Application of the Renormalization-group Method for the Reduction of

Transport Equations

Teiji Kunihiro(YITP, Kyoto)

Renormalization Group 2005Aug. 29 – Sep. 3, 2005Helsinki, 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 energyscale.

g

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)

manifolddimM m n

Y.Kuramoto(’89)

c.f. N.N.Bogoliubov(a) Notion of inv. manifold

(b) Derivation of Boltzmannequation from the Liouville

equation.(c) Fluid dyn. from Boltzmann

A geometrical interpretation:construction of the envelope of the perturbative solutions

: ( , , , ( )) 0C F x y C

The envelope of CE:

?

The envelop equation: the solution is inserted to F with the condition

0 0x ( , ) ( , , ( ))G x y F x y x C

the tangent point

RG eq.0/ 0dF d

T.K. (’95)

G=0

0( ) 0F 0( ') 0F

x0x

0y

A simple example:resummation and extracting slowdynamics T.K. (’95)

a secular term appears, invalidating P.T.

the dumped oscillator!

; parameterized by the functions,0 0 0 0( ), ( ) ( )A t t t t

:

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 functionof the set of locally divergent functions,Parameterized by t0 !

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. P the projection onto the kernel ker A

1P Q

Parameterized with variables,Instead of !

mn

The would-be rapidly changing terms can be eliminated by thechoice;

Then, the secular term appears only the P space;a deformation ofthe manifold.

Deformed (invariant) slow manifold:

The RG/E equation00/ t tt u 0 gives the envelope, which is

The global solution (the invariant manifod):

We have derived the invariant manifold and the slow dynamicson the manifold by the RG method.

Extension; A(a) Is not semi-simple. (2) Higher orders.

A set of locally divergent functions parameterized by !0t

globally valid:

(Ei,Fujii and T.K.Ann.Phys.(’00))Layered pulse dynamics for TDGL and NLS.

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:

The collision invariant:

The conservation laws:

the particle number the momentum the kin. energy

Notice; this isonly formal,because thedistribution function is not solved!

H-function and the equilibrium

If is collision invariant, the entropy (-H) doesnot change.This is the case when is a local equilibriumFunction.

ln f

( , , )f tr v

The reduction of Boltzmann eq. toFluid dynamical equation

Suppose that the system is an old system and theSpace-time dependence of the distribution functionis 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:

P :the projection onto Ker A. 1Q P

Def. the inn. prod.

The 1st order solution:

the secular term Deformation from thelocal 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 elementsof Ker A, i.e.,

, Euler Eq.

with

The higher order:

Navier-Stokes equation with a dissipation.

Interesting to apply to derive the relativisticFluid 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.