Hydrodynamical Simulation of Relativistic Heavy Ion Collisions

1
Hydrodynamical Simulation of Relativistic Heavy
Ion Collisions
Strongly Coupled Plasmas Electromagnetic,
Nuclear and Atomic

• Tetsufumi Hirano

2
Introduction

• Features of heavy ion collision at RHIC
• System of strongly interacting particles
• Quantum ChromoDynamics
• Quarks Gluons / Hadrons
• Phase transition from Quark Gluon Plasma to
  hadrons
• Dynamically evolving system
• Transient state (life time 10 fm/c 10-23 sec)
• No heat bath. Control parameters collision
  energy and the size of nucleus.
• The number of observed hadrons lt5000
• Impact parameter can be used to categorize
  events through the number of observed hadrons.

3
Introduction (contd.)

• Need dynamical modeling of heavy ion collisions ?
  How?
• Local thermal equilibrium? Non-equilibrium?
• Fluids (hydrodynamics)? Gases (Boltzmann)?
• Perfect? Viscous?
• Lots of stages in collision (next slide)
• Ultimate purpose Dynamical description of the
  whole stage
• Current status Description of intermediate
  stage based on hydrodynamics

4
Space-Time Evolution of Relativistic Heavy Ion
Collisions
t
Thermal freezeout
Chemical freezeout
QCD phase transtion (1st or crossover?)
t -z/c
Thermalized matter QGP?
t z/c
Chemical equilibration (Quark Gluon Plasma)
Time scale 10 fm/c10-23sec Temperature
scale 100MeV/kB1010K
Local thermalization (Gluon Plasma)
zcollision axis
0
Parton distribution function in colliding nuclei
Gold nucleus
Gold nucleus
v0.99c
5
Dynamical Modeling Based on Hydrodynamics
6
Rapidity and Boost Invariant Ansatz
forward rapidity ygt0
t
midrapidityy0
yinfinity
tconst.
t, z
hsconst.
Rapidity as a relativistic velocity
z
0

• Boost invariant ansatz Bjorken (83)
• Dynamics depends on t, not on hs.

7
Hydrodynamic Equationsfor a Perfect Fluid
four velocity
e energy density,
P pressure,
Energy
Momentum
Baryon number
8
Inputs for Hydrodynamic Simulations
Final stage Free streaming particles ? Need
decoupling prescription
t
Intermediate stage Hydrodynamics can be valid as
far as local thermalization is achieved. ? Need
EoS P(e,n)
z

• Initial stage
• Particle production,
• pre-thermalization, instability?
• Instead, initial conditions
• for hydro simulations

0
Need modeling (1) EoS, (2) Initial cond., and (3)
Decoupling
9
Main Ingredient Equation of State
One can test many kinds of EoS in hydrodynamics.
Typical EoS in hydro model
Lattice QCD simulations
H resonance gas(RG)
Q QGPRG
P.Kolb and U.Heinz(03)
F.Karsch et al. (00)
pe/3
Latent heat
Lattice QCD predicts cross over phase
transition. Nevertheless, energy density
explosively increases in the vicinity of Tc. ?
Looks like 1st order.
10
Interface 1 Initial Condition

• Need initial conditions (energy density, flow
  velocity,)

Initial time t0 thermalization time
Energy density distribution
Rapidity distribution of produced charged hadrons
Perpendicular to the collision axis
Reaction plane (Note Vertical axis represents
expanding coordinate hs)
T.H. and Y.Nara(04)
mean energy density 5.5-6.0GeV/fm3
(Lorentz-contracted) nucleus
11
Interface 2 Freezeout
(1) Sudden freezeout
(2) Transport of hadrons via Boltzman eq. (hybrid)
At TTf, l0 (ideal fluid) ? linfinity (free
stream)
Continuum approximation no longer valid at the
late stage ?Molecular dynamic approach for
hadrons (p,K,p,)
TTf
t
Hadron fluid
QGP fluid
QGP fluid
z
0
12
Observable Elliptic Flow
13
Anisotropic Flow in Atomic Physics

• Fermionic 6Li atoms in an optical trap
• Interaction strength controlled via Feshbach
  resonance
• Releasing the cloud from the trap
• Superfluid? Or collisional hydrodynamics?

K.M.OHara et al., Science298(2002)2179
How can we see anisotropic flow in heavy ion
collisions?
14
Elliptic Flow
Ollitrault (92)
Response of the system to initial spatial
anisotropy
Hydrodynamic behavior
No secondary interaction
Input
Spatial anisotropy e
Interaction among produced particles
2v2
Output
Momentum anisotropy v2
dN/df
dN/df
f
0
2p
f
0
2p
15
Elliptic Flow from a Parton Cascade Model
Time evolution of v2
hydro limit
View from collision axis
b 7.5fm

• Gluons uniformly distributed
• in the overlap region
• dN/dy 300 for b 0 fm
• Thermal distribution with
• T 500 MeV/kB

Zhang et al.(99)
generated through secondary collisions
saturated in the early stage sensitive to cross
section (viscosity)
v2 is
16
Comparison of Hydro Results with Experimental Data
17
Particle Density Dependence of Elliptic Flow
NA49(03)
Kolb, Sollfrank, Heinz (00)

• Dimension
• 2Dboost inv.
• EoS
• QGP hadrons (chem. eq.)
• Decoupling
• Sudden freezeout

(response)(output)/(input)

• Hydrodynamic response is
• const. v2/e 0.2 _at_ RHIC
• Exp. data reach hydrodynamic
• limit at RHIC for the first time.

Number density per unit transverse area
Dawn of the hydro age?
18
Wave Length Dependence
T.H.(04)

• Dimension
• Full 3D (t-hs coordinate)
• EoS
• QGP hadrons (chem. frozen)
• Decoupling
• Sudden freezeout

Short wave length
(response)(output)/(input)
Long wave length

• Long wave length components (small transverse
  momentum)
• obey hydrodynamics scaling
• Short wave length components (large transverse
  momentum)
• deviate from hydro scaling.

particle density
low
high
spatial anisotropy
large
small
19
Particle Density Dependence of Elliptic Flow
(contd.)
Teaney, Lauret, Shuryak(01)

• Dimension
• 2Dboost inv.
• EoS
• Parametrized by latent heat
• (LH8, LH16, LH-infinity)
• Hadrons
• QGPhadrons (chem. eq.)
• Decoupling
• Hybrid (Boltzmann eq.)

• Deviation at lower energies can be filled by
  viscosity in hadron gases
• Latent heat 0.8 GeV/fm3 is favored.

20
Rapidity Dependence of Elliptic Flow

• Density ? low
• ? Deviation from hydro
• Forward rapidity at RHIC
• Midrapidity at SPS?
• Heinz and Kolb (04)

T.H. and K.Tsuda(02)

• Dimension
• Full 3D (t-hs coordinate)
• EoS
• QGP hadrons (chem. eq.)
• QGP hadrons (chem. frozen)
• Decoupling
• Sudden freezeout

21
Fine Structure of v2 Transverse Momentum
Dependence
PHENIX(03)
STAR(03)

• Correct pT dependence
• up to pT1-1.5 GeV/c
• Mass ordering
• Deviation in small wave length regions
• ? Effects other than hydro

Huovinen et al.(01)

• Dimension
• 2Dboost inv.
• EoS
• QGP RG (chem. eq.)
• Decoupling
• Sudden freezeout

22
Viscous Effect on Distribution
Parametrization of hydro field dist. fn. with
viscous correction

• 1st order correction to dist. fn.

Sound attenuation length
Tensor part of thermodynamic force

• Reynolds number in boost invariant scaling flow

G.Baym(84)
D.Teaney(03)
Nearly perfect fluid !?
23
Summary, Discussion and Outlook

• Large magnitude of v2, observed at RHIC, is
  consistent with hydrodynamic prediction.
• Long wave length components obey hydrodynamics
  scaling.
• Hybrid approach gives a good description (v2 at
  midrapidity, mass splitting, density dependence).
• Ideal hydro for the QGP liquid
• Molecular dynamics for the hadron gas
• No full 3D hybrid, viscous hydro model yet.

24
Summary A Probable Scenario
Almost Perfect Fluid of quark-gluon matter
Gas of Hadrons
pre-thermalization?
proper time t
Thermalization time 0.5-1.0fm/c Mean energy
density 5.5-6 GeV/fm3 _at_1fm/c
Colliding nuclei
Latent heat 0.8 GeV/fm3
25
BACKUP SLIDES
26
Coupling Parameter
S.Ichimaru et al.(87)
Plasma Physics
G (Average Coulomb Energy)/(Average Kinetic
Energy)
G O(10-4) for laser plasma O(0.1) for
interior of Sun O(50) for interior of
Jupiter O(100) for white dwarf
Quark Gluon Plasma near Tc
M.H.Thoma (04)
C Casimir (4/3 for quark or 3 for gluon) g
strong coupling constant T Temperature d
Distance between partons
27
Hydro or Boltzmann ?
Molnar and Huovinen (04)

• Comparison between hydro and Boltzmann
• Pure gluon system
• Elastic scattering
• (gg??gg)
• Number conservation in hydro
• Need to check more realistic model

elastic cross section
At the initial stage, interaction among gluons
are so strong that many body correlation could be
important. ?Almost perfect fluid?
Knudsen number (mean free path)/(typical size)
10-4 _at_ t 0.1 fm/c (initial time) 10-1 _at_ t
10 fm/c (final time)
28
Discussion and Outlook
29
Hydrodynamic Simulations for Viscous Fluids
Non-relativistic case (Based on discussion by
Cattaneo (1948))
Balance eq.
Constitutive eq.
Fouriers law
t ?0
t relaxation time
Parabolic equation (heat equation) ?ACAUSAL!!
(Similar difficulty is known in relativistic
hydrodynamic equations.)
finite t Hyperbolic equation (telegraph
equation) No full 3D
calculation yet. (D.Teaney, A.Muronga)
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Hydro Rate Eq. in the QGP phase
T.S.Biro et al.,Phys.Rev.C48(93)1275.
Including gg??qqbar and gg??ggg
Collision term
Assuming multiplicative fugacity, EoS is
unchanged.
31
2nd order formula
Balance eqs.
How obtain additional equations?
1st order
2nd order
In order to ensure the second law of
thermodynamics , one can choose
Constitutive eqs.
14 equations