Hydrodynamic fluctuations and dissipation in an integrated dynamical model

0.18 MB·

by  Koichi Murase

Nuclear
Physics A
NuclearPhysicsA00(2016)1–4
www.elsevier.com/locate/procedia
Hydrodynamic fluctuations and dissipation in an integrated
dynamical model
Koichi Murasea,b,c, Tetsufumi Hiranoa
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1
aDepartmentofPhysics,SophiaUniversity,Tokyo102-8554,Japan
0 bDepartmentofPhysics,theUniversityofTokyo113-0033,Japan
2 cTheoreticalResearchDivision,NishinaCenter,Riken,Wako351-0198,Japan

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a
J

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1 Abstract

] Wedevelopanewintegrateddynamicalmodeltoinvestigatetheeffectsofthehydrodynamicfluctuationsonobserv-
h
ables in high-energy nuclear collisions. We implement hydrodynamic fluctuations in a fully 3-D dynamical model
t
- consistingofthehydrodynamicinitializationmodelsoftheMonte-CarloKharzeev-Levin-Nardimodel,causaldissipa-
cl tivehydrodynamicsandthesubsequenthadroniccascades. Byanalyzingthehadrondistributionsobtainedbymassive
u event-by-eventsimulationswithbothofhydrodynamicfluctuationsandinitial-statefluctuations,wediscusstheeffects
n ofhydrodynamicfluctuationsontheflowharmonics,v andtheirfluctuations.
n
[
Keywords: Quarkgluonplasma,Relativisticfluctuatinghydrodynamics,Event-by-eventsimulation

1
v
0 1. Introduction
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2 In high-energy nuclear collisions, initial-state fluctuations are the major origin of event-by-event fluc-
2 tuationsobservedinthehigher-orderazimuthalanisotropiesv oftheobservedhadrons. Sincethehydro-
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dynamicresponseofthematterconvertstheinitialspatialanisotropiestothefinalmomentumanisotropies,
.
1 we can extract transport properties of the created matter, such as viscosity and relaxation times, by com-
0
paringtheinitial-statefluctuationsandthefinalobservables. Howevertheinitial-statefluctuationsarenot
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theonlysourceoftheevent-by-eventfluctuations[1]. Forexample,hydrodynamicfluctuationsgeneratead-
1
: ditionalflowfluctuationsduringthehydrodynamicevolutionofthematter. Hydrodynamicfluctuationsare
v thethermalfluctuationsofthehydrodynamicfieldswhosepowerisrelatedtotransportcoefficientsthrough
i
X thefluctuation-dissipationrelation(FDR)[2]. Topreciselydeterminethetransportpropertiesofthematter,
r we need to quantify the effects from the hydrodynamic fluctuations on the observables such as the flow
a coefficientsv . Weimplementhydrodynamicfluctuationsintoourdynamicalmodelofhigh-energynuclear
n
collisionstostudytheeffects.
2. Integrateddynamicalmodelwithhydrodynamicfluctuations
Weuseanextendedversionofanintegrateddynamicalmodel[3]consistingoffourparts. First,event-
by-event initial conditions are generated using the Monte-Carlo Kharzeev-Levin-Nardi (MC-KLN) model
2 K.Muraseetal./NuclearPhysicsA00(2016)1–4
[4].Subsequenthydrodynamicexpansioniscalculatedusingnewlydevelopedcodesofthesecond-orderrel-
ativisticfluctuatinghydrodynamics,namelyviscoushydrodynamicswithhydrodynamicfluctuations. Then
we sample hadronson the isothermal hypersurface atthe temperature of 155 MeVusing the Cooper-Frye
formula[5]includingtheviscouscorrectionofphase-spacedistributionδf [6].Finallyweperformhadronic
cascadesusingJAM[7]toobtainhadronspectra.
Inourhydrodynamiccalculationsthetimeevolutionofthecreatedmatterissolvedintheτ-η coordi-
s
nates: (τ,η , x )wheret = τcoshη andz = τsinhη . Anessentialpartofhydrodynamicequationsisthe
s ⊥ s s
conservationlawoftheenergy-momentumtensor:
∂¯ Tµν =0, Tµν =euµuν−P∆µν+πµν, (1)
µ
where∂¯ isthecovariantderivative, ande, Pandπµν aretheenergydensity, theequilibriumpressureand
µ
theshearstress,respectively. Thebulkpressureisnotconsideredhere. TheLandauframeisadoptedfora
fluidvelocityuµ (u uµ = 1), whereTµ uν = euµ. Herethesignsofmetrictensorg are(+,−,−,−). The
µ ν µν
tensor∆µν = gµν −uµuν istheprojectorontothespaceoffour-vectorstransversetotheflowvelocity. For
the equation of state (EoS), P(e), we adopt s95p-v1.1 [8] which smoothly connects a lattice EoS at high
temperatureandahadrongasEoSatlowtemperature. Fortheevolutionoftheshearstress, thefollowing
constitutiveequationisused:
τ ∆µν uλ∂¯ παβ+πµν[1+(4/3)τ ∂¯ uλ]=2η∆µν ∂¯αuβ+ξµν, (2)
π αβ λ π λ αβ
where ∆µν = 1(∆µ ∆ν + ∆µ ∆ν ) − 1∆µν∆ is the projector onto the tensor components which are
αβ 2 α β β α 3 αβ
symmetric, traceless and transverse to the flow velocity. In this study, the shear viscosity coefficient η
is fixed to be the KSS bound: η/s = 1/4π [9]. The relaxation time is chosen as τ = 3/4πT [10, 11].
π
Hydrodynamicfluctuationsappearasthetermξµν whichistheGaussianwhitenoisewithautocorrelations
determinedbytheFDR[12]:
(cid:104)ξµν(x)ξαβ(x(cid:48))(cid:105)=4Tη∆µναβδ(4)(x−x(cid:48)). (3)
Noise values are numerically generated using Gaussian random numbers. The above autocorrelations are
diagonalized by considering proper linear combinations of noise components: ξ(i) (i = 1,...,5), so that
correlations among different components are disentangled. Then independent noise components ξ(i) are
generatedas:
(cid:113)
ξ(i)(τ,η ,x )= 4Tη∆(i)/τw(τ,η ,x ), (4)
s ⊥ i s ⊥
wherew(x)arethenormalGaussiannoisefields,and∆(i) arecoefficientscomingfromtheprojector. The
i √
Jacobian1/τ=1/ −gfortheτ-η coordinatescomesfromacoordinatetransformofthedeltafunction.
s
Here we introduce a coarse-graining scale or a cutoff length scale of the hydrodynamic fluctuations.
Thehydrodynamicdescriptionofphysicalsystemshasalwaysalower-boundscaleofthedescriptionwhere
the assumption of the local equilibrium becomes invalid. If hydrodynamic fluctuations with arbitrarily
smallwavelengthareconsidered,themagnitudeoffluctuationsdivergessuchthatthestresstensorbecomes
unphysicallylargeandbreakscalculations. Thereforeweintroduceacoarse-grainingscaleofthehydrody-
namicfluctuationsbyreplacingthenormalnoisefieldsw(x)in(4)withsmearedonesw(cid:48)(x):
i i
w(cid:48)i(τ,ηs,x⊥)=(cid:90) dη(cid:48)sd2x⊥(cid:48) (cid:112)(2π)13σησ2⊥ exp−(ηs2−ση2η(cid:48)s)2 − (x⊥2−σ2⊥x(cid:48)⊥)2wi(τ,η(cid:48)s,x(cid:48)⊥). (5)
Forthisstudy,wechoosesmearingscalesasσ =1fm,andσ =1.
⊥ η
3. Results
Toquantifytheeffectsofhydrodynamicfluctuationsonobservables,weperformevent-by-eventsimu-
lationsusingtheintegrateddynamicalmodel. Forthecollisionsystemweconsiderminimum-biasAu+Au
K.Muraseetal./NuclearPhysicsA00(2016)1–4 3
√
collisionsatthecollisionenergy s =200GeV.Weconsidertwotypesofhydrodynamiccalculations:
NN
fluctuatinghydrodynamicswiththeshearviscosityandthecorrespondinghydrodynamicfluctuations, and
conventionalviscoushydrodynamicswithouthydrodynamicfluctuations. Theinitial-stateentropydensities
are scaled for each type of hydrodynamics to reproduce experimental charged particle multiplicities [13].
For each type of hydrodynamics we perform 105 events of hydrodynamic simulations. Then one hundred
eventsofhadroniccascadesareperformedforeachhydrodynamicevent,i.e.,105×100=107cascadesare
performedintotalforeachtypeofhydrodynamics.
0.07 0.2
v2 w/ HF v2 w/ HF
0.06 w/o HF 0.18 w/o HF
v3 w/ HF 0.16 v3 w/ HF
w/o HF w/o HF
0.05 v4 w/ HF 0.14 v4 w/ HF
P} 0.04 w/o HF P} 0.12 w/o HF
E E 0.1
v{n 0.03 v{n 0.08
0.02 0.06
0.04
0.01
0.02
0 0
0 0.5 1 1.5 2 0 0.5 1 1.5 2
p (GeV/c) p (GeV/c)
T T
Fig.1. The pT differentialflowsofthechargedhadronsareshownforcentralcollisions0-5%(theleftpanel),andfornon-central
collisions20-30%(therightpanel). Theflowcoefficientsarecalculatedusingtheη-submethodwithtwosubeventsatthepseudora-
pidityrange1<|η|<2.8. Thesolidlinesaretheresultsfromfluctuatinghydrodynamicswithhydrodynamicfluctuations(HF),and
thedashedonesarefromconventionalviscoushydrodynamics.
In the left panel of Fig. 1, we see increase of the flow harmonics v in central collisions 0-5% due to
n
hydrodynamicfluctuations. Theamountofincreaseislargerforhigherorderoftheharmonics,whichcan
√
beunderstoodbytheFDR:Theeffectivesizeofthefluctuationsscalesasξ ∝1/ V sincethedeltafunction
intheFDRbecomes1/V througheffectivecoarsegraininginthevolumeV. Asahigherorderofharmonics
correspondstosmallerstructuresofthecreatedmatter,theeffectsarelargerforthehigherorder.
The right panel of Fig. 1 shows the same results for non-central collisions 20-30%. Unlike the case
of the central collisions, the elliptic flows v are almost identical with each other in both cases, which
2
can be explained by a difference of the origin of elliptic flows. In central collisions, the origin of elliptic
flowsisdominatedbyfluctuations. Innon-centralcollisions,thereisanadditionaloriginfromthecollision
geometry.Smallflowfluctuationscausedbyhydrodynamicfluctuationsaroundthelargegeometricalelliptic
flowdonotchangetheaveragemagnitudeoftheflows. Neverthelessthedistributionoftheflowsischanged
byhydrodynamicfluctuations. ThiscanbeconfirmedinFig.2inwhichtheevent-by-eventdistributionof
v inthenon-centralcollisionsisbroadenedbyhydrodynamicfluctuations.
2
Anothereffectcausedbyhydrodynamicfluctuationsisthedecorrelationoftheflowanglesatdifferent
rapiditieswhichhaveastrongcorrelationduetotheinitial-statefluctuationsfromnucleonpositiondistribu-
tions. Thedecorrelationscanbecausedbyseveralmechanisms[15]includinghydrodynamicfluctuations.
In the right panel of Fig. 2 we see the decorrelation by hydrodynamic fluctuations. The flow angles are
disturbedbythehydrodynamicfluctuationswhichhaveshortlongitudinalcorrelationsduetotheFDR.
4. Summaryanddiscussions
Hydrodynamic fluctuations have a non-negligible amount of effects on observables of flow harmonics
compared to that of the initial-state fluctuations. In central collisions hydrodynamic fluctuations increase
the flow coefficients. The increase is larger in higher order of harmonics as expected from the FDR. In
non-central collisions with a large geometrical origin of v , hydrodynamic fluctuations do not change v
2 2
of the event-plane method which roughly corresponds to the average of event-by-event v . Nevertheless
2
4 K.Muraseetal./NuclearPhysicsA00(2016)1–4
10 1.05
w/ HF w/ HF
w/o HF
w/o HF
1
1
0.95
prob. 0.1 ηη,)ab 0.9
r(2
0.85
0.01
0.8
0.001 0.75
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0 0.5 1 1.5 2 2.5
v2 ∆η/2
Fig.2. Intheleftpanel,theevent-by-eventdistributionsofthe pT-integratedv2 fornon-centralcollisions20-30%areshown. The
verticalaxisindicatestheprobabilitydensityofvn.Intherightpanel,η-dependentfactorizationratior2(ηa,ηb)proposedinRef.[14]
isshownfornon-centralcollisions.Thehorizontalaxisisηa=∆η/2.Therapidityrangeforηbisfixedas3<ηb<3.5.
thedistributionoftheevent-by-eventv isbroadenedbythefluctuations. Also,hydrodynamicfluctuations
2
contributetothedecorrelationoftheflowanglesinthelongitudinaldirection.
Theeffectsofhydrodynamicfluctuationsaretoolargewiththecurrentparametersetoftheshearviscos-
ity,η/s,andthecutoffscaleofhydrodynamicfluctuations,σ andσ ,Weneedtofixtherealisticvaluesof
η ⊥
thoseparametersbyscanningtheparametersinevent-by-eventcalculations. Alsothereisroomtoimprove
themodelbytakingintoaccountthecorrectionsofvariousquantitiesbythecutoffscaleofhydrodynamic
fluctuations. For example, transport coefficients should be renormalized for each cutoff scale of hydrody-
namic fluctuations [16]. The equation of state, the phasespace distribution of hadrons on the switching
hypersurfaceandeventheinitialentropydensitycouldbesubjecttochangeduetorenormalization.
ThisworkwassupportedbyJSPSKAKENHIGrantNumbers12J08554(K.M.)and25400269(T.H.).
References
[1] J.I.Kapusta,B.Muller,M.Stephanov,Phys.Rev.C85(2012)054906. arXiv:1112.6405,doi:10.1103/PhysRevC.85.
054906.
[2] L.D.Landau,E.M.Lifshitz,FluidMechanics,Vol.6ofCourseofTheoreticalPhysics,PergamonPress,NewYork,1959.
[3] T.Hirano,P.Huovinen,K.Murase,Y.Nara,Prog.Part.Nucl.Phys.70(2013)108–158.arXiv:1204.5814,doi:10.1016/j.
ppnp.2013.02.002.
[4] H.-J.Drescher,Y.Nara,Phys.Rev.C76(2007)041903.arXiv:0707.0249,doi:10.1103/PhysRevC.76.041903.
[5] F.Cooper,G.Frye,Phys.Rev.D10(1974)186.doi:10.1103/PhysRevD.10.186.
[6] D.Teaney,Phys.Rev.C68(2003)034913.arXiv:nucl-th/0301099,doi:10.1103/PhysRevC.68.034913.
[7] Y.Nara,N.Otuka,A.Ohnishi,K.Niita,S.Chiba,Phys.Rev.C61(2000)024901.arXiv:nucl-th/9904059,doi:10.1103/
PhysRevC.61.024901.
[8] P.Huovinen,P.Petreczky,Nucl.Phys.A837(2010)26–53.arXiv:0912.2541,doi:10.1016/j.nuclphysa.2010.02.015.
[9] P. Kovtun, D. T. Son, A. O. Starinets, Phys. Rev. Lett. 94 (2005) 111601. arXiv:hep-th/0405231, doi:10.1103/
PhysRevLett.94.111601.
[10] H.Song,Ph.D.thesis,OhioStateU.(2009).arXiv:0908.3656,[link].
URLhttp://inspirehep.net/record/829461/files/arXiv:0908.3656.pdf
[11] R.Baier,P.Romatschke,D.T.Son,A.O.Starinets,M.A.Stephanov,JHEP04(2008)100.arXiv:0712.2451,doi:10.1088/
1126-6708/2008/04/100.
[12] K.Murase,T.Hirano,Unpublishedresults(2013).arXiv:1304.3243.
[13] S.S.Adler,etal.,Phys.Rev.C69(2004)034909.arXiv:nucl-ex/0307022,doi:10.1103/PhysRevC.69.034909.
[14] V.Khachatryan,etal.,Phys.Rev.C92(3)(2015)034911.arXiv:1503.01692,doi:10.1103/PhysRevC.92.034911.
[15] J.Jia,P.Huo,Phys.Rev.C90(3)(2014)034915.arXiv:1403.6077,doi:10.1103/PhysRevC.90.034915.
[16] P.Kovtun,J.Phys.A45(2012)473001.arXiv:1205.5040,doi:10.1088/1751-8113/45/47/473001.