黏滞性流体动力学

第31卷第12期高能物理与核物理VoL 31, No. 122007年12月HIGH ENERGY PHYSICS AND NUCLEAR PHYSICSDec,2007Viscous Fluid dynamicsA. K. Chaudhuri)Variable Energy Cyclotron Centre, 1/AF, Bidhan Nagar, Kolkata 700 064, India)Abstract We briefly discuss the phenomenological theory of dissipative fuid. We also present some numericalresults for hydrodynamic evolution of QGP Auid with dissipation due to shear viscosity only. Its effect onparticle production is also studiedKey words QGP, dissipative fluid, viscosity1 Introductionlogical theory of dissipative hydrodynamics. Moredetailed exposition can be found in Ref. 5A large volume of experimental data from Au+Au A simple fuid, in an arbitrary state, is fully spec-eLisions at RHC are successfully analysed in an ified by primary variables: particle current(N")ideal Buid dynamic model".However, experimen- energy-momentum tensor(Tu )and entropy currenttal data do show deviation from ideal behavior. The (S" )and a number of additional(unknown)variablesideal fuid description works well in almost central Primary variables satisfies the conservation lawsAu+Au collisions near mid-rapidity at top rhC energy, but gradually breaks down in more peripheralaN“=0,collisions, at forward rapidity, or at lower collisiond, Tuv =0energies), indicating the onset of dissipative effend the 2nd law of thermodynamicsTo describe such deviations from ideal auid dynamcs, quantitatively, requires the numerical implemen-S"≥0tation of dissipative relativistic fuid dynamicsIn relativistic Auid dynamics, one defines a time-Though the theories of dissipative hydrodynamicsl5-6l has been known for more than 30 yearslike hydrodynamic 4-velocity, u"(normalised as u2significant progress towards its numerical implemen-1). One also define a projector, A=9'tation has only been made very recently 6-10. In the orthogonal to the velocity(4,=O). In equi-librium, an unique 4-velocity(u")exists such thatfollowing, I will briefly review the phenomenologicaltheory of dissipative hydrodynamics. Some numeri-entropy density(s)can be obtained fromI results comparing the ideal and 1st order disscode AZHYDRO-KOLKATA will be shown.2 Phenomenological theory of dissipa-tive hydrodynamics中国煤化工 to be fully speci-In this section, I briefly discuss the phenomeno- fied bHaquivalently by theCNMHGReceived 25 June 20071Er-mail: akc@ veccal ernet in1157一1161高能物理与核物理(HEP&NP)第31卷thermal potential, a=u/ r (u being the chemical po- Only in a state, close to a equilibrium one, such atential)and inverse temperature, B=u"/T. Given relation can be established. Assuming that the equi-a equation of state, 8=8(E, n), pressure p can be ob- librium relation Eq. (8)remains valid in a"near equi-tained from the generalised thermodynamic relation, librium state"also the entropy current can be gen-SH=pB-aN# +BTa(7 praised asUsing the Gibbs-Duhem relation, d(pBaS=S+dS"=p-aN“+aT“+Q",(12)Ne dar-TapdBa, following relations can be establishedon the equilibrium hyper-surface zeq(a, B")where Q is an undetermined quantity in 2nd orderin deviations SN=NA-Nu and STAv=TAV-TEUdse=-adNP+ BadA(8) Detail form of Q is constrained by the 2nd lawIn a non-equilibrium system, no 4-velocity can be a,S">0. With the help of conservation laws andfound such that Eqs. (4)--(6)remain valid. Tensor Gibbs-Duhem relation, entropy prodrdecomposition leads to additional termsbe written as,N“=N+bN“=n“+V“,(9)aS“=-N“∂4a+8T“∂-,+∂Q".(13)r=T+87“=Choice of Q leads to lst order or 2nd order the-eu"u-p△“]+ⅡΔw+r“+ories of dissipative hydrodynamics In 1st order the-(W“u"+W"u"),(10) ories Q=0,entropy current contains terms upto lstorder in deviations, SN and ST. Entropy produc-S"=S+55"=8u“+φ,(11) tion rate can be written asThe new terms describe a net Aow of chargeTS“=ⅡX-qXa+mX(14)△wN, heat flow,Wa=(e+p)/mnV“+q(where q" is the heat flow vector), and entropy fow where,xp“.Ⅱ=-;△T- p is the bulk viscous pressureThe2 nd law,aS“≥0 can be satisfied by postulating a linear relation between the dissipative flows=12(+44“-34」Tnand and thermodynamic forcesis the shear stress tensor. Hydrodynamic 4-velocity(15)can be chosen to eliminate either V(the Eckartframe, u" is parallel to particle flow)or the heat AowAV“(叫/T)(16)Etpq"(the Landau frame, u"is parallel to energy flow)(17)In relativistic heavy ion collisions, Landau's frame ismore appropriate than the eckart's frame. Dissipa- where s, A and n are the positive transport coef-tive flows are transverse to u and additionally, shear ficients, bulk viscosity, heat conductivity and shearstress tensor is traceless. Thus a non-equilibrium viscosity.state require 1+3+5=9 additional quantities, the dis- In lst order theories, causality is violated. If,sipative flows I, g(or V)and TH. In kinetic in a given fuid cell, at a certain time, thermodytheory, N" and Tw are lst and 2nd moment of the namic forces vanish, corresponding dissipative Auxesdistribution function. Unless the function is known also中国煤化工 cted in2 nd ordera-priori, two moments do not furnish enough infor- theorN MH Gntain terms uptomation to enumerate the microscopic states required 2nd order inon,"≠0. The most gen-to determine S", and in an arbitrary non-equilibrium eral Q containing terms upto 2nd order in deviationsstate, no relation exists between, Nv, Twv and S". can be written as第12期A. K. Chaudhuri: Viscous Fluid DynamicsQ=-(BoIr2-B1gqu +BT AT)TCentre Kolkata, we have developed a code for solv-Hor+aT(18) due to shear viscosity only, in 1+1 dimension(as-As before, one can cast the entropy production suming boost-invariance and cylindrical symmetry)rate(Ta, s)in the form of Eq. (14). Neglecting and presently extending the code in 2+1 dimensionsthe terms involving dissipative flows with gradients (with boost-invariance only ). We have completed theof equilibrium thermodynamic quantities(both are coding for the 1st order theory. Here, we present someassumed to be small)and demanding that a linear numerical resultsrelation exists between the dissipative fows and therAssuming boost-invariance, we have solved Ist order viscous hydrodynamics for initial state QGP inche dissipative fows can be obtained2+1 dimension, in(T, I, y,n )coordinate. We restrictourselves to central rapidity region, where the QGPⅡ=-(+ADm),fuid is essentially baryon free and to keep the calculaAD|,(20)tions simple we consider the most important dissipative term, the shear viscosity and neglect the otherwhere d= ua is the convective time derivativecosity. For viscosity, we have used two values, theUnlike in the 1st order theories, in 2nd order the-ADS/CFT motivated value n/s=0.08, and the per-ories, dynamical equations controls the dissipatiturbative estimate n/s=0. 135. Details of the equa-flows. Even if thermodynamic forces vanish, dissi-tions solved can be found in Ref [10 We just men-ative Rows do not vanish instantly.tion that we have used the equation of state EoS-QBefore we proceed further, it may be mentioned developed in Ref [1], with 1st order phase transitionwith critical temperature T=164Mev. As mentionedthat the parameters, a and Ba are not connected tothe actual state(N", TH ) The pressure p in To d Flier, Ist order dissipative hydrodynamics violateis also not the actual"thermodynamics pressurecausality and can lead to unphysical effects like earlyi.e. not the work done in an isentropic expansionreheating. But for QGP Auid, which is close to anChemical potential a and 4-inverse temperature Bxideal Auid, such effects can be minimised with ap-has meaning only for the equilibrium state. Their propriate initial conditions, and we did not find anymeaning need not be extended to non-equilibrium evidence of early reheatingstates also. However, it is possible to define a fic- 3.1 Evolution of the viscous fuidtitius"local equilibrium"state, point by point, suchIn the following we will show the results obtain inthat pressure p in Eq.(12 )can be identified with the aut au collision at impact parameter b=6. 8fm, whichthermodynamic pressure, at least upto lst order. In approximately corresponds to 16%--24% centrality2nd order in deviations, such an identification is not Au+Au collisions. With the same initial conditions,we have solved the energy-momentum conservationequations for ideal Auid and viscous Auid. In Fig. 13 Viscous hydrodynamics for QGP in we have shown the constant energy density contour2+1 dimensionsplot in r-y plane, after an evolution of 5fm. The blacklines are for ideal fluid evolution. The red and blueNumerically, causal dissipative hydrodynamics is lines中国煤化工/8=0.08and0.135a challenging problem. One needs to solve simultane- respeCNMH Got of temperatureously 14 partial differential equations(5 conservation in a-T plane is shown. It 18 evident that comparedequations and 9 additional equations for the dissipa to ideal fuid, viscous fuid cools slowly. Transversetive fows, I, q"and T). Recently, at the Cyclotron expansion is also enhanced in a viscous fuid. It is not1160高能物理与核物理(HEP&N第31卷shown, but fluid velocity grow faster in viscous fow.Au+Au @=6.8fm: f=5.2fm0000号10010-50510Fig 3. PT spectra of t for ideal and viscousFig. 1. Contour plots of energy density at(proper)time T=5. 2fm. (see arX: nucl-th0二歌3m0703027025Aut Au@b=6.8fm: r52fmPr/GevFig. 4. Pr dependence of elliptic fow for nLastly, in Fig. 5, we have shown a comparison ofPr spectra obtained in ideal hydrodynamics with ini-Fig. 2. Contour plots of temperature at y=0 intial entropy density Sin:=110fm- with pr spectra ob-x-T plane.(see arX: nucl-th/0703027)tained in viscous hydrodynamics obtained with initiaentropy density, 60, 80 and 110. Pr spectra from vis-3.2 Particle spectracous Auid initialised with Sini=60-80 compare wellViscosity infuences the particle production by with the spectra from ideal fluid, initialised at higher(i)extending the freeze-out surface and (i) by in- entropy density. To produce the same spectra,vis-troducing a correction to the equilibrium distribution cous fluid require 20%-30% less initial temperaturefunction(see Ref [10 ) In Fig 3, transverse momentum distribution of t" from ideal and viscous hydro-n=0,S10dynamics are compared. Freeze-out temperature isTr=0. 158GeV. Pion production is increased in vis-10=1cous dynamics. We also note thaat eaec号10-3is more prominent at large pr than at low pr. Pr是10spectra of pions are flattened with viscosityEffect of viscosity is also prominent on elliptic iow(Fig 4). In ideal dynamics, elliptic flow continues toincrease with pr, In viscous dynamics, on the oth-"凵中国煤化工deal fluid witherhard, elliptic Alow tends to saturate. The result isCNMHG is comparevery encouraging, as experimentally also elliptic flowhi viscous dynamics with different initial en-tends to saturate at large pr第12期A K. Chaudhuri: Viscous Fluid Dynamics1161We have also studied the effect of vi4 Summary and conclusionsticle production. Viscosity generates entropy leadingto enhanced particle production. Particle productionWe have briefly reviewed the phenomenological is increased due to()extended freeze-out surface andtheory of dissipative hydrodynamics and presented (i)non-equilibrium correction to equilibrium distrisome numerical results from 1st order dissipative hy- bution function. With ADS/CFT (perturbativedrodynamic of QGP fluid with only shear viscosity. timate of viscosity at pr=3Gev, pion production isIst order theories suffer from the problem of causal- increased by a factor 3(5). Increase is even more atity, signal can travel faster than light. Unphysical large Pr. While viscosity enhances particle produc-ffects like reheating of the fuid, early in the evolu- tion, it reduces the elliptic Aow. At pr=3GeV, fortion,can occur. In this model study, we have con- ADS /CFT (perturbative )estimate of viscosity, ellip-sidered two values of viscosity, the ADS/CFT moti- tic flow is reduced by a factor of 2 (3). We also findvated value, n/s 80.08 and perturbatively estimated that at large pr elliptic flow tends to saturateviscosity, n/s80 135. We did not find any indicationTo conclude, present study shows viscosity, evenof unphysical reheating. Explicit simulation of ideal if small, can be very important in analysis of RHICand viscous fuids confirms that energy density of a Au+Au collisions. Currently accepted initial temperviscous fuid evolve slowly than its ideal counterpart. ature of hot dense matter produced in RHIC Au+AuThe fluid velocities on the other hand evolve faster in collisions, obtained from ideal fluid analysis can beviscous dynamics than in ideal dynamics. Transverse changed by 20% or more with dissipative dynamicsReferencesStewart J M. Ann. Phys.(NY ) 1979, 118: 3496 Teaney D. Phys. Rev., 2003, C68: 034913. ar Xiv:nucl-1 Kolb P F, Heinz U. In Quark-Gluon Plasma 3, Edited byth/0301099Hwa R C,WANG XN. 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