the henryk niewodniczański institute of nuclear physics polish academy of sciences cracow, poland
DESCRIPTION
Characteristic form of 2+1 relativistic hydrodynamic equations. Mikołaj Chojnacki. The Henryk Niewodniczański Institute of Nuclear Physics Polish Academy of Sciences Cracow, Poland. Based on paper M.Ch., W. Florkowski nucl-th/0603065. Cracow School of Theoretical Physics - PowerPoint PPT PresentationTRANSCRIPT
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).