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

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

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

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.

The merits of the Renormalization Group/Flow eq:

The purpose of the 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 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!

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

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

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

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

; 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 !

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)

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

Def. the projection onto the kernel ker

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.

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.

The fluid dynamics limit of the Boltzmann equation Liouville equationBoltzman equationHydro dyn. Slower dynamics The basics of Boltzmann equation: the coll. Integral: the symmetry of the cross section:

The collision invariant: The conservation laws: the particle numberthe momentumthe kin. energy

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

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.

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

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

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

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

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

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

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.