RG Approach from Ultra Cold Atoms to the Hot QGP YITP, Aug. 22 –Sep.9, 2011

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

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

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

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

7. The RG/flow equation The yet unknown function issolved 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. even for evolution equations appearing other fields! The purpose of this talk: • 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. • 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.

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

11. Secular terms appear again! With I.C.: ; parameterized by the functions, Let us try to construct the envelope function of the set of locally divergent functions, Parameterized by t0 ! The secular terms invalidate the pert. theroy, like the log-divergence in QFT! :

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

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

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

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

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

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

21. Summary of !st part and remarks 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. c) PD equations; Layered pulse dynamics for TDGL and NLS. d) Reduction of stocastic equation with several variables S. Ei, K. Fujii and T.K. , Ann.Phys.(’00) Y. Hatta and T.K. Ann. Phys. (2002)

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

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

24. References for RG method • 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 • 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] (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 Navier-Stokes eq. 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) Hydrodynamics is the effective dynamics of the kinetic (Boltzmann) equation in the infrared refime.

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

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

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

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

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; de Groot et al (1980), Cercignani and Kremer (2002) 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 No dissipation! Local equilibrium D. H. Rischke, nucl-th/9809044

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

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

34. 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 Ansatz of the origin of the dissipation=the spatial inhomogeneity, leading toNavier-Stokes in the non-rel. case . would become a macro flow-velocity Coarse graining of space-time may not be time-like derivative space-like derivative Rewrite the Boltzmann equation as, perturbation Only spatial inhomogeneity leads to dissipation. RG gives a resummed distribution function, from which and are obtained. Chen-Goldenfeld-Oono(’95),T.K.(’95), S.-I. Ei, K. Fujii and T.K. (2000)

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

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

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

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

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

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: RG/E equation The meaning of found to be the coarse graining condition The novel feature in the relativistic case; ; eg. Choice of the flow

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

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

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

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

47. Landau equation: Eckart (particle-flow) frame: Setting = with = i.e., (i) This satisfies the GMS constraints but not the Eckart’s. (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. Grad-Marle-Stewart constraints c.f. Grad-Marle-Stewart equation;

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

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, Matching condition! i.e., the Eckart frame, 4, (C) Constraints 2, 3 Contradiction! Constraint 1 See next page.

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