Application of the Renormalization-group Method for the Reduction of Transport Equations

1. Application of the Renormalization-group Method for the Reduction of Transport Equations Teiji Kunihiro(YITP, Kyoto) Renormalization Group 2005 Aug. 29 – Sep. 3, 2005 Helsinki, Finland

2. Based on: • 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), 236 • Y. Hatta and T. Kunihiro, Ann. Phys. 298 (2002), 24

3. Contents • Introduction; merits of RG • RG equation v.s. envelope eq. • A simple example for RG resummation and derivation of the slow (amplitude and phase) dynamics • A generic example • Fluid dynamic limit of Boltzmann eq. • Brief summary and concluding remarks

4. 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.

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

6. even for evolution equations appearing other fields! The purpose of the 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 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.

7. Geometrical Image of the Reduction of Dynamics invariant (attractive) manifold

8. Y.Kuramoto(’89) c.f. N.N.Bogoliubov (a) Notion of inv. manifold (b) Derivation of Boltzmann equation from the Liouville equation. (c) Fluid dyn. from Boltzmann

9. 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

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. 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!

13. More generic example S.Ei, K. Fujii & T.K.(’00)

14. ker Def. the projection onto the kernel

15. 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.

16. 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.

17. The fluid dynamics limit ofthe Boltzmann equation Liouville equation Boltzman equation Hydro dyn. Slower dynamics The basics of Boltzmann equation: the coll. Integral: the symmetry of the cross section:

18. The conservation laws: the particle number the momentum the kin. energy The collision invariant:

19. Notice; this is only formal, because the distribution function is not solved!

20. H-function and the equilibrium If is collision invariant, the entropy (-H) does not change. This is the case when is a local equilibrium Function.

21. The reduction of Boltzmann eq. toFluid dynamical equation T.K. (’99);Y.Hatta and T.K.(’02) Suppose that the system is an old system and the Space-time dependence of the distribution function is now slow.

22. I.C. Pert. Exp. The 0-th order: We choose the stationary solution: Local Maxwellian!

23. The first order eq.: Def. of the lin.op.A: Def. the inn. prod. :the projection onto Ker A. The 1st order solution: the secular term Deformation from the local equilibrium dist.

24. Applying the RG/E equation, This is the master equation giving the time evolution of which constitute the fluid dynamic equation! In fact, taking the inner product with the elements of Ker A, i.e., , Euler Eq. with

25. The higher order: Navier-Stokes equation with a dissipation. Y.Hatta and T.K. Ann. Phys.298,24 (2002) Interesting to apply to derive the relativistic Fluid dynamics with dissipations.

26. Brief Summary and concluding remarks • The RG v.s. the envelop equation • 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

27. Some references • 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), 236 • Y. Hatta and T. Kunihiro, Ann. Phys. 298 (2002), 24 • L.Y.Chen, N. Goldenfeld and Y.Oono, PRL.72(’95),376; Phys. Rev. E54 (’96),376.