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Anisotropic hydrodynamics – basic concepts 3 1 WojciechFlorkowski ∗ 0 InstituteofPhysics,JanKochanowskiUniversity,ul.Swietokrzyska15,PL-25406Kielce,Poland 2 andTheH.NiewodniczanskiInstituteofNuclearPhysics,PolishAcademyofSciences, n PL-31342Krakow,Poland a J E-mail:[email protected] 1 3 Mauricio Martinez DepartamentodeFísicadePartículasandIGFAE,UniversidadedeSantiagodeCompostela, ] h E-15782SantiagodeCompostela,Galicia,Spain t E-mail:[email protected] - l c RadoslawRyblewski u n TheH.NiewodniczanskiInstituteofNuclearPhysics,PolishAcademyofSciences,PL-31342 [ Krakow,Poland 1 E-mail:[email protected] v 9 MichaelStrickland 3 DepartmentofPhysics,KentStateUniversity,Kent,OH44242UnitedStates 5 andFrankfurtInstituteforAdvancedStudies,Ruth-Moufang-Strasse1,D-60438,Frankfurtam 7 . Main,Germany 1 0 E-mail:[email protected] 3 1 Due to the rapid longitudinalexpansionof the quark-gluonplasma created in relativistic heavy : v ioncollisions,potentiallylargelocalrestframemomentum-spaceanisotropiesaregenerated.The i X magnitude of these momentum-space anisotropies can be so large as to violate the central as- r sumption of canonical viscous hydrodynamicaltreatments which linearize around an isotropic a background. Inordertobetterdescribetheearly-timedynamicsofthequarkgluonplasma,one canconsiderinsteadexpandingaroundalocallyanisotropicbackgroundwhichresultsinadynam- icalframeworkcalledanisotropichydrodynamics.Inthisproceedingscontributionwereviewthe basicconceptsoftheanisotropichydrodynamicsframeworkpresentingviewpointsfromboththe phenomenologicalandmicroscopicpointsofview. XthQuarkConfinementandtheHadronSpectrum, October8-12,2012 TUMCampusGarching,Munich,Germany Speaker. ∗ (cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ Anisotropichydrodynamics WojciechFlorkowski 1. Introduction Thespace-timeevolutionofmattercreatedinrelativistic heavy-ioncollisionsseemstobevery welldescribedbyrelativisticviscoushydrodynamics[1–13]. Nevertheless,intheearlystagesofthe collisions,duetolargeflowgradientstheviscouscorrectionstotheperfect-fluidenergy-momentum tensor become very large and the system becomes highly anisotropic in momentum space (in the local restframe). Typically, thetransverse pressure, P , ismuchlarger than the longitudinal pres- ⊥ sure,P ,wherethedirectionsarespecifiedwithrespecttothebeamaxis. Generallyspeaking,large k momentum-spaceanisotropiesposeaproblemfor2nd-orderviscoushydrodynamics, sinceitrelies on a linearization around an isotropic background. It has been shown that large linear corrections generate unphysical results such as negative particle pressures, negative one-particle distribution functions, etc.[14]. A similar aspect of the same problem is that microscopic models of the early stages of relativistic heavy-ion collisions suggest that the produced system is highly anisotropic [15]. In the theory of the Color Glass Condensate (CGC), at very early proper-times, t 1/Q , where Q is s s ≪ the saturation scale, the classical gluon fields lead to an energy-momentum tensor which has the form Tmn =diag(e ,e ,e , e )[16,17]. Thisformimpliesanegativevalueofthelongitudinal pres- − sure with the transverse pressure equal to the energy density. At later proper times, t 1/Q , s ≫ boththeanalytical perturbative approaches [18]andthefullnumericalsimulations[19]leadtothe form Tmn =diag(e ,e /2,e /2,0). This form implies that the longitudinal pressure is zero. Conse- quently, propermatching oftheresults ofthemicroscopic modelswiththehydrodynamic descrip- tion (where the energy-momentum tensor should be close tothe isotropic form)is not straightfor- ward. In addition to the issues faced at early times, one finds that as one approaches the transverse edge of the interaction region, corrections to ideal hydrodynamics can be large at all times. As a result, one expects large momentum-space anisotropies to be present at the edges [14] and, as a result, one is motivated to find better approximation schemes to describe the dynamics in these regions. Finally,andperhapsmostimportantly,physicallyoneexpectsentropyproductiontovanish in two limits: the ideal hydrodynamical limit (vanishing shear viscosity) and the free streaming limit (infinite shear viscosity); however, within viscous hydrodynamics, entropy production is a monotonically increasing function of the shear viscosity. In the large shear viscosity limit, as one might expect, viscous hydrodynamics becomes a poor approximation and one is, once again, motivatedtofindanalternative framework. The difficulties eluded to above motivated the development of a reorganization of viscous hydrodynamics inwhichoneincorporates thepossibilityof largemomentum-spaceanisotropies at leading order. Theinclusion oflarge anisotropies alsoallowsfordirect matchingwiththetheories such as CGC. This framework has been dubbed anisotropic hydrodynamics [20–26]. Below we will briefly review the basic ideas underpinning the framework and direct the reader to various references for more details. A more self-contained presentation of the concepts of anisotropic hydrodynamics canbefound,e.g.,inRef.[27]. 2 Anisotropichydrodynamics WojciechFlorkowski 2. Anisotropichydrodynamics The framework of anisotropic hydrodynamics may be introduced in (at least) two ways. The firstmethod isphenomenological and refers directly totensor structures describing ananisotropic fluid [20,22,24,26]. The second method is microscopic; one uses the Boltzman equation for the parton (gluon) distribution function and makes an expansion around the anisotropic background [21,23,25]. In this Section we outline the first method. The microscopic approach will be dis- cussed inthenextSection. Thedynamicequations ofanisotropic hydrodynamics follow fromtheequations ¶ m Tmn = 0, (2.1) ¶ m s m = S , (2.2) whichexpresstheenergy-momentum conservation andentropyproduction laws,respectively. The mn energy-momentum tensor T hastheform Tmn =(e +P )Um Un P gmn (P P )Vm Vn . (2.3) ⊥ − ⊥ − ⊥− k WenotethatEq.(2.3)reproducestheenergy-momentum tensoroftheperfectfluidifthetwopres- sures are equal. Similarly, in the case S = 0 the entropy production law in (2.2) is reduced to m m the entropy conservation law. The four-vector U describes the flow of matter and V defines the beam (z) axis. Inthe general (3+1)-dimensional case, weusethe following parameterizations: Um =(u coshJ ,u ,u ,u sinhJ ) and Vm =(sinhJ ,0,0,coshJ ), where u and u are the trans- 0 x y 0 x y verse components of the four-velocity field and J is the longitudinal fluid rapidity. The entropy fluxs m equalss Um ,wheres isthenon-equilibrium entropydensity. One can show [20] that instead of P and P it is more convenient to use the entropy density s and the anisotropy parameter x as twok indep⊥endent variables (one may use the approximation P /P x 3/4). Similarly to standard hydrodynamics with vanishing baryon chemical potential, − thkee⊥ne≈rgy density e ,theentropy density s ,andtheanisotropy parameter xarerelatedthroughthe generalized equationofstate[22] e (x,s ) = e (s )r(x), (2.4) qgp P (x,s ) = P (s ) r(x)+3xr (x) , qgp ′ ⊥ P (x,s ) = P (s )(cid:2)r(x) 6xr (x)(cid:3), qgp ′ k − (cid:2) (cid:3) where the functions e and P correspond to a realistic isotropic QGP equation of state, see qgp qgp e.g. Ref. [28]. The function r(x) characterizes properties of the fluid which exhibits the pressure anisotropy specifiedbythevariable x[20,29,30] x−13 xarctan√x 1 r(x)= 1+ − . (2.5) 2 (cid:20) √x 1 (cid:21) − Intheisotropiccasex=1,r(1)=1,r(1)=0,andEq.(2.4)isreducedtothestandardequationof ′ stateusedin[28]. 3 Anisotropichydrodynamics WojciechFlorkowski ThefunctionS appearinginEq.(2.2)definestheentropyproduction. Inthephenomenological approach [20]thefollowingformofS hasbeenused (1 √x)2 s S (s ,x)= − , (2.6) √x t eq wherethetime-scaleparametert controlstherateofequilibration. InRefs. [21,23,25]amedium eq dependentt wasused,whichwasinverselyproportionaltothetypicaltransversemomentumscale eq in the system. In this case, as wewilldiscuss in some detail below, the late time dynamics is that of second order viscous hydrodynamics. If a constant value of t is used, the system instead eq approaches perfectfluidbehaviorfor t t . eq ≫ If the generalized equation of state and the entropy production term are defined, the dynamical equations of anisotropic hydrodynamics become 5 equations for 5 unknown functions: three independent components of the fluid four-velocity Um , the entropy density s , and the anisotropy parameter x. In each case one has to additionally specify the value of the relaxation time or its dependence onotherthermodynamics-like variables suchas s orx. 3. Microscopicapproach Inthemicroscopicapproach[21,23,25]oneconsiderslocallyanisotropicmomentumdistribu- tionsofpartonsthatarecharacterized bytwodifferentscales: l andl . Theymaybeinterpreted ⊥ k asthetransverse andlongitudinal temperatures (LRF standsbelowforthelocalrestframe) p p fLRF = f(cid:18)l ⊥,|l k|(cid:19). (3.1) ⊥ k In the studies of anisotropic quark-gluon plasma one often uses the Romatschke-Strickland form [31] p2 p2 1 fLRF = f−vul ⊥2 +l k2= f(cid:18)−l p2 +x p2(cid:19), (3.2) u q ⊥ k  t ⊥ k  ⊥ where f is an arbitrary isotropic distribution function which is set by the underlying equilibrium distribution functionfortheparticles beingconsidered. Thecovariant versionof(3.2)reads 1 f = f (p U)2+x (p V)2 , (3.3) (cid:18)−l q · · (cid:19) ⊥ where x =1 x. Using Eq. (3.3) in the standard definitions of the energy-momentum tensor and − theentropyfluxoneobtainstheexpressionsforTmn ands m oftheformintroducedintheprevious Section. In order to obtain the dynamical equations in the microscopic approach, one uses the Boltz- mannequation. ThezerothmomentoftheBoltzmannequation, pm ¶ m f =C,withthecollisionterm treated intherelaxation-time approximation, C= p UG (f f ),gives eq − · − 1 G ¶ m (s Um )= dPC=S = (neq n). (3.4) 4Z 4 − 4 Anisotropichydrodynamics WojciechFlorkowski Here G is the inverse relaxation time (G is inversely proportional to t introduced earlier in the eq entropy source, however, the proportionality constant might be different from unity), f is the eq background equilibrium distribution function, and nisthepartondensity(n isthecorresponding eq equilibrium partondensity). WenotethatfortheRomatschke-Strickland formonefindss =4n. ThefirstmomentoftheBoltzmannequation yieldstheenergy-momentum conservation law ¶ m dPpn pm f =¶ m Tnm =0. (3.5) Z Thesefourequations aresatisfiedonlyifthecollisionterm fulfillsthecondition dPpn C= dPp U pn G (f f )=0. (3.6) eq Z −Z · − Thelastequation istheLandaumatching condition thatspecifies thetemperature, T,appearing in thebackgroundequilibriumdistributions f . IftheLandaumatchingisusedtoeliminateT,weare eq leftwithEqs.(3.4)and(3.5)whichare5equationsfor5unknowns,similarlytothecasedescribed inthepreviousSection. Thestructureoftheequationsoftheanisotropichydrodynamicsobtainedfromthemicroscopic considerations is thesame asthat used inthe phenomenological approach. Onthe otherhand, the advantage ofthemicroscopic approach isthattheexplicitformofthedistribution function defines theentropysourcetermS andthegeneralizedequationofstate. IftheRomatschke-Stricklandform isused, thegeneralized equation ofstate hastheform (2.4)–(2.5), however, thefunctions e and qgp P describe a massless parton gas (with Bose or Boltzmann statistics). The phenomenological qgp andmicroscopic approaches havebeencomparedinmoredetailin[34]. Within the microscopic approach one canalso perform anexplicit matching toIsrael-Stewart second order viscous hydrodynamics in the limit of small anisotropy. In Ref. [21] it was shown that bylinearizing the(0+1)-dimensional anisotropic hydrodynamics dynamical equations andre- quiring thattheyreproduce second-order viscous hydrodynamics, onecanfixtherelaxation rate G appearing inEq.(3.4)tobe 2 5 h G ≡ tp , tp ≡ 4Peq, (3.7) whichforanidealequation ofstateresults in 2T G = , (3.8) 5h¯ where h¯ =h /S withh beingtheshearviscosity, S istheequilibrium entropy density, and T is eq eq theeffectivetemperature determined viaLandaumatching inthefirstmomentasdescribed above. Two points can be made using this result: (1) the relaxation rate G is proportional to the effective local temperature (fourth root of the energy density) and hence in regions of lower temperature, the relaxation to isotropy will be slower and (2) the relaxation rate G is inversely proportional to the shear viscosity to entropy ratio and hence when the shear viscosity increases, one expects the relaxation to isotropy will be slower. Both of these effects are important to understanding the dynamics ofisotropization inthequarkgluonplasma. 5 Anisotropichydrodynamics WojciechFlorkowski Wewant to emphasize that the microscopic approach presented here uses the zeroth and first moments of the Boltzmann equation. The zeroth moment in the standard approach to relativistic fluiddynamicsisusedtointroduce theparticle (orbaryon) numberconservation law. Thisisdone by introducing a chemical potential m together with the appropriate Landau matching condition which can beused toestablish thevalue of m (inasimilarmanner asthe temperature was defined above). In our opinion this procedure is not appropriate for the systems produced in the early stagesofrelativistic heavy-ion collisions. Suchsystemsconsistprimarilyofgluonswhosenumber is not conserved. In fact, the radiation of soft gluons leads to the increase of their numbers and to entropy production. Hence, the production of gluons is the main physical process which leads to equilibration of the system. Interestingly, such a physical picture is qualitatively similar to the bottom-up thermalization scenario [35]. 4. Summary Inthis brief proceedings contribution wehave attempted to review the motivation forand the foundational ideasbehindanisotropic hydrodynamics. Thedevelopmentofthisframeworkhasoc- curred using two separate methods, dubbed the phenomenological approach and the microscopic approach herein. We have reviewed both methods pointing out the fundamental connections that existbetweenthetwoapproaches. Inbothcasespresented, theframeworkofanisotropic hydrodynamics is constructed in such a way that the pressure (momentum) anisotropies of the system are builtdirectlyintotheformalism. Inouropinionthisframeworkformsanalternativetothestandard viscous-hydrodynamics approach where one starts with the isotropic distribution and introduces potentially largecorrections intheexpansion. The use of anisotropic hydrodynamics may help us to circumvent the early thermalization puzzle. It turns out that all soft hadronic observables may be successfully reproduced in the ap- proachwherethesystemisinitiallyhighlyanisotropic[26]. Itturnsoutthattheeffectsoftheinitial anisotropymaybecompensatedbyachangeoftheinitialconditions. Thispointstoapossibleuni- versality of the results for flow with respect to initial condition variation (at least with respect to theinitialmomentum-space anisotropy whichisassumed). There have been many other recent developments which we were not able to go into much detail about herein. The approach presented has been generalized to incorporate the presence of dynamicallongitudinalchromoelectricoscillations[32]andtoincludeamixtureoftwointeracting anisotropic(quarkandgluon)fluidswhicharecharacterizedbydifferentanisotropyparametersand equilibrationtimes[33]. 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