Hydrodynamical Hydrodynamical Simulation of Relativistic Simulation of Relativistic

Heavy Ion CollisionsHeavy Ion CollisionsTetsufumi HiranoTetsufumi Hirano

Strongly Coupled Plasmas:Strongly Coupled Plasmas:Electromagnetic, Nuclear and AtomicElectromagnetic, Nuclear and Atomic

IntroductionIntroduction• 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 ~ <5000– “Impact parameter” can be used to categorize events

through the number of observed hadrons.

Introduction (contd.)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

Space-Time Evolution of Space-Time Evolution of Relativistic Heavy Ion Relativistic Heavy Ion

Collisions Collisions

Parton distributionfunction in collidingnuclei

Local thermalization(Gluon Plasma)

Chemical equilibration(Quark Gluon Plasma)

QCD phase transtion(1st or crossover?)

Chemical freezeout

Thermal freezeout

Goldnucleus

Gold nucleus

v~0.99c

0 z:collision axis

t

Time scale10 fm/c~10-23secTemperature scale100MeV/kB~1010K

t = z/

ct = -z/c Thermalized matter

QGP?

Dynamical ModelingDynamical ModelingBased onBased on

HydrodynamicsHydrodynamics

Rapidity and Boost Rapidity and Boost Invariant AnsatzInvariant Ansatz

z

t

0

midrapidity:y=0forward rapidity

y>0 y=infinity

Rapidity as a“relativistic velocity”

Boost invariant ansatz Bjorken (’83) Dynamics depends on , not on s.

=const.

s=

cons

t. t, z

Hydrodynamic EquationsHydrodynamic Equationsfor a Perfect Fluidfor a Perfect Fluid

Baryon number

Energy

Momentum

e : energy density, P : pressure, : four velocity

Inputs for Hydrodynamic Inputs for Hydrodynamic SimulationsSimulations

Final stage:Free streaming particles Need decoupling prescription

Intermediate stage:Hydrodynamics can be validas far as local thermalization isachieved. Need EoS P(e,n)

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

Need modeling(1) EoS, (2) Initial cond., and (3) Decoupling

0z

t

Main Ingredient: Equation Main Ingredient: Equation of Stateof State

Latent heat

One can test many kinds of EoS in hydrodynamics.

Lattice QCD predicts cross over phase transition.Nevertheless, energy density explosively increases in the vicinity of Tc. Looks like 1st order.

Lattice QCD simulationsLattice QCD simulationsLattice QCD simulationsLattice QCD simulations Typical EoS in hydro modelTypical EoS in hydro modelTypical EoS in hydro modelTypical EoS in hydro model

H: resonance gas(RG)

p=e/3

Q: QGP+RG

F.K

arsch et al. (’00)

P.K

olb and U.H

einz(’03)

Interface 1: Initial Interface 1: Initial ConditionCondition

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

Initial time 0 ~ thermalization time

Perpendicular tothe collision axis

Reaction plane(Note: Vertical axis

represents expanding coordinate s)

Energy density distributionEnergy density distribution Rapidity distribution ofRapidity distribution ofproduced charged hadronsproduced charged hadrons

(Lorentz-contracted) nucleus

T.H. and Y.Nara(’04)

mean energy density~5.5-6.0GeV/fm3

Interface 2: FreezeoutInterface 2: Freezeout(1) Sudden freezeout (2) Transport of hadrons

via Boltzman eq. (hybrid)

Continuum approximation no longer valid at the late stageMolecular dynamic approach for hadrons (,K,p,…)

0z

t

0z

t

At T=Tf,=0 (ideal fluid) =infinity (free stream)

T=Tf

QGP fluid

Hadron fluid

QGP fluid

Observable:Observable:Elliptic FlowElliptic Flow

Anisotropic Flow in Atomic Anisotropic Flow in Atomic PhysicsPhysics

• Fermionic 6Li atoms in an optical trap• Interaction strength controlled via

Feshbach resonance• Releasing the “cloud” from the trap• Superfluid? Or collisional hydrodynamics?

How can we “see” anisotropic How can we “see” anisotropic flow in heavy ion collisions?flow in heavy ion collisions?

K.M

.O’H

ara

et a

l., S

cien

ce29

8(20

02)2

179

Elliptic FlowElliptic FlowResponse of the system to initial spatial anisotropy

Ollitrault (’92)

Hydrodynamic behavior

Spatial anisotropy

Momentumanisotropy v2

Input

Output

Interaction amongproduced particles

dN

/d

No secondary interaction

0 2

dN

/d

0 2

2v2

x

y

Elliptic Flow from a Elliptic Flow from a Parton Cascade ModelParton Cascade Model

b = 7.5fm

Time evolution of v2

generated through secondary collisions saturated in the early stage sensitive to cross section (~viscosity)

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

v2 is

Zhang et al.(’99)

View from collision axishydro limit

Comparison ofComparison ofHydro Results withHydro Results withExperimental DataExperimental Data

Particle Density Particle Density Dependence of Elliptic FlowDependence of Elliptic Flow

•Hydrodynamic response isconst. v2/ ~ 0.2 @ RHIC•Exp. data reach hydrodynamiclimit at RHIC for the first time.

(re

spo

nse

)=(o

utp

ut)/

(inp

ut)

Number density per unit transverse area

• Dimension• 2D+boost inv.

• EoS• QGP + hadrons (chem.

eq.)• Decoupling

• Sudden freezeout

NA49(’03) Kolb, Sollfrank, Heinz (’00)

Dawn of the hydro age?Dawn of the hydro age?

““Wave Length” Wave Length” DependenceDependence

Short wave length

Long wave length

• Dimension• Full 3D (s coordinate)

• EoS• QGP + hadrons (chem.

frozen)• Decoupling

• Sudden freezeout

T.H.(’04)

particledensity lowhigh

spatialanisotropy

largesmall

•Long wave length components (small transverse momentum)obey “hydrodynamics scaling”

•Short wave length components (large transverse momentum)deviate from hydro scaling.

(re

spo

nse

)=(o

utp

ut)/

(inp

ut)

Particle Density Particle Density Dependence of Elliptic Flow Dependence of Elliptic Flow

(contd.) (contd.) • Dimension

• 2D+boost inv.• EoS

• Parametrized by latent heat (LH8, LH16, LH-infinity)• Hadrons• QGP+hadrons (chem. eq.)

• Decoupling• Hybrid (Boltzmann eq.)

Teaney, Lauret, Shuryak(’01)

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

Rapidity Dependence of Rapidity Dependence of Elliptic FlowElliptic Flow

• Dimension• Full 3D (s coordinate)

• EoS1. QGP + hadrons (chem. eq.)2. QGP + hadrons (chem. frozen)

• Decoupling• Sudden freezeout

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

T.H. and K.Tsuda(’02)

““Fine Structure” of vFine Structure” of v22:: Transverse Momentum Transverse Momentum

Dependence Dependence

• Dimension• 2D+boost inv.

• EoS• QGP + RG (chem. eq.)

• Decoupling• Sudden freezeout

PHENIX(’03)

• Correct pT dependence up to pT=1-1.5 GeV/c• Mass ordering• Deviation in small wave length regions

Effects other than hydro

Huovinen et al.(’01)

STAR(’03)

Viscous Effect on Viscous Effect on DistributionDistribution

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

Nearly perfect fluid !?Nearly perfect fluid !?D.Teaney(’03)

G.Baym(’84)

Summary, Discussion and Summary, Discussion and OutlookOutlook

• 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.

Summary: A Probable Summary: A Probable ScenarioScenario

Collidingnuclei

proper time t

Almost PerfectFluid of

quark-gluon matter

pre

-th

erm

aliz

ati

on?

Thermalization time~0.5-1.0fm/cMean energy density~5.5-6 GeV/fm3 @1fm/c

“Latent heat”~0.8 GeV/fm3

Gas

of

Hadro

ns

BACKUPSLIDES

““Coupling Parameter”Coupling Parameter”S.Ichimaru et al.(’87)

(Average Coulomb Energy)/(Average Kinetic Energy)

Plasma Physics

=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

C: Casimir (4/3 for quark or 3 for gluon)g: strong coupling constantT: Temperatured: Distance between partons

M.H.Thoma (’04)

Hydro or Boltzmann ?Hydro or Boltzmann ?Molnar and Huovinen (’04)

ela

stic cross se

ction

At the initial stage, interaction among gluons are so strong that many body correlation couldbe important.Almost perfect fluid?

Comparison between hydro and Boltzmann

•Pure gluon system•Elastic scattering(gggg)•Number conservation in hydro•Need to check more realistic model

Knudsen number=(mean free path)/(typical size) ~10-4 @ = 0.1 fm/c (~initial time) ~10-1 @ = 10 fm/c (~final time)

DiscussionDiscussionandand

OutlookOutlook

Hydrodynamic Simulations Hydrodynamic Simulations for Viscous Fluidsfor Viscous Fluids

Non-relativistic case (Based on discussion by Cattaneo (1948))

Fourier’s law

: “relaxation time”

Parabolic equation (heat equation) ACAUSAL!! (Similar difficulty is known in relativistic hydrodynamic equations.)

finite Hyperbolic equation (telegraph equation)

No full 3D calculation yet. (D.Teaney, A.Muronga…)

Balance eq.:

Constitutive eq.: 0

Hydro + Rate Eq. in the QGP phase

Including ggqqbar and ggggg

Collision term:

T.S.Biro et al.,Phys.Rev.C48(’93)1275.

Assuming “multiplicative” fugacity, EoS is unchanged.

2nd order formula…

14 equations…

1st order 2nd order

How obtain additional equations?

In order to ensure the second law of thermodynamics , one can choose

Balance eqs.

Constitutive eqs.