1. Hydro + Cascade Model at RHIC
Duke University, Chiho Nonaka
Introduction
3-d Hydrodynamic Model
Hydro + Cascade Model
Results (hadron spectra, elliptic flow)
Summary
Contents
In Collaboration with
Steffen A. Bass (Duke University & RIKEN)
October 30, Chicago

2. Introduction Hydrodynamic Model at RHIC Success Failure?
Single particle spectra
Elliptic flow
Morita, Muroya, CN and Hirano,
PRC66:054904,2002
Hirano and Tsuda, PRC66
HBT
Elliptic flow
Morita et al., PRC66
Huovinen et.al, PLB503
Possible solution?

3. Freeze-out, Final Interactions
universal Tf for all hadrons in Hydro
Freeze-out
Tf & 
Final interactions
Thermal + radial flow fit
Thermal model
T = 177 MeV
 = 29 MeV
UrQMD
STAR, nucl-ex/
Freeze-out is not universal for all hadron spectra.
Rescattering and regeneration is important.

4. Hydro + Cascade Model Hydro + hadron transport model Key:
Bass and Dumitru, PRC61,064909(2000)
Teaney et al, nucl-th/
Hydro + hadron transport model
Key:
Freeze-out condition ex. Chemical and kinetic freeze-out
Final interactions
Hirano and Tsuda, PRC66(2002)054905
Treatment of freeze-out in transport model is determined
by mean free path.
Full 3-d Hydrodynamics
EoS 1st order phase transition
QGP + excluded volume model
Hadronization
UrQMD
( Improved)
Cooper-Frye
formula (Reco)
Final
interactions
Monte Carlo
t fm/c

5. 3-d Hydrodynamic Model Hydrodynamic equation Coordinates
Baryon number density conservation
Coordinates
Lagrangian hydrodynamics
Tracing the adiabatic path of each
volume element
Effects of phase transition of observables
Algorithm
Focusing on conservation law
Lagrangian hydrodynamics
Discretized grids move along the expansion of the fluid; therefore, we can perform
The calculation at all stages on the lattice points which we prepare under the initial
Condisions.
Flux of
fluid

6. Trajectories on the Phase Diagram
Lagrangian hydrodynamics
temperature and chemical
potential of volume element
of fluid
effect of
phase transition
C.N et al., Eur. Phys.J C17,663(2000)

7. Parameters Initial Conditions Equation of State Hydro UrQMD
Energy density
Baryon number density
Parameters
Flow
longitudinal: Bjorken’s Solution
Equation of State
1st order phase transition
QGP phase (Bag model), mixed phase,
hadron phase (up to 2GeV)
(excluded volume model)
Bag constant:
Hydro UrQMD

8. Hadron Spectra (I) Pure Hydro Hydro works well up to PT ~ 2 GeV
Central collision
Parameters
Hydro works well up to PT ~ 2 GeV

9. Hadron Spectra (II) Hydro + UrQMD Many pions are produced in UrQMD.
Low PT resonances
High PT interactions
Transition temperature is too low.
PT slope becomes flatter.
Extra radial flow in UrQMD
The initial condition for Hydro + UrQMD
Is different from that for pure hydro.

10. Elliptic Flow Pure Hydro Centrality 5-10 % Centrality dependence
Hydro works well.
Centrality dependence

11. Elliptic Flow (II) Hydro + UrQMD preliminary
In UrQMD elliptic flow becomes small ?
Shape of elliptic flow as a function of 

12. Summary Hydro + Cascade Model Work in progress
Hadron Spectra, elliptic flow
Effect of resonances, final interactions in experimental data
Work in progress
Parameter
Initial Conditions
EoS (QCD critical point) ,CN and Asakawa nucl-th/
Parton Cascade Model
Hadronization mechanism
Recombination + Fragmentation model, Duke Group

13. BACK UP

14. Numerical Calculation
Step 1.
Step 2.
Step 3.
Coordinates move in parallel
with baryon number current
and entropy density current.
local velocity:
from hydro eq.
temperature and chemical potential
CPU time is almost
proportional of # of lattice points.