相対論的流體力學(xué)方程式ppt課件

1、相対論的流膂力學(xué)方程式相対論的流膂力學(xué)方程式微視的導(dǎo)出関最近試微視的導(dǎo)出関最近試13th Heavy-Ion Caf相対論的流膂力學(xué)相対論的流膂力學(xué) 高高重重反応：來反応：來 方行方行 末末 2021年月日年月日, 東大理東大理國(guó)広悌二京大理國(guó)広悌二京大理- 津村-大西氏共同研討中心-IntroductionRelativistic hydrodynamics for a perfect fluid is widely and successfully used in the RHIC phenomenology. T. Hirano, D.Teaney, .A growing interes

2、t in dissipative hydrodynamics. hadron corona (rarefied states); Hirano et al Generically, an analysis using dissipative hydrodynamics is needed even to show the dissipative effects are small. A.Muronga and D. Rischke; A. K. Chaudhuri and U. Heinz,; R. Baier, P. Romatschke and U. A. Wiedemann; R. Ba

3、ier and P. Romatschke (2019)and the references cited in the last paper. is the theory of relativistic hydrodynamics for a viscous fluidfully established?However,The answer isNo!unfortunately.Fundamental problems with relativistic hydro-dynamical equations for viscous fluid Ambiguities in the form of

4、 the equation, even in the same frame and equally derived from Boltzmann equation: Landau frame; unique, Eckart frame; Eckart eq. v.s. Grad-Marle-Stewart eq.; Muronga v.s. R. Baier et al b. Instability of the equilibrium state in the eq.s in the Eckart frame, which affects even the solutions of the

5、causal equations, say, by Israel-Stewart. W. A. Hiscock and L. Lindblom (85, 87); R. Baier et al (06, 07) c. Usual 1st-order equations are acausal as the diffusion eq. is, except for Israel-Stewart and those based on the extended thermodynamics with relaxation times, but the form of causal equations

6、 is still controversial.- The purpose of the present talk -For analyzing the problems a and b first,we derive hydrodynaical equations for a viscous fluid from Boltzmann equationon the basis of a mechanical reduction theory (so called the RG method) and a natural ansatz on the origin of dissipation.W

7、e also show that the new equation in the Eckart frame is stable.We emphasize that the definition of the flow and the physical nature of therespective local rest frame is not trivial as is taken for granted in the literature,which is also true even in the second-order equations. Typical hydrodynamic

8、equations for a viscous fluid Fluid dynamics = a system of balance equationsEckart eq.energy-momentum：number：Landau-Lifshits no dissipation in the number flow;no dissipation in energy flowDescribing the flow of matter. describing the energy flow.with transport coefficients:Dissipative partwith- Invo

9、lving time-like derivative -.- Involving only space-like derivatives -; Bulk viscocity,;Heat conductivity; Shear viscocity- Choice of the frame and ambiguities in the form -0,Tu0uNNo dissipativeenergy-densitynor energy-flow.No dissipativeparticle density Compatibility of the definitions of the flow

10、and the nature of the Local Rest FrameIn the kinetic approach, one needs a matching condition.Seemingly plausible ansatz are;Distribution function in LRF:Non-local distribution function;Is this always correct, irrespective of the frames?In particular,is particle frame the same local equilibrium stat

11、e as the energy frame?c.f. D. Rischke nucl-th/9809044J. M. Stewart, Non-Equilibrium Relativistic Kinetic Theory, Lecture Notes in Physics 10 (Springer-Verlag), 1971c.f. D. Rischke nucl-th/9809044These issues have not been seriously considered and are obscure in the existing literature.C. Marle, A.I.

12、H.Poincare, 10 (1969)The separation of scales in the relativistic heavy-ion collisions Liouville Boltzmann Fluid dyn. HamiltonianSlower dynamicson the basis of the RG method; Chen-Goldenfeld-Oono(95),T.K.(95)C.f. Y. Hatta and T.K. (02) , K.Tsumura and TK (05); Tsumura, Ohnishi, T.K. (07)Navier-Stoke

13、s eq.力學(xué)系縮約Relativistic Boltzmann equationRelativistic Boltzmann equationConservation law of the particle number and the energy-momentumH-theorem.The collision invariants, the system is local equilibriumMaxwell distribution (N.R.)Juettner distribution (Rel.)perturbationAnsatz of the origin of the dis

14、sipation= the spatial inhomogeneity, leading to Navier-Stokes in the non-rel. case . would become a macro flow-velocityDerivation of the relativistic hydrodynamic equation from the rel. Boltzmann eq. - an RG-reduction of the dynamicsK. Tsumura, T.K. K. Ohnishi;Phys. Lett. B646 (2019) 134-140c.f. Non

15、-rel. Y.Hatta and T.K., Ann. Phys. 298 (02), 24; T.K. and K. Tsumura, J.Phys. A:39 (2019), 8089time-like derivativespace-like derivativeRewrite the Boltzmann equation as,Only spatial inhomogeneity leads to dissipation.Coarse graining of space-timeRG gives a resummed distribution function, from which

16、andare obtained.Chen-Goldenfeld-Oono(95),T.K.(95), S.-I. Ei, K. Fujii and T.K. (2000)Solution by the perturbation theory0th0th invariant manifold“slowFive conserved quantitiesm = 5Local equilibriumreduced degrees of freedomwritten in terms of the hydrodynamic variables.Asymptotically, the solution c

17、an be written solelyin terms of the hydrodynamic variables.1st Evolution op.：inhomogeneous：The lin. op. has good properties:Collision operatorCollision operator1.Self-adjoint2.Semi-negative definite3. has 5 zero modes、other eigenvalues are negative.Def.inner product: metricfast motionto be avoided T

18、he initial value yet not determined The initial value yet not determined Modification of the manifold：Def. Projection operators:eliminated by the choicefast motionThe initial value not yet determinedSecond order solutionswithModification of the invariant manifold in the 2nd order;eliminated by the c

19、hoiceApplication of RG/E equation to derive slow dynamicsCollecting all the terms, we have;Invariant manifold (hydro dynamical coordinates) as the initial value:The perturbative solution with secular terms:found to be the coarse graining conditionChoice of the flow RG/E equationThe meaning of The no

20、vel feature in the relativistic case; eg.References on the RG/E method: T.K. Prog. Theor. Phys. 94 (95), 503; 95(97), 179 T.K.,Jpn. J. Ind. Appl. Math. 14 (97), 51 T.K.,Phys. Rev. D57 (98),R2035 T.K. and J. Matsukidaira, Phys. Rev. E57 (98), 4817 S.-I. Ei, K. Fujii and T.K., Ann. Phys. 280 (2000), 2

21、36 Y. Hatta and T. Kunihiro, Ann. Phys. 298 (2019), 24 T.K. and K. Tsumura, J. Phys. A: Math. Gen. 39 (2019), 8089 (hep-th/0512108) K. Tsumura, K. Ohnishi and T.K., Phys. Lett. B646 (2019), 134 L.Y.Chen, N. Goldenfeld and Y.Oono, PRL.72(95),376; Phys. Rev. E54 (96),376.C.f.The distribution function;

22、produce the dissipative terms!Notice that the distribution function as the solution is representedsolely by the hydrodynamic quantities! A generic form of the flow vector ：a parameter1223P PPP Projection op. onto space-like traceless second-rank tensor; Landau frameand Landau eq.!ExamplesTsatisfies

23、the Landau constraints0,0u uTuT0uNBulk viscosityHeat conductivityShear viscosityC.f. Bulk viscosity may play a role in determining the acceleration of the expansion of the universe, and hence the dark energy!-independentpc.f.()ppaIn a Kubo-type form;with the microscopic expressions for the transport

24、 coefficients;Eckart (particle-flow) frame:Setting =with(ii) Notice that only the space-like derivative is incorporated.(iii) This form is different from Eckarts and Grad-Marle-Stewarts, both of which involve the time-like derivative.c.f. Grad-Marle-Stewart equation;(i) This satisfies the GMS constr

25、aints but not the Eckarts.i.e.,Grad-Marle-Stewart constraints Landau equation:3uTuXWhich equation is better, Stewart et als or ours?The linear stability analysis around the thermal equilibrium state.c.f. Ladau equation is stable. (Hiscock and Lindblom (85)K.Tsumura and T.K. ;Phys. Lett. B 668, 425 (

26、2019).The stability of the equations in the “Eckart(particle) frameThe Eckart and Grad-Marle-Stewart equations gives an instability, which has been known, and is now found to be attributed to the fluctuation-induced dissipation, proportional to .(ii) Our equation (TKO equation) seems to be stable, b

27、eing dependent on the values of the transport coefficients and the EOS.K. Tsumura and T.K. ,PLB 668, 425 (2019).The stability of the solutions in the “Eckart (particle) frame:DuThe numerical analysis shows that, the solution to our equation is stable at least for rarefied gasses. See also,Y. Minami

28、and T.K., Prog. Theor. Phys. 122, No.4 (2021); arXiv:0904.2270 hep-th Compatibility with the underlying kinetic equations? Eckart constraints are not compatible with the Boltzmann equation, as proved in K.Tsumura, T.K. and K.Ohnishi;PLB646 (06), 134.Collision operator has 5 zero modes:Proof that the

29、 Eckart equation constraints can not be compatible with the Boltzmann eq.Preliminaries:The dissipative part;= withwheredue to the Q operator.The orthogonality condition due to the projection operator exactly corresponds to the constraints for the dissipative part of the energy-momentum tensor and th

30、e particle current!i.e., Landau frame,i.e., the Eckart frame,4,(C) Constraints 2, 3Constraint 1Contradiction!Matching condition!Phenomenological Derivationparticle frameenergy frameGeneric form of energy-momentum tensor and flow velocity:withNotice;natural choice and parametrization3eXpXcf. K. Tsumu

31、ra and T.K., arXiv:0906.0079hep-phFromIn particle frame;With the choice,we havef_e, f_n can be finite,not in contradiction withthe fundamental laws! Energy frame:coincide with the Landau equation with f_e=f_n=0.Microscopic derivation gives the explicit form of f_e and f_n in each frame:particle fram

32、e;energy frame;Israel-Stewart equations fromKinetic equation on the basis of the RG methodK. Tsumura and T.K., arXiv:0906.0079hep-phGeometrical image of reductionof dynamicsnRtXMdimMmndim Xn( ) tsOdim msInvariant and attractive manifold( )ddtXF X( )ddtsG sM=( )X XX s( , )fXr p; distribution function

33、 in the phase space (infinite dimensions), , uT ns; the hydrodinamic quantities (5 dimensions), conserved quantities.eg.The viscocities are frame-independent, in accordance with Lin. Res. Theory.However, the relaxation times and legths are frame-dependent.The form is totally different from the previ

34、ous ones like I-Ss,And contains many additional terms.contains a zero mode of the linearizedcollision operator.2p pmConformal non-inv.gives the ambiguity.00uuK. Tsumura and T.K., arXiv:0906.0079hep-phFor the details, seeSummary The (dynamical) RG method is applied to derive generic 1st- and 2nd-orde

35、r fluiddynamic equations, giving new constraints in the particle frame, consistent with a general phenomenological derivation. The new equation in the particle frame does not show a pathological behavior as Eckart eq. does. This means that the acausality problem and instability problem are due to di

36、fferent origins, respectively.Backups/vCTtqx Fouriers law; /qTx Then2/vCTtTCausality is broken; the signal propagate with an infinitespeed.Modification;NonlocalthermodynamicsMemory effects; i.e., non-MarkovianDerivation(Israel-Stewart)： Grads 14-moments method+ ansats so that Landau/Eckart eq.s are

37、derived.ProblematicThe problem of acausality:Extended thermodynamicsFive integral consts；zero modepseudo zero mode sol.Init. valueConstraints;Orthogonality condition with the zero modeszero mode pseudo zero modeEq. governing the pseudo zero mode;collision invariantseqqpqeqppqfAfL1Lin. Operator;andDe

38、rivation of the secnd-order equationsK. Tsumura and T.K., arXiv:0906.0079hep-phwith the initial cond.;Def.Projection to the pseudo zero modes；Thus,Up to 1st order;Initial condition；Invariant manifoldRG/E equationSlow dynamics (Hydro dynamicsInclude relaxation equationsExplicitly；Specifically,Def.New

39、!For the veocity field,0; Landau,/ 2;EckartIntegrals given in termsof the distribution functionExample: Energy frameFrame dependence of the relaxation timesCalculated for relativistic ideal gas with; frame independentB0y(v%s#oXlUiQfNbK8H5D2A+x*u$qZnWkShPeMaJ7F4C1z)w&t!pYmVjRgOcL9I6E3B0y(v%r#oXlT

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