# Presentation: 506048

## Hydrodynamical Simulation of Relativistic Heavy Ion Collisions

**Tetsufumi Hirano**

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

## 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 ~ <5000
- “Impact parameter” can be used to categorize events through the number of observed hadrons.

## 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 Relativistic Heavy Ion Collisions 

- Parton distribution
- function in colliding
- nuclei

- Local thermalization
- (Gluon Plasma)

- Chemical equilibration
- (Quark Gluon Plasma)

- QCD phase transtion
- (1st or crossover?)

- Chemical freezeout

- Thermal freezeout

- Gold
- nucleus

- Gold
-  nucleus

*v*~0.99*c*

- 0

*z*:collision axis

*t*

- Time scale
- 10 fm/*c*~10-23sec
- Temperature scale
- 100MeV/*k**B*~1010K

*t *= *z*/*c*

*t *= -*z*/*c*

- Thermalized matter
- QGP?

## Dynamical Modeling

**Based on**
**Hydrodynamics**

## Rapidity and Boost Invariant Ansatz

*z*

*t*

- 0

- midrapidity:*y*=0

- forward rapidity
*y*>0

*y*=infinity

- Rapidity as a
- “relativistic velocity”

- Boost invariant ansatz     Bjorken (’83)
-  Dynamics depends on **, not on **s.

**=const.

**s=const.

*t*, z

## Hydrodynamic Equationsfor a Perfect Fluid

- _Baryon number_

- _Energy_

- _Momentum_

*e *: energy density,

*P* : pressure,

- : four velocity

## Inputs for Hydrodynamic Simulations

- Final stage:
- Free streaming particles
-  Need decoupling prescription

- Intermediate stage:
- Hydrodynamics can be valid
- as far as local thermalization is
- achieved.  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

- 0

*z*

*t*

## Main Ingredient: Equation of 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 simulations

- Typical EoS in hydro model

- H: resonance gas(RG)

- p=e/3

- Q: QGP+RG

- F.Karsch *et al.* (’00)

- P.Kolb and U.Heinz(’03)

## Interface 1: Initial Condition

- Need initial conditions (energy density, flow velocity,...)

- Initial time **0 ~ thermalization time

- Perpendicular to
- the collision axis

- Reaction plane
- (Note: Vertical axis
-  represents expanding
-  coordinate **s)

- Energy density distribution

- Rapidity distribution of
- produced charged hadrons

- (Lorentz-contracted) nucleus

- T.H. and Y.Nara(’04)

- mean energy density
- ~5.5-6.0GeV/fm3

## Interface 2: Freezeout

**Interface 2: Freezeout**

- (1) Sudden freezeout

- (2) Transport of hadrons
- via Boltzman eq. (hybrid)

- Continuum approximation no longer valid at the late stage
- Molecular dynamic approach for hadrons (,K,p,...)

- 0

*z*

*t*

- At *T*=*T*f,
**=0 (ideal fluid)
-  **=infinity (free stream)

*T*=*T*f

- QGP fluid

- Hadron fluid

- QGP fluid

**Notes:**
Expansion of the system is too fast to keep adiabatic expansion.

## Observable:

**Elliptic Flow**

## 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?

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

- K.M.O’Hara et al., Science298(2002)2179

## Elliptic Flow

- Response of the system to initial spatial anisotropy

- Ollitrault (’92)

- Hydrodynamic behavior

- Spatial anisotropy 

- Momentum
- anisotropy v2

- Input

- Output

- Interaction among
- produced particles

- dN/d

**

- No secondary interaction

- 0

- 2

- dN/d

**

- 0

- 2

- 2*v*2

## Elliptic Flow from a Parton Cascade Model

*b *= 7.5fm

- Time evolution of *v*2

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

-  Gluons uniformly distributed
- in the overlap region
-  *dN*/*dy *~ 300 for *b *= 0 fm
-  Thermal distribution with
*T *= 500 MeV/*k*B

- v2 is

- Zhang et al.(’99)

- View from collision axis

- hydro limit

## Comparison of

**Hydro Results with**
**Experimental Data**

## Particle Density Dependence of Elliptic Flow

- Hydrodynamic response is
- const. *v*2/** ~ 0.2 @ RHIC
- Exp. data reach hydrodynamic
- limit at RHIC for the first time.

- (response)=(output)/(input)

- 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?

**Notes:**
Check excitation function from hydro

## “Wave Length” Dependence

- Short wave length

- Long wave length

_**Dimension**_
**Full 3D** **(***********s***** coordinate)**
_**EoS**_
**QGP + hadrons (chem. frozen)**
_**Decoupling**_
- Sudden freezeout

- T.H.(’04)

- particle
- density

- low

- high

- spatial
- anisotropy

- large

- small

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

- (response)=(output)/(input)

## Particle Density Dependence of Elliptic Flow (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 Elliptic Flow

**Rapidity Dependence of ****Elliptic Flow**

_**Dimension**_
**Full 3D** **(***********s***** coordinate)**
_**EoS**_
- QGP + hadrons (chem. eq.)
**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)

**Notes:**
Check dN/deta at eta=3 at RHIC and compare it with dN/deta at SPS.

## “Fine Structure” of v2: Transverse Momentum Dependence 

_**Dimension**_
- 2D+boost inv.
_**EoS**_
- QGP + RG (chem. eq.)
_**Decoupling**_
- Sudden freezeout

- PHENIX(’03)

-  Correct *p**T* dependence
-  up to *p**T*=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 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

- Nearly perfect fluid !?

- D.Teaney(’03)

- G.Baym(’84)

## 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.

## Summary: A Probable Scenario

- Colliding
- nuclei

- proper time t

- Almost Perfect
- Fluid of
- quark-gluon
-  matter

- pre-thermalization?

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

- “Latent heat”
- ~0.8 GeV/fm3

- Gas of Hadrons

## BACKUP

- SLIDES

## “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 *T*c

- C: Casimir (4/3 for quark or 3 for gluon)
- g: strong coupling constant
- T: Temperature
- d: Distance between partons

- M.H.Thoma (’04)

## Hydro or Boltzmann ?

- Molnar and Huovinen (’04)

- elastic cross section

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

- Comparison between hydro and Boltzmann
- Pure gluon system
- Elastic scattering
- (gggg)
- 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)

## Discussion

**and**
**Outlook**

## Hydrodynamic Simulations for 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 gg********qqbar and gg********ggg**

**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.