       decay

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IJCER (www.ijceronline.com) International Journal of computational Engineerin...
 

Numerical Technique, Initial Conditions, Eos,

  • 1. Numerics, Initial Conditions, EOS, Freeze-out and the Role of Dissipation in Relativistic Hydro. Udo Ornik SoulTek GbR Homberg/Ohm
  • 2. Content  Numerical Issues of HYLANDER  Solution of PDE-System  Grid  Numerical PT as LHC  Stabilty, Quality, Tests  Initial Conditions  Partonic or Collective  Collision of an Ideal Fluid  Taking into account Uncertainty  Baryon-Model  Navier-Stokes Solutions  EOS & Freeze-Out Problems  Inconsistencies between Fluid and Free Flow Stage  Absorption Problem  Resonances and Pion chemical Potential  Leakage in Hydro  How to deal with Hyper surfaces
  • 3. Solution of the PDE-System I M i  1   ( M iv1  P )  ( rM i v 2 ) t x r r E  1   ( v1 ( E  P ))  ( rv 2 ( E  P )) t x r r B  1   ( Bv 1 )  ( rBv 2 ) t x r r E   (   Pv ) 2 2 M i   (  P ) v i 2 B  b P  c ( , b ) 2 Requirement: spatial and time derivatives have to be separable!
  • 4. 1d Navier-Stokes   D q  v t ( ) z( ) T T In 1d Viscosity leads to an    t    z ( v ) effective Pressure Solutions require implicit algorithms or iteration procedure e.g. ~ n 1 Dq  Dq n Approximation: n 1 ~ n 1  0 .5 ( D q  D q ) n Dq
  • 5. Solution of the PDE-System II Numerical Operations f g :  for a simple example t x  k 1 f ( x, t )  fi k fi  First Iteration t t  k 1 t f   f  1 k k f ( x, t ) fi  fi k 1  ( f i  f i 1 ) k k  x x x   mod( k , 2 )  i  k 1 , p : f ( x, t ) fi k 1  0 .5 ( f i k  fi )  Second Iteration t t k 1 k 1 f ( x, t ) fp  f p 1  k 1  t  k 1  k 1  fi k 1  0 .5 ( f i k 1  fi )  ( f p  f p 1 ) x x x alternating forward/ p  mod(   i  1, 2 )  i backward differences Algorithm generalized from nonrelativistic Flow-Problems
  • 6. Solution of the PDE-System III tk+1 tk+1  k 1  k 1 k 1  k 1  0 . 5 ( E ij  E ij ) k t r F Fi  1 j  Fi , j E E  ij  z z t t Forward Derivative z tk  k 1 F F E ij  E ij k k k F i 1 , j E  ij  z z t t Backward Derivative
  • 7. Grid Growth for high density problems  x i 1  c  x i c 1 Very slow cooling of edges (relativistic effect)
  • 8. Numerical „Phase Transition“ at LHC Real*4  Real *8 v cm  0.9999999  cm  3000 To reduce computing time: grid-Size has to be enlarged during collision
  • 9. Stability, Quality & Tests  v   c0 Stability Condition: t  x 1  v  Tests:1d Khalatnikov, Shock, 1d Weiner/Masuda (asymmetric) 1d rarefaction wave Global E,P,S conservation At least 10 points for Accuracy: 1d < 0.1% profile 3d 0.5-5% Duration: 1minute/[fm/c] on a 2 GH Celeron 650x350 Grid cells
  • 10. Initial Conditions I  PDE-System needs initial condition  Long History of Thermal/Hydro-Models by Fermi/Landau (1950+X).  Yet unsolved question if initial state dominated by parton interactions ( as<1 ) or collective (as>1 )  If parton dominated then initial collision cannot described by hydro, shocks  But what happens before local equilibrium?  However Landau was invented for cosmic rays and also works on earth until Tevatron energies  Shows an extended 1d stage  Hydro Collisions violates Quantum Mechanics at SPS
  • 11. 1st Hydro Models Landau/Khalatnikov 3d radial Extended 1d Reason for success of Landau up to cosmic ray is extended 1d stage. 3d from beginning fails with too narrow distributions
  • 12. Space-time of Collision Freeze-out Isotherm  f Tf 0 sudden freeze-out Quantum Limit 1  xi  Ti Local equilibrium Hydro t0 Hydro Pre-equilibrium c2 Shock 2-Fluid Transport t0 Collision RQMD …
  • 13. Uncertainty-model pi   p  p   f (T i ) for a system in local eq. 2 2 1  xi   i   p i Fluid Cell 1 Inelasticity c  2 K s 3 i  R  i i 2 B part bi  R  i 2 i Ti ( i , bi ) Ideal gas: Self consistent set of equations to determine  i i  0 .1 T [ GeV ]
  • 15. Initial Conditions III Initial longitudinal Extension [fm] 1,4 px  1 1,2 EOS: Lattice 1 0,8 0 .1  x  Landau UC-Model 0,6 Hydro Collisions violate 0,4 Quantum Limits T 0,2 0 AGS SPS RHIC LHC At Energies > SPS Landau and all Hydro Codes with collision violates QM
  • 16. Baryonmodel Parametrization for SPS Pb+Pb Data Initial Parametrization taking into account: Landau, QM, Leading Particle/ Inelasticity 2 Fluid effects
  • 17. Shock Solution of RNS-Equation Collision of 2 infinite extended nuclei Ornik,Mornas (1994)
  • 18. Dissipation and Transparency RNS-shock possible no RNS-shock
  • 20. 1d Navier Stokes vs. 1d Euler
  • 21. Navier-Stokes in the Expansion-Phase analytical solution: Remarkable: •Thermal conductivity terms disappear in 1d •Terms containing derivatives with viscosity disappear
  • 22. 1d Navier-Stokes for Bjorken-like Expansion Strong effect Weak effect for for collision expansion
  • 23. EOS Nucleonic Interaction inspired -> extendet to small baryon densities Hadron Gas/QGP inspired -> extended to large baryon densities Best results with Lattice and Hagedorn Gas Ornik, Mornas 1994
  • 24. EOS II Newer results, invented by Blaschke, Bugaev show a deeper relation between old Hagedorn and Lattice QCD Actual calculation by D. Blaschke and V. Yudichev. However: Problem for freeze-out  HG  7   Programm ist to extend this to finite Baryon no. and recalculate up to LHC
  • 25. Freeze-out I  Simplest Ansatz: Final Particle Spektra at T(tf,x),b(tf,x) , sudden freeze-out  violates causality  Better: Freeze-out Isotherm T = m Integration on isotherms.  More accurate:Dynamical criterions  1  coll  n   i v   exp  (   u ) chemical freeze-out for different particle species  Final State interactions-decay of resoances  very time consuming (e.g. Parsifal Program)
  • 27. EOS at freeze-out EOS -Fluid Free Particles ~ ~ ~ , P ,b  , P, b f f f In general: Fluid EOS contains Interaction, mean Fields, etc free Particle State: all final particles one likes to take into account ~ ~ ~ , P ,b   , P, b   f f f  No energy conservation or unphysical shocks instead of smooth transition
  • 28. Resonances •Soft pt behavior was successfully explained by resonance decays •The decays can be effectively described by pion chemical potential •It also helps to overcome the smooth freeze-out problem: ~    f    ( u  , T )   Strange ( u s , T )   Baryons ( u s , T )  .... Resonances ~ P  P f  P ( u  , T )  PStrange ( u s , T )  PBaryons ( u s , T )  .... ~ b  b  b (u b , u b , T ) Pion s~ 0 s ground state … and enormously speeds up the calculation!
  • 29. Cooper, Fry Schoenberg Formula i dN  E 3 d p   fi ( x, p ) p d  i-particle species Problem: Not coupled to Hydro-Expansion unphysical particles, crossing the fluid without absorption.
  • 30. Freeze-out III Usually Hydro-Expansion and Freeze-Out treated independently Absorption Processes in the space-like area are not  correctly treated! Small  p d  0 (surface) effect – only in the rarefaction region  [From PHD Thesis 1990]
  • 31. Leakage Term dN    f ( x, p ) p d   ( p  d ) u E 3 d p     T  S d   ( p  d ) Astrophysical Pendant: Neutrino Cooling Faster Cooling by Leakage Numerical Advantage. Leaked part don’t have to be calculated any more  1  coll  n   i v   exp  (   u ) t Fluid Free Freeze-Out Surface x
  • 32. Freeze-Out How to connect and cover the Hyper surface t 15 14 13 12 11 10 9 8 7  6 dx 2  ( t i  1  t i , z i  1  z i ) 5 4 i  1,..,15 dx 2  3 2 3 1 d 2  dx 2 z 2 1 d 2  ( z 2  z 1 , t1  t 2 )
  • 34. Conclusions I  Initial State for E>AGS only as input for Eulerian-Hydro (Preequilibrium models, Relativistic Navier-Stokes, 2 Fluid Hydro or appropriate Parameterizations: e.g. Baryon-Model)  Influence of Transport Terms low during the Expansion Good Approximation to use 1d -Rel. Navier-Stokes during collision
  • 35. Conclusions II  Freeze-Out on Hyper surface (T isotherm) requires consistency of fluid and particle densities  Absorption effects on large volumes might be important  Description by a Leakage Term ->faster cooling
  • 36. Conclusions III  At LHC „Phase Transition“ to double precision for the initial state  To avoid time consuming Calculations:  1d Approximation (and RNS) for the first fermi  Grid-space adaptation steps to reduce Data  Pion spectra by Pion chemical Potential instead of resonance production and decay