The document discusses several key issues in relativistic hydrodynamics simulations:
1) Numerical methods for solving the relativistic hydrodynamic equations, including grid structure, stability, and accuracy tests.
2) Initial conditions for relativistic heavy ion collisions, including whether the initial state is partonic or collective, and accounting for uncertainties.
3) The equation of state and challenges connecting the fluid and free streaming stages, such as inconsistencies between the fluid and particle descriptions and absorption problems.
4) Modeling freeze-out and particle production, including treatment of resonances, pion chemical potentials, and leakage terms to couple the fluid to free-streaming particles.
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 ]
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