1. RG derivation of relativistic hydrodynamic equations as the infrared dynamics of Boltzmann equation RG Approach from Ultra Cold Atoms to the Hot QGP YITP, Aug. 22 –Sep.9, 2011 T. Kunihiro (Kyoto)

2. Contents Introduction: separation of scales in (non-eq. ) physics/need of reduction theory RG method as a reduction theory Simple examples for RG resummation and derivation of the slow dynamics A generic example with zero modes A foundation of the method by Wilson-Wegner like argument Fluid dynamical limit of Boltzmann eq. Brief summary

3. The separation of scales in the relativistic heavy-ion collisions Liouville Boltzmann Fluid dyn. Hamiltonian Slower dynamics Hydrodynamics is the effective dynamics with fewer variables of the kinetic (Boltzmann) equation in the infrared refime. Introduction (BBGKY hierarchy) One-body dist. fn. T,n, Fewer d.o.f

4. Def of coarse-grained differentiation (H. Mori, 1956, 1858, 1959) an intermidiate scale time Time-derivative in transport coeff. Is an average Of microscopic derivatives. Averaging ： Set-up of Initial condition ： invariant (or attractive) manifold Eg: Boltzmann: mol. chaos Take I.C. with no two-body correl.. Bogoliubov (1946), Kubo et al (Iwanami, Springer) J.L. Lebowitz, Physica A 194 (1993),1. K. Kawasaki (Asakura, 2000), chap. 7. Construction of the invariant manifold Basic notions for reduction of dynamics

5. Geometrical image of reduction of dynamics X M O Invariant and attractive manifold eg. In Field theory, renormalizableInvariant manifold Mdim M n-dimensional dynamical system:

6. Y.Kuramoto(’89) X M Geometrical image of perturbative reduction of dynamics Perturbative reduction of dynamics O Perturbative Approach:

7. The RG/flow equation The yet unknown function is solved exactly and inserted into, which then becomes valid in a global domain of the energy scale.

8. The merits of the Renormalization Group/Flow eq:

9. The purpose of this talk: (1)Show that the RG gives a powerful and systematic method for the reduction of dynamics; useful for construction of the attractive slow manifold. (2) Apply the method to reduce the relativistic fluid dynamics from the Boltzmann equation. (3)An emphasis put on the relation to the classical theory of envelopes ； the resummed solution obtained through the RG is the envelope of the set of solutions given in the perturbation theory. even for evolution equations appearing other fields!

10. A simple example:resummation and extracting slowdynamics T.K. (’95) a secular term appears, invalidating P.T. the dumped oscillator!

11. ; parameterized by the functions, : Secular terms appear again! With I.C.: The secular terms invalidate the pert. theroy, like the log-divergence in QFT! Let us try to construct the envelope function of the set of locally divergent functions, Parameterized by t 0 !

12. A geometrical interpretation: construction of the envelope of the perturbative solutions The envelope of E: ? The envelop equation: the solution is inserted to F with the condition the tangent point RG eq. T.K. (’95) G=0

13. The envelop function an approximate but global solution in contrast to the pertubative solutions which have secular terms and valid only in local domains. Notice also the resummed nature! c.f. Chen et al (’95) Extracted the amplitude and phase equations, separately!

14. Limit cycle in van der Pol eq. Expand: I.C. ;supposed to be an exact sol.

15. Limit cycle with a radius 2! An approximate but globaly valid sol. as the envelope; Extracted the asymptotic dynamics that is a slow dynamics.

16. Generic example with zero modes S.Ei, K. Fujii & T.K.(’00)

17. Def. the projection onto the kernel

18. Parameterized with variables, Instead of ! The would-be rapidly changing terms can be eliminated by the choice; Then, the secular term appears only the P space; a deformation of the manifold

19. Deformed (invariant) slow manifold: The RG/E equation gives the envelope, which is The global solution (the invariant manifod): We have derived the invariant manifold and the slow dynamics on the manifold by the RG method. Extension; (a)Is not semi-simple. (2) Higher orders. A set of locally divergent functions parameterized by ! globally valid: (Ei,Fujii and T.K. Ann.Phys.(’00)) Layered pulse dynamics for TDGL and NLS.

20. The RG/E equation gives the envelope, which is The global solution (the invariant manifod): globally valid: M u c.f. Polchinski theorem in renormalization theory In QFT.

21. We have derived the invariant manifold and the slow dynamics on the manifold by the RG method. Extension; a)Is not semi-simple. b) Higher orders. S. Ei, K. Fujii and T.K., Ann.Phys.(’00) Y. Hatta and T.K. Ann. Phys. (2002) Layered pulse dynamics for TDGL and NLS. c) PD equations; Summary of !st part and remarks d) Reduction of stocastic equation with several variables

22. Pert. Theory: with RG equation! A foundation of the RG method a la FRG.T.K. (1998)

23. Let showing that our envelope function satisfies the original equation (B.11) in the global domain uniformly.

24. References for RG 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. (Non-pert. Foundation) T.K. and J. Matsukidaira, Phys. Rev. E57 (’98), 4817 S.-I. Ei, K. Fujii and T.K., Ann. Phys. 280 (2000), 236 (foundation) Y. Hatta and T. Kunihiro, Ann. Phys. 298 (2002), 24 T.K. and K. Tsumura, J. Phys. A: Math. Gen. 39 (2006), 8089 （ N-S) K. Tsumura, K. Ohnishi and T.K., Phys. Lett. B646 (2007), 134, K. Tsumura and T.K., arXiv:1108.1519 [hep-ph] L.Y.Chen, N. Goldenfeld and Y.Oono, PRL.72(’95),376; Phys. Rev. E54 (’96),376. Discovery that allowing secular terms appear, and then applying RG- like eq gives `all’ basic equations of exiting asymptotic methods (envelope th.)

25. Hydrodynamic limit of Boltzmann equation ---RG method---

26. The separation of scales in the relativistic heavy-ion collisions Liouville Boltzmann Fluid dyn. Hamiltonian Slower dynamics on 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) Navier-Stokes eq. Hydrodynamics is the effective dynamics of the kinetic (Boltzmann) equation in the infrared refime.

27. Geometrical image of reduction of dynamics X M O Invariant and attractive manifold ; distribution function in the phase space (infinite dimensions) ; the hydrodinamic quantities (5 dimensions), conserved quantities. eg.

28. Relativistic Boltzmann equation --- (1) Collision integral: Symm. property of the transition probability: Energy-mom. conservation; --- (2) Owing to (1), --- (3) Collision Invariant : the general form of a collision invariant; which can be x-dependent! Eq.’s (3) and (2) tell us that

29. The entropy current: Conservation of entropy i.e., the local equilibrium distribution fn; (Maxwell-Juettner dist. fn.) Local equilibrium distribution Owing to the energy-momentum conservation, the collision integral also vanishes for the local equilibrium distribution fn.; Remark:

30. Typical hydrodynamic equations for a viscous fluid Fluid dynamics = a system of balance equations Eckart eq. energy- momentum ： number ： Landau-Lifshits no dissipation in the number flow; no dissipation in energy flow Describing the flow of matter. describing the energy flow. with transport coefficients: Dissipative part with --- Involving time-like derivative ---. --- Involving only space-like derivatives --- ; Bulk viscocity, ;Heat conductivity ; Shear viscocity --- Choice of the frame and ambiguities in the form --- No dissipative energy-density nor energy-flow. No dissipative particle density

31. Ambiguities of the definition of the flow and the LRF In the kinetic approach, one needs conditions of fit or matching conditions., irrespective of Chapman-Enskog or Maxwell-Grad moment methods: In the literature, the following plausible ansatz are taken; Is this always correct, irrespective of the frames? Particle frame is the same local equilibrium state as the energy frame? Note that the distribution function in non-eq. state should be quite different from that in eq. state. Eg. the bulk viscosity Local equilibrium No dissipation! D. H. Rischke, nucl-th/9809044 de Groot et al (1980), Cercignani and Kremer (2002)

32. An exaple of the problem with the ambiguity of LRF: = = with i.e., Grad-Marle-Stewart constraints K. Tsumura, K.Ohnishi, T.K. (2007) still employed in the literature including I-S trivial Landau c.f.

33. Previous attempts to derive the dissipative hydrodynamics as a reduction of the dynamics N.G. van Kampen, J. Stat. Phys. 46(1987), 709 unique but non-covariant form and hence not Landau either Eckart! Here, In the covariant formalism, in a unified way and systematically derive dissipative rel. hydrodynamics at once! Cf. Chapman-Enskog method to derive Landau and Eckart eq.’s; see, eg, de Groot et al (‘80)

34. perturbation Ansatz of the origin of the dissipation= the spatial inhomogeneity, leading to Navier-Stokes in the non-rel. case. would become a macro flow-velocity Derivation of the relativistic hydrodynamic equation from the rel. Boltzmann eq. --- an RG-reduction of the dynamics K. Tsumura, T.K. K. Ohnishi;Phys. Lett. B646 (2007) 134-140 c.f. Non-rel. Y.Hatta and T.K., Ann. Phys. 298 (’02), 24; T.K. and K. Tsumura, J.Phys. A:39 (2006), 8089 time-like derivative space-like derivative Rewrite the Boltzmann equation as, Only spatial inhomogeneity leads to dissipation. Coarse graining of space-time RG gives a resummed distribution function, from whichandare obtained. Chen-Goldenfeld-Oono(’95), T.K.(’95), S.-I. Ei, K. Fujii and T.K. (2000) may not be

35. Solution by the perturbation theory 0th 0th invariant manifold “slow” Five conserved quantities m = 5 Local equilibrium reduced degrees of freedom written in terms of the hydrodynamic variables. Asymptotically, the solution can be written solely in terms of the hydrodynamic variables.

36. 1st Evolution op. ： inhomogeneous ： The lin. op. has good properties: Collision operator 1. Self-adjoint 2. Semi-negative definite 3. has 5 zero modes 、 other eigenvalues are negative. Def.inner product:

37. 1. Proof of self-adjointness 2. Semi-negativeness of the L 3.Zero modes Collision invariants! or conserved quantities. en-mom. Particle #

38. metric fast motion to be avoided fast motion to be avoided The initial value yet not determined Modification of the manifold ： Def. Projection operators: eliminated by the choice

39. fast motion The initial value not yet determined Second order solutions with Modification of the invariant manifold in the 2 nd order; eliminated by the choice

40. Application of RG/E equation to derive slow dynamics Collecting 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 condition Choice of the flow RG/E equation The meaning of The novel feature in the relativistic case; ; eg.

41. The distribution function; produce the dissipative terms! Notice that the distribution function as the solution is represented solely by the hydrodynamic quantities!

42. A generic form of the flow vector ： a parameter Projection op. onto space-like traceless second-rank tensor;

43. bulk pressure heat flow The microscopic representation of the dissipative flow: stress tensor de Groot et al Evaluation of the dissipative term thermal forces All quantities are functions of the hydrodynamic variables, and Reduced enthalpy Ratio of specific heats

44. Thermal forces Volume change Shear flow Temperature and pressure gradient

45. Landau frame and Landau eq.! Landau frame and Landau eq.! Examples satisfies the Landau constraints

46. Bulk viscosity Heat conductivity Shear viscosity C.f. Bulk viscosity may play a role in determining the acceleration of the expansion of the universe, and hence the dark energy! -independent c.f. In a Kubo-type form; with the microscopic expressions for the transport coefficients;

47. Eckart (particle-flow) frame: Setting = = with (ii) Notice that only the space-like derivative is incorporated. (iii) This form is different from Eckart’s and Grad-Marle-Stewart’s, both of which involve the time-like derivative. c.f. Grad-Marle-Stewart equation; (i) This satisfies the GMS constraints but not the Eckart’s. i.e., Grad-Marle-Stewart constraints Landau equation:

48. Collision operator has 5 zero modes: Preliminaries: The dissipative part; = with where due to the Q operator. Conditions of fit v.s. orthogonality condition

49. The orthogonality condition due to the projection operator exactly corresponds to the constraints for the dissipative part of the energy-momentum tensor and the particle current! i.e., Landau frame, i.e., the Eckart frame, 4, (C) Constraints 2, 3 Constraint 1 Contradiction! Matching condition! See next page.

50. (C) Constraints 2, 3 Constraint 1 Contradiction!

51. Which equation is better, Stewart et al’s or ours? The linear stability analysis around the thermal equilibrium state. c.f. Ladau equation is stable.(Hiscock and Lindblom (’85)) The stability of the equations in the “Eckart(particle)” frame K.Tsumura and T.K. ; Phys. Lett. B 668, 425 (2008).; arXiv:1107.1519 See also, Y. Minami and T.K., Prog. Theor. Phys.122, 881 (2010)

52. (i)The 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, being dependent on the values of the transport coefficients and the EOS. K.Tsumura and T.K. (2008) The stability of the solutions in the particle frame: The numerical analysis shows that, the solution to our equation is stable at least for rarefied gasses. A comment: The stability of our equations derived with the RG method is proved to be stable without recourse to any numerical calculations; this is a consequence of the positive-definiteness of the inner product. (K. Tsumura and T.K., (2011))

53. Brief Summary and concluding remarks (1)The RG v.s. the envelop equation (2) The RG eq. gives the reduction of dynamics and the invariant manifold. (3) The RG eq. was applied to reduce the Boltzmann eq. to the fluid dynamics in the limit of a small space variation. Other applications: a. the elimination of the rapid variable from Focker- Planck eq. b. Derivation of Boltzmann eq. from Liouvill eq. c. Derivation of the slow dynamics around bifurcations and so on. See for the details, Y.Hatta and T.K., Ann. Phys. 298(2002),24 T.K. and K. Tsumura, J. Phys. A: Math. Gen. 39 (2006), 8089 (hep-th/0512108)

54. Summary of second-half part Eckart equation, which and a causal extension of which are widely used, is not compatible with the underlying Relativistic Boltsmann equation. The RG method gives a consistent fluid dynamical equation for the particle (Eckart) frame as well as other frames, which is new and has no time-like derivative for thermal forces. The linear analysis shows that the new equation in the Eckart (particle) frame can be stable in contrast to the Eckart and (Grad)-Marle-Stewart equations which involve dissipative terms proportional to. The RG method is a mechanical way for the construction of the invariant manifold of the dynamics and can be applied to derive a causal fluid dynamics, a la Grad 14-moment method. (K. Tsumura and T.K., in prep.) According to the present analysis, even the causal (Israel-Stewart) equation which is an extension of Eckart equation should be modified. There are still many fundamental isseus to clarify for establishing the relativistic fluid dynamics for a viscous fluid.