Non-ideal Particle Distributions from Kinetic Freeze Out Models

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Xiv:nucl-th/9808024v1 11 Aug 1998Non-idealParticleDistributionsfromKineticFreezeOutModels

Cs.Anderlik,1Zs.I.L´az´ar,1V.K.Magas,1

L.P.Csernai,1,2H.St¨ocker3andW.Greiner3

1SectionforTheoreticalPhysics,DepartmentofPhysics

UniversityofBergen,Allegaten55,5007Bergen,Norway

2KFKIResearchInstituteforParticleandNuclearPhysics

P.O.Box49,1525Budapest,Hungary

3Institutf¨urTheoretischePhysik,Universit¨atFrankfurt

Robert-Mayer-Str.8-10,D-60054FrankfurtamMain,Germany

Abstract:Influiddynamicalmodelsthefreezeoutofparticlesacrossathreedimensionalspace-timehypersurface

isdiscussed.Thecalculationoffinalmomentumdistributionofemittedparticlesisdescribedforfreezeoutsurfaces,

withbothspace-likeandtime-likenormals,takingintoaccountconservationlawsacrossthefreezeoutdiscontinuity.

1.Introduction

Thefreezeoutofparticledistributionsisanessentialpartofcontinuumorfluiddynamicalreactionmodels.From

thepointofviewofobservableconsequencesthisisoneofthemostessentialpartsofthemodel.Ontheotherhand

thisstepisnotbasedonfluiddynamicalprinciplesandgovernedbyalargevariatyofadhocassumptions.The

freezeoutcanbeconsideredasadiscontinuityacrossahypersurfaceinspace-time.

Thegeneraltheoryofdiscontinuitiesinrelativisticflowwasnotworkedoutforalongtime,andthe1948work

ofA.Taub1discusseddiscontinuitiesacrosspropagatinghypersurfacesonly(whichhaveaspace-likenormalvector,

dσµdσµ=−1).Eventshappeningonapropagating,(2dimensional)surfacebelongtothiscategory.

Anothertypeofchangeinacontinuumisanoverallsuddenchangeinafinitevolume.Thisisrepresentedbya

hypersurfacewithatime-likenormal,dσµdσµ=+1,calledconfusinglybothspace-likeandtime-likesurfaceinthe

literature.In1987Taubsapproachwasgeneralizedtobothtypesofsurfaces,2makingitpossibletotakeintoaccount

conservationlawsexactlyacrossanysurfaceofdiscontinuityinrelativisticflow.Thisapproachalsoeliminatesthe

imaginaryparticlecurrentsarisingfromtheequationoftheRayleighline.WhentheEoSisdifferentonthetwosides

ofthefreezeoutfronttheseconservationlawsyieldchangingtemperature,density,flowvelocityacrossthefront.

Infactthefreezeoutsurfaceisanidealizationofalayeroffinitethicknesswherethefrozenoutparticlesare

formed,andtheinteractionsinthematterbecomegraduallynegligible.Thedynamicsofthislayercanbedescribed

indifferentkineticmodelsorfour-volumeemissionmodels.3Thezerothicknesslimitofsuchalayeristheidealized

freezeoutsurface.

Theinvariantnumberofconservedparticles(worldlines)crossingasurfaceelement,dσµ,isdN=Nµdσµ,and

thetotalnumberofalltheparticlescrossingtheFOhyper-surface,S,isN=󰀑

SNµdσµ.Thistotalnumber,N,

andthetotalenergyandmomentumareofcoursethesameatbothsidesofthefreezeoutsurface.Ifweinsertthe

kineticdefinitionofNµ

Nµ=󰀊d3p

d3p=󰀊

fFO(x,p;T,n,uν)pµdσµ,(1)

wherefFO(x,p;T,n,uν)isthepostFOphasespacedistributionoffrozen-outparticleswhichisnotknownfrom

thefluiddynamicalmodel.Problemsusuallyarisefromthebadchoiceofthisdistribution.Firstofall,toevaluate

measurableswehavetousethecorrectparametersofthematteraftertheFOdiscontinuity!

Ifweknowtheprefreezeoutbaryoncurrentandenergy-momentumtensor,Nµ0andTµν0,wecancalculatelocally,

acrossasurfaceelementofnormalvectordσµthepostfreezeoutquantities,NµandTµν,fromtherelations1,2:

1[Nµdσµ]=0and[Tµνdσµ]=0,where[A]≡A−A0.Innumericalcalculationsthelocalfreezeoutsurfacecan

bedeterminedmostaccuratelyviaself-consistentiteration.7,9ThisfixestheparametersofourpostFOmomentum

distribution,fFO(x,p;T,n,uν).

Forexamplewecanillustratetheeffectofconservationlawsforasituationwherethefrozenoutmatterismassless,

baryonfreeBosegas.Then,theconservationlawsacrossthefreeze-outsurfacewithtimelikenormalvectordσµare

[Tµνdσν]=0,Inthemostgeneral(threedimensional)casetherearefourparameterstobedeterminedfromthe

conservationlaws:Thefinal,postFOtemperature,T,andthreecomponentsofthevelocity,u.Theenergy-

momentumtensorontheprefreeze-outside,andthenormaltothesurfacearegiven.Thepostfreeze-outenergy-

momentumtensorisoftheformTµν=(e+p)uµuν−pgµν,wheretheenergydensity,pressure,andtemperatureare

connectedbytheEoS:e=σSBT4=3p,whereσSBistheStefan-Boltzmannconstant.ThenTµν=(e+p)uµuν−pgµν,

canbewrittenasavectorequation:

(4uµuνdσν−dσµ)=xaµ,(2)

where

x=󰀏1

(aµdσµ)2+3aµaµ2󰀉

p0f∗FO(x,p;T,n,uν,dσγ)pµ󰀒

dσµ=󰀊

SNµ0(x)dσµ,(4)󰀊

S󰀏󰀊d3p

†Ontheotherhand,ifkineticfreezeoutcoincideswitharapidphasetransition,likeinthecaseofrapiddeconfinement

transitionofsupercooledquark-gluonplasma,theshortfreezeouthypersurfaceidealizationmaystillbeapplicableevenforheavyionreactions.Itis,however,beyondthescopeofthisworktostudythefreezeoutdynamicsandkineticsinthislattercase.

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