Initial conditons, equations of state and final state in hydrodynamics Hydro modelsHydro models IS,...

Initial conditons, equations of Initial conditons, equations of

state and final state in state and final state in

hydrodynamicshydrodynamics

•Hydro modelsHydro models•IS, EoS, FOC and FSIS, EoS, FOC and FS•ObservablesObservables

Csanád MátéCsanád MátéEötvös UniversityEötvös University

Department of Atomic Department of Atomic PhysicsPhysics

11 September, 11 September, 20082008

M. Csanád, WPCF08 KrakowM. Csanád, WPCF08 Krakow 22

Little vocabulary of Little vocabulary of hydrodynamicshydrodynamics

• Exact solutionExact solution– Solution of hydro equations Solution of hydro equations analytically, analytically,

without approximationwithout approximation

• Parametric solutionParametric solution– Exact solution, that has fit parametersExact solution, that has fit parameters

• Hydro inspired parameterizationHydro inspired parameterization– Distribution determined at freeze-out only, Distribution determined at freeze-out only,

their time dependence is not consideredtheir time dependence is not considered

• Numerical solutionNumerical solution– Solution of hydro equations numericallySolution of hydro equations numerically



How analytic hydro worksHow analytic hydro works• Take hydro equations and EoSTake hydro equations and EoS• Find a solutionFind a solution

– Will contain parameters (like Friedmann, Will contain parameters (like Friedmann, Schwarzschild etc.)Schwarzschild etc.)

– Will use a possible set of initial conditionsWill use a possible set of initial conditions• Use a freeze-out conditionUse a freeze-out condition

– Eg fixed proper time or fixed temperatureEg fixed proper time or fixed temperature– Generally a hyper-surfaceGenerally a hyper-surface

• Calculate the hadron source functionCalculate the hadron source function• Calculate observablesCalculate observables

– E.g. spectra, flow, correlationsE.g. spectra, flow, correlations– Straightforward calculationStraightforward calculation

• HydrodynamicsHydrodynamics:: Initial conditions Initial conditions dynamical equations dynamical equations freeze-out freeze-out conditionsconditions

1N ( , , )

HBT( , , )t

t

p

p

1N ( , , )

HBT( , , )t

t

p

p



Famous solutionsFamous solutions• Landau’s solution (1D, developed for p+p):Landau’s solution (1D, developed for p+p):

– Accelerating, implicit, complicated, 1DAccelerating, implicit, complicated, 1D

– L.D. Landau, Izv. Acad. Nauk SSSR 81 (1953) 51L.D. Landau, Izv. Acad. Nauk SSSR 81 (1953) 51

– I.M. Khalatnikov, Zhur. Eksp.Teor.Fiz. 27 (1954) 529I.M. Khalatnikov, Zhur. Eksp.Teor.Fiz. 27 (1954) 529

– L.D.Landau L.D.Landau andand S.Z.Belenkij, Usp. Fiz. Nauk 56 (1955) S.Z.Belenkij, Usp. Fiz. Nauk 56 (1955) 309309

• Hwa-Bjorken solution:Hwa-Bjorken solution:– Non-accelerating, explicit, simple, 1D, boost-invariantNon-accelerating, explicit, simple, 1D, boost-invariant

– R.C. HwaR.C. Hwa, Phys. Rev. D, Phys. Rev. D10, 2260 (1974)10, 2260 (1974)

– J.D. Bjorken, Phys. Rev. D27, 40(1983)J.D. Bjorken, Phys. Rev. D27, 40(1983)

• OthersOthers– Chiu, Sudarshan and WangChiu, Sudarshan and Wang– Baym, Friman, Blaizot, Soyeur and CzyzBaym, Friman, Blaizot, Soyeur and Czyz– Srivastava, Alam, Chakrabarty, Raha and SinhaSrivastava, Alam, Chakrabarty, Raha and Sinha



3D solutions3D solutions• Nonrelativistic, spherically symmetric solutionNonrelativistic, spherically symmetric solution

– P. Csizmadia, T. Csörgő, B. Lukács, nucl-th/9805006P. Csizmadia, T. Csörgő, B. Lukács, nucl-th/9805006

• Relativistic, spherically symmetric solutionRelativistic, spherically symmetric solution– T. Csörgő, L. Csernai, Y. Hama, T. Kodama, T. Csörgő, L. Csernai, Y. Hama, T. Kodama,

nucl-th/0306004nucl-th/0306004– AccelerationlessAccelerationless– Hubble flow profile (flow proportional to distance)Hubble flow profile (flow proportional to distance)

• Relativistic, spherically symmetric solutionRelativistic, spherically symmetric solution– T. Csörgő, M. Nagy, M. Csanád, nucl-th/0605070T. Csörgő, M. Nagy, M. Csanád, nucl-th/0605070– AcceleratingAccelerating– Realistic rapidity distributions (data described by it)Realistic rapidity distributions (data described by it)– Advanced energy and lifetime estimateAdvanced energy and lifetime estimate

• All describe expanding fireballsAll describe expanding fireballs– Sometimes: rings/shells of fireSometimes: rings/shells of fire



Where we areWhere we are• Other accelerationless solutions:Other accelerationless solutions:

– T. S. Biró, Phys. Lett. B 474, 21 (2000)T. S. Biró, Phys. Lett. B 474, 21 (2000)– Yu. M. Sinyukov and I. A. Karpenko, nucl-th/0505041Yu. M. Sinyukov and I. A. Karpenko, nucl-th/0505041

• Solutions by coordinate transformations:Solutions by coordinate transformations:– S. Pratt, nucl-th/0612010S. Pratt, nucl-th/0612010

• Revival of interestRevival of interest– Bialas, Janik, Peschanski: Phys.Rev.C76:054901,2007Bialas, Janik, Peschanski: Phys.Rev.C76:054901,2007– Borsch, Zhdanov: SIGMA 3:116,2007Borsch, Zhdanov: SIGMA 3:116,2007

• There are some exotic solutions as well There are some exotic solutions as well • Need for solutions that are:Need for solutions that are:

– explicitexplicit– simplesimple– acceleratingaccelerating– relativisticrelativistic– realistic / compatible with the datarealistic / compatible with the data

• Buda-Lund type of solutions: each fulfilledBuda-Lund type of solutions: each fulfilled– but not simultaneouslybut not simultaneously



A Buda-Lund type of solutionA Buda-Lund type of solution• For sake of simplicity, take the following nonrel. For sake of simplicity, take the following nonrel.

solutionsolution– Csörgő, Akkelin, Hama, Lukács, Sinyukov, Csörgő, Akkelin, Hama, Lukács, Sinyukov,

Phys.Rev.C67:034904,2003Phys.Rev.C67:034904,2003

• Self similarly expanding ellipsoid, Gaussian ICSelf similarly expanding ellipsoid, Gaussian IC

• Flow profile: directional HubbleFlow profile: directional Hubble

• Equation of motion for principal axes:Equation of motion for principal axes:

• Freeze-out at constant temperature assumedFreeze-out at constant temperature assumed

1/

00

2 2 20 2

0 2 2 2

, , , ,

e , with ) 0s

t

VX Y Zv x y z T T V XYZ

X Y Z V

V x y zn n s v s s

V X Y Z

1/

00

2 2 20 2

0 2 2 2

, , , ,

e , with ) 0s

t

VX Y Zv x y z T T V XYZ

X Y Z V

V x y zn n s v s s

V X Y Z

TXX YY ZZ

m T

XX YY ZZm



Dependence on IC+EoS Dependence on IC+EoS (nonrel)(nonrel)

• Evolution of principal axes of the ellipsoidEvolution of principal axes of the ellipsoid

0 constT

XX YY ZZ X X X X tm

0 constT

XX YY ZZ X X X X tm




• Evolution of expansion ratesEvolution of expansion rates




• Time evolution of temperatureTime evolution of temperature



Same in relativistic hydroSame in relativistic hydro

Nagy, Csörgő, Csanád, Phys.Rev.C77:024908,2008, Csanád, Nagy, Csörgő, Eur.Phys.J.ST 155:19-Nagy, Csörgő, Csanád, Phys.Rev.C77:024908,2008, Csanád, Nagy, Csörgő, Eur.Phys.J.ST 155:19-26,200826,2008

Same final state for different evolutions, even with viscosity (see T. Same final state for different evolutions, even with viscosity (see T. Csörgő, WPCF’07)Csörgő, WPCF’07)



Conjectured EoS dependence Conjectured EoS dependence of of 00

• Relativistic, accelerating solution → describe dn/dRelativistic, accelerating solution → describe dn/d• Energy density modified compared to BjörkenEnergy density modified compared to Björken

• With With ff//00 = 10, c = 10, css = 0.35 [nucl-ex/0608033], = 0.35 [nucl-ex/0608033], correction to correction to is about 2.9 is about 2.9××

• = 14.5 GeV/fm= 14.5 GeV/fm33 in 200 GeV, 0-5 %Au+Au at RHIC in 200 GeV, 0-5 %Au+Au at RHIC



Predictions of the Buda-Lund Predictions of the Buda-Lund modelsmodels

• Hydro predicts scaling (even Hydro predicts scaling (even viscous) viscous)

• What does a scaling mean? What does a scaling mean? – See Hubble’s law – or Newtonian See Hubble’s law – or Newtonian

gravity:gravity:– Data collapseData collapse

• Collective, thermal behavior →Collective, thermal behavior →

Loss of informationLoss of information• Spectra slopes:Spectra slopes:

• Elliptic flow:Elliptic flow:

• HBT radii:HBT radii:

2v gh 2v gh

2eff 0 tT T mu 2eff 0 tT T mu

12

0

( ), ~

( ) T

I wv w KE

I w 1

20

( ), ~

( ) T

I wv w KE

I w

2 2 2side long out

1~

t

R R Rm

2 2 2side long out

1~

t

R R Rm



Elliptic flowElliptic flow• Prediction of Prediction of

2003: scaling 2003: scaling variablevariable

• If plotted against If plotted against ‘w’, data collapse:‘w’, data collapse:– From 20 to 200 From 20 to 200

GeVGeV– All centralitiesAll centralities– Pion, kaon, protonPion, kaon, proton

– pptt and and dependencedependence

• Prediction:Prediction:Csanád, Csörgő, Lörstad, Ster et al. nucl-Csanád, Csörgő, Lörstad, Ster et al. nucl-th/0512078th/0512078

4 24

wv v4 24

wv v

momkin

eff

w ET

mom

kineff

w ET



Prediction for HBT radiiPrediction for HBT radii• Exact hydro result (nonrel shown)Exact hydro result (nonrel shown)• Correlation radii = geometrical Correlation radii = geometrical thermal thermal

– Harmonic squared sum: Harmonic squared sum: 1/R1/R22corrcorr= = 1/R1/R22

geomgeom+ + 1/R1/R22thermtherm

• Geom.: Geom.: RRgeomgeom = X = X

• Thermal:Thermal:

• Hubble-profile → Hubble-profile → XXthth==YYthth

• RRout out RRsideside RRlonglong

0therm

TXR

X m

0therm

TXR

X m

Correlation radii

Geometrical radii

Thermal radii



Azimuthal HBT and elliptic Azimuthal HBT and elliptic flowflow

• AsHBT data describedAsHBT data described

• Both governed by asymmetriesBoth governed by asymmetries– aass: coordinate-space: coordinate-space– 22: momentum-space: momentum-space– vv22 depends only on depends only on 22

• Csanád, Tomasik, CsörgőCsanád, Tomasik, CsörgőEur. Phys. J. A 37,111 (2008)Eur. Phys. J. A 37,111 (2008)

2 2 2,0 ,2

2 2 2,0 ,2

2 2,0

2 2 2,0 ,2

cos 2

cos 2

sin 2

o o o

s s s

l l

os os os

R R R

R R R

R R

R R R

2 2 2,0 ,2

2 2 2,0 ,2

2 2,0

2 2 2,0 ,2

cos 2

cos 2

sin 2

o o o

s s s

l l

os os os

R R R

R R R

R R

R R R



Azimuthal HBT and elliptic Azimuthal HBT and elliptic flowflow

• Simultaneous descriptionSimultaneous description• Slopes as before (slide 12)Slopes as before (slide 12)• Elliptic flow as before (slide 13)Elliptic flow as before (slide 13)• Correlation radiiCorrelation radii

• Asymmetry parameters used:Asymmetry parameters used:22=0.17, a=0.17, ass=0.997=0.997

• Csanád, Tomasik, CsörgőCsanád, Tomasik, Csörgő

Eur. Phys. J. A 37,111 (2008)Eur. Phys. J. A 37,111 (2008)

2 2 2 22 2,0 ,2

2

2 2 2 2 20

,2 2

1 1 1 1

x y y xs s

t

x geom therm

R R R RR R

m X

R R R X T X

2 2 2 22 2,0 ,2

2

2 2 2 2 20

,2 2

1 1 1 1

x y y xs s

t

x geom therm

R R R RR R

m X

R R R X T X



Prediction for kaon HBTPrediction for kaon HBT• Transverse mass scaling → same curve for Transverse mass scaling → same curve for

pions and kaons if plotted versus mpions and kaons if plotted versus m tt

• Other models?Other models?

K



Beyond hydro: long source Beyond hydro: long source tailstails

• HRC reproduces HBT (hydro as well)HRC reproduces HBT (hydro as well)• But also long tails in two-pion source!But also long tails in two-pion source!• Anomalous diffusion (rescattering)Anomalous diffusion (rescattering)• This goes beyond hydroThis goes beyond hydro

– Hydro: regular mHydro: regular mtt scaling scaling

– Lévy-tails important here!Lévy-tails important here!

• Tail depends on m.f.p., Tail depends on m.f.p.,

thus the cross-sectionthus the cross-section– Kaons: lowest cross-Kaons: lowest cross-

section → heaviest tailsection → heaviest tail

T. Humanic, Int. J. Mod. Phys. E15 T. Humanic, Int. J. Mod. Phys. E15 197 (2006)197 (2006)

Csörgő, Braz.J.Phys.37:1002-Csörgő, Braz.J.Phys.37:1002-1013,20071013,2007



The HBT testThe HBT test• Models with acceptable results:Models with acceptable results:

– nucl-th/0204054nucl-th/0204054 Multiphase Trasport model (AMPT)Multiphase Trasport model (AMPT)Z. Lin, C. M. Ko, S. PalZ. Lin, C. M. Ko, S. Pal

– nucl-th/0205053nucl-th/0205053 Hadron cascade modelHadron cascade modelT. HumanicT. Humanic

– hep-ph/9509213 hep-ph/9509213 Family of BFamily of Buda-Lund hydro modeluda-Lund hydro modelssT. Csörgő, B. Lörstad, A. SterT. Csörgő, B. Lörstad, A. Ster

– hep-ph/0209054hep-ph/0209054 Cracow (single freeze-out, thermal) Cracow (single freeze-out, thermal) W. Broniowski, W. FlorkowskiW. Broniowski, W. Florkowski

– nucl-ex/0307026nucl-ex/0307026 Blast wave modelBlast wave modelF. Retiére for STARF. Retiére for STAR

– 0801.4361 0801.4361 2 + 1 boost invariant rel. hydro2 + 1 boost invariant rel. hydro,, W. Broniowski, M. Chojnacki, W. Broniowski, M. Chojnacki,

W. Florkowski, A. KisielW. Florkowski, A. Kisiel



ConclusionsConclusions• Several types of hydro modelsSeveral types of hydro models

– Success in spectra and flowSuccess in spectra and flow– Few describe vFew describe v22(() or HBT) or HBT

• Hadronic final state: combination of Hadronic final state: combination of IC, EoS and FCIC, EoS and FC– Penetrating probes requiredPenetrating probes required

• Similarities of successful models?Similarities of successful models?– Gaussian IC, Hubble flow etc.Gaussian IC, Hubble flow etc.– Compare Hubble-coefficients in models!Compare Hubble-coefficients in models!– Search for decisive tests!Search for decisive tests!

Thank you for your Thank you for your attentionattention

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