The Henryk NiewodniczańskiThe Henryk NiewodniczańskiInstitute of Nuclear PhysicsInstitute of Nuclear Physics

Polish Academy of SciencesPolish Academy of SciencesCracow, PolandCracow, Poland

Based on paperBased on paperM.Ch., W. FlorkowskiM.Ch., W. Florkowski

nucl-th/0603065nucl-th/0603065

Characteristic form of 2+1Characteristic form of 2+1relativistic hydrodynamic equationsrelativistic hydrodynamic equations

Mikołaj ChojnackiMikołaj Chojnacki

Cracow School of Theoretical PhysicsCracow School of Theoretical Physics May 27 - June 5, 2006, Zakopane, POLANDMay 27 - June 5, 2006, Zakopane, POLAND

22

OutlineOutline

Angular asymmetry in non-central collisionsAngular asymmetry in non-central collisions

2+1 Hydrodynamic equationsHydrodynamic equations

Boundary and initial conditions

Results from hydrodynamics

Freeze-out hypersurface and v2

Conclusions

33

Angular asymmetry in non-central collisionsAngular asymmetry in non-central collisions

x

y

Space asymmetries transform to momentum space asymmetriesSpace asymmetries transform to momentum space asymmetriesIndirect proof that particle interactions take placeIndirect proof that particle interactions take place

44

Equations of relativistic hydrodynamicsEquations of relativistic hydrodynamics

Energy and momentum conservation law:Energy and momentum conservation law:

0 T

gPuuPT

energy-momentum energy-momentum tensortensor

at midrapidity (y=0) for RHIC energiesat midrapidity (y=0) for RHIC energies

0Btemperature is the only temperature is the only thermodynamic parameterthermodynamic parameter

thermodynamic relationsthermodynamic relations

sdTdP Tdsd TsP

55

System geometrySystem geometry

Cylindrical coordinates ( r, Cylindrical coordinates ( r, ))

r

vR

vT

v

x

y

z = 0

x

y

yxr

arctan

22

R

T

RT

v

v

vvv

arctan

22

Boost – invariant symmetryBoost – invariant symmetry

Values of physical quantities at z Values of physical quantities at z ≠ 0 may be calculated by Lorentz transformation≠ 0 may be calculated by Lorentz transformation

21

21

vLorentz factor : Lorentz factor :

66

Equations in covariant formEquations in covariant form 0

su

TTuu

Non-covariant notationNon-covariant notation Dyrek + Florkowski, Dyrek + Florkowski, Acta Phys.Acta Phys. Polon.Polon. BB1515 (1984) (1984) 653653

0cos

sinsin

0sincos

0sincos

2

T

rr

T

r

v

dt

dvT

TTr

rTvrt

vstvsrtr

srtt

r

v

rv

tdt

d sincos

77

Temperature dependent sound velocityTemperature dependent sound velocity c css(T)(T)

s

T

T

sPTcs

2

Relation between T and s needed Relation between T and s needed to close the set of three equations.to close the set of three equations.

Potential Potential ΦΦ

sdcTdc

d ss

lnln1

Lattice QCD model by MohantyLattice QCD model by Mohanty and and Alam AlamPhys. Rev. Phys. Rev. CC68 (2003) 06490368 (2003) 064903

TTCC = 170 [MeV] = 170 [MeV]

0

0'

'lnT

Tc

TdT

T

T sT

Potential Φ dependent oPotential Φ dependent onn T T Temperature T dependent oTemperature T dependent onn Φ Φ

T inverse function ofinverse function of TT

88

Semifinal form of 2 + 1 hydrodynamic equations Semifinal form of 2 + 1 hydrodynamic equations in the transverse directionin the transverse direction

auxiliary functionsauxiliary functions:: expavtanh transverse rapiditytransverse rapiditywherewhere

01cos

1

cossin

1

1

sin

1cos

atr

v

vc

ca

rrvc

vc

a

vc

cv

rr

a

vc

cv

t

a

s

s

s

s

s

s

s

s

0cos

sin1

sinsincos

2

rrc

v

v

rrrv

t

s

99

Generalization of 1+1 hydrodynamic equationsGeneralization of 1+1 hydrodynamic equationsby Baym, Friman, Blaizot, Soyeur, Czyzby Baym, Friman, Blaizot, Soyeur, Czyz

Nucl. Phys. A407 (1983) 541Nucl. Phys. A407 (1983) 541

2 + 1 hydrodynamic equations reduce to 12 + 1 hydrodynamic equations reduce to 1 ++ 1 case 1 case

0,1

1,

1,

tratr

v

cv

ctra

rcv

cvtra

tr

sr

s

sr

sr

0,, 0 ttr

angular isotropy in initial conditionsangular isotropy in initial conditions

0

potential potential Φ independent of Φ independent of

0ln21

aaa

r

1010

Observables as functions of aObservables as functions of a±± and and

velocity velocity

aa

aav

potential potential ΦΦ aaln21

sound velocitysound velocity aaTcc ss ln21

temperaturetemperature aaTT ln21

solutions solutions

tr

traa

,,

,,

1111

Boundary conditionsBoundary conditions

Automatically fulfilled boundary conditions at r = 0Automatically fulfilled boundary conditions at r = 0

0

,,0

,,0,,0

00

rr dr

trd

dr

trdTtrv

rr

aa±±, ,

a+(r,,t)

a-(r,,t)

(r,,t)

0,,,,

0,,,,

rtratra

rtratra

0,,,, rtrtr

Single function a to describe aSingle function a to describe a±±

FFunction unction symmetrically symmetrically extended to negative values of rextended to negative values of r

a(r,,t)

(-r,,t)

Equal values at Equal values at = 0 and = 0 and = 2 = 2ππ

tratra

tratra

,2,,0,

,2,,0,

trtr ,2,,0,

1212

Initial conditions - TemperatureInitial conditions - Temperature

Initial temperature is connected with Initial temperature is connected with the number of participating nucleonsthe number of participating nucleons

3

1

0 const,,

dxdy

dNttrT p

22 11 22

bAin

bAin xTb

AxTb

AABp exTexTxT

dxdy

dN

0

0

0222

exp12,

a

rzyxA dzyxT

Teaney,Lauret and Shuryak Teaney,Lauret and Shuryak nucl-th/0110037nucl-th/0110037

xx

yy

AA BBbb

Values of parametersValues of parameters

fmafmr

fmmbin

54.037.6

17.040

0

30

1313

Initial conditions – velocity fieldInitial conditions – velocity field

0,,,

1,,,

00

220

000

ttrr

rH

rHttrvrv

Isotropic Hubble-like flowIsotropic Hubble-like flow

Final form of the aFinal form of the a±± initial conditions initial conditions

3

1

0

000

exp,

,1

,1,,,,

dxdy

dNconstra

rv

rvrattrara

pTT

T

1414

ResultsResults

Impact parameter b and centrality classesImpact parameter b and centrality classes

hydrodynamic evolution initial timehydrodynamic evolution initial time tt00 = 1 [fm] = 1 [fm]

sound velocity based on Lattice QCD calculationssound velocity based on Lattice QCD calculations

initial central temperatureinitial central temperature TT00 = 2 T = 2 TCC = 340 [MeV] = 340 [MeV]

initial flownitial flow HH00 = 0.0 = 0.0001 [fm1 [fm-1-1]]

minmax

minmax2

020

minmax

23

23

max

min3

44

1

cc

ccrdccr

ccb

c

c

1515

Centrality class 0 - 20%Centrality class 0 - 20%b = 3.9 [fm]b = 3.9 [fm]

1616

Centrality class 0 - 20%Centrality class 0 - 20%b = 3.9 [fm]b = 3.9 [fm]

1717

Centrality class 0 - 20%Centrality class 0 - 20%b = 3.9 [fm]b = 3.9 [fm]

1818

Centrality class 20 - 40%Centrality class 20 - 40%b = 7.1 [fm]b = 7.1 [fm]

1919

Centrality class 20 - 40%Centrality class 20 - 40%b = 7.1 [fm]b = 7.1 [fm]

2020

Centrality class 20 - 40%Centrality class 20 - 40%b = 7.1 [fm]b = 7.1 [fm]

2121

Centrality class 40 - 60%Centrality class 40 - 60%b = 9.2 [fm]b = 9.2 [fm]

2222

Centrality class 40 - 60%Centrality class 40 - 60%b = 9.2 [fm]b = 9.2 [fm]

2323

Centrality class 40 - 60%Centrality class 40 - 60%b = 9.2 [fm]b = 9.2 [fm]

2424

Freeze-outFreeze-out

upfpd

dydpdp

Nd

pTT

3

3

2

1 Cooper-Frye formulaCooper-Frye formula

Hydro initial parametersHydro initial parameters

• ccSS from Lattice QCD data from Lattice QCD data

• centrality: 0 - 80%centrality: 0 - 80% mean impact parameter mean impact parameter b = 7.6 fmb = 7.6 fm

• HH00 = 0.001 fm = 0.001 fm-1-1

• TT00 = 2.5 T = 2.5 TCC = 425 MeV = 425 MeV

Freeze-out temperatureFreeze-out temperature

• TTFOFO = 165 MeV = 165 MeV

2525

Freeze-out hypersurfaceFreeze-out hypersurface

2626

Azimuthal flow of Azimuthal flow of ΩΩ

Data points from STAR for Data points from STAR for ΩΩ + + ΩΩPhys. Rev. Lett. 95 (2005) 122301Phys. Rev. Lett. 95 (2005) 122301

_

2727

ConclusionsConclusions New and elegant approach to old problem:New and elegant approach to old problem: we have generalized the equations we have generalized the equations of 1+1 hydrodynamics to the case of angular asymmetry using the method of Baym of 1+1 hydrodynamics to the case of angular asymmetry using the method of Baym et al. (this is possible for the crossover phase transition, recently suggested by the et al. (this is possible for the crossover phase transition, recently suggested by the lattice simulations of QCD, only 2 equations in the extended r-space, automatically lattice simulations of QCD, only 2 equations in the extended r-space, automatically fulfilled boundary conditions at r=0)fulfilled boundary conditions at r=0)

Velocity field is developed that tends to transform the initial almond shape to a Velocity field is developed that tends to transform the initial almond shape to a cylindrically symmetric shape. As expected, the magnitude of the flow is greater in cylindrically symmetric shape. As expected, the magnitude of the flow is greater in the in-plane direction than in the out-of-plane direction. The direction of the flow the in-plane direction than in the out-of-plane direction. The direction of the flow changes in time and helps the system to restore a cylindrically symmetric shape.changes in time and helps the system to restore a cylindrically symmetric shape.

For most peripheral collisions the flow changes the central hot region to a For most peripheral collisions the flow changes the central hot region to a pumpkin-like form – as the system cools down this effect vanishes.pumpkin-like form – as the system cools down this effect vanishes.

Edge of the system preserves the almond shape but the relative asymmetry is Edge of the system preserves the almond shape but the relative asymmetry is decreasing with time as the system grows.decreasing with time as the system grows.

Presented results may be used to calculate the particle spectra and the vPresented results may be used to calculate the particle spectra and the v22

parameter when supplemented with the freeze-out model (THERMINATOR).parameter when supplemented with the freeze-out model (THERMINATOR).