Date post: | 26-Dec-2015 |
Category: |
Documents |
Upload: | linette-harmon |
View: | 221 times |
Download: | 1 times |
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.