Relativistic Hydrodynamics

1
Part III
Relativistic Hydrodynamics For Modeling
Ultra-Relativistic Heavy Ion Reactions
2
Multi Module Modeling

• Initial state - pre-equilibrium Parton
  Cascade Coherent Yang-Mills Magas
• Local Equilibrium ? Hydro, EoS
• Final Freeze-out Kinetic models, measurables
• If QGP ? Sudden and simultaneous hadronization
  and freeze out (indicated by HBT, Strangeness,
  Entropy puzzle)

Landau (1953), Milekhin (1958), Cooper Frye
(1974)
3
Matching Conditions

• Conservation laws
• Nondecreasing entropy

4
Initial stage Coherent Yang-Mills model
Magas, Csernai, Strottman, NEW2001
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Expanding string ropes Full energy conservation
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Initial state
3rd flow component
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Freeze out
L Bravina et al.
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Hypersurface
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Cooper-Frye formula
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Conservation Laws across hypersurface
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Matching Conditions

• Conservation laws
• Nondecreasing entropy

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Consequences of conservation laws Problem I

• Non-decreasing entropy current across front!

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Aside Taub-adiabat and Rayleigh line
Perfect fluid on both sides of the front!
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Aside Taub-adiabat and Rayleigh line
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Aside Taub-adiabat and Rayleigh line
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Aside Taub-adiabat and Rayleigh line
Goal scalar equations
I Parallel Projection
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Aside Taub-adiabat and Rayleigh line
Taub 48 missed the sign ? was not applicable
for freeze-out.
The Rayleigh line is a straight line in the P,X
plane. It gives the locus of final states 2 if
the initial state 1 is known. The slope, j,
is given by the current across the front.
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Aside Taub-adiabat and Rayleigh line
II Orthogonal Projection
? To obtain scalar
.
Then to obtain scalar (one of the cross terms
and the last term cancel)
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Aside Taub-adiabat and Rayleigh line
Comparing the two equations for the current, j ,

So, we obtain the Taub adiabat
The locus of the possible final states, 2, lies
on the Taub adiabat. If the initial state and the
EoS of the final state is known the Taub adiabat
with the Rayleigh line determine the final state.
If the final state is out of equilibrium, I.e.
not a perfect fluid, this is not applicable!
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Aside Taub-adiabat and Rayleigh line
Taub-adiabat for final states E.g. Bag Model
EoS P (4B Po )/3 (X Xo /3)
wo 4(B P0 )/3 Xo /3 Eg. Ideal gas EoS
(P 2Po /3) (X 2Xo /3) (eo - 2Po /3)
2Xo /3
P
space-like
Rayleigh-line
time-like
space-like
1
2 Taub-adiabat
X
Problem I is Solved
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Space-like hypersurface - Problem II
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Space-like hypersurface II
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Matching Conditions

• Conservation laws
• Nondecreasing entropy

If the final state is out of Eq., the
energy-momentum tensor has to be evaluated, and
the above eqs. solved!!!
e.g.. Anderlik et al. Phys.Rev.C 59 (99) 3309
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Post F.O. - Cut-Jüttner distribution
Bugaev, Nucl.Phys.A606(96)559
Proposed by
p
x
Anderlik et al., Phys.Rev.C59(99)3309
Solved
Post F.O. distribution ??pmLm??f(p)
p
y
V-flow
Matching conditions determine 5 parameters only .
Ansatz in needed for final f(x,p) !
V-parameter
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Cut Jüttner distribution
T(p.ds) f(x,p)
Problem II is partly solved
Pre FO velocity
Anderlik et al., Phys.Rev.C59(99)3309
Better non-eq. ansatz Tamousiunas in pr.
Bugaev, Nucl.Phys.A606(96)559
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Kinetic freeze-out models

• Kinetic approach
• f (x,p) out of equilibrium
• Asymmetry

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Freeze out model with rescattering
Anderlik et al., Phys.Rev.C59 (1999) 388-394
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Freeze out distribution with rescattering
V0
V. Magas, et al., Heavy Ion Phys.9193-216,1999
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Freeze out model with rescattering
V 0.5
V. Magas, et al. Heavy Ion Phys.9193-216,1999
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Change of the rest temperature in FO
V. Magas, et al., Heavy Ion Phys.9193-216,1999
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Change of the rest velocity during FO
V. Magas, et al., Heavy Ion Phys.9193-216,1999
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P-t distribution (T130 MeV)
V. Magas et al., Phys.Lett.B459(99)33
Croonin effect ?
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Conclusion

• Hydro works amazingly well! Stronger and stronger
  hydro effects are observed!
• ? Equilibrium and EoS exists ( in part of the
  reaction )
• We have a good possibility to learn more and more
  about the EoS, with improved experimental and
  theoretical accuracy!