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)

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

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, u

Fewer d.o.f

Def of coarse-grained differentiation 　 (H. Mori, 1956, 1858, 1959)

　 an intermidiate scale time

Time-derivative in transport coeff. Is an averageOf 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

Geometrical image of reductionof dynamics

nR

t X

M

dim M m n

dim X n

( )ts

O dim ms

Invariant and attractive manifold

( )d

dt

XF X

( )d

dtsG s

M={ ( )}X X X s

eg.

1 2 1 2, ,..., ) ( , ,..., )n mg g g s s s X = ( g s =In Field theory,

renormalizable Invariant manifold M dim M m n

n-dimensional dynamical system:

Y.Kuramoto(’89)

( )ts0G (s)

XG(s) M

0M

( ) s

Geometrical image ofperturbative reductionof dynamics

Perturbative reductionof dynamics

O

Perturbative Approach:

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 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!

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 !

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

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!

Limit cycle in van der Pol eq.

Expand:

I.C. ;supposed to be an exact sol.

0( )O

1( )O

Limit cycle with a radius 2!

An approximate but globaly valid sol. as the envelope;

Extracted the asymptotic dynamics thatis a slow dynamics.

Generic example with zero modesS.Ei, K. Fujii & T.K.(’00)

Def. P the projection onto the kernel

1P Q ker A

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

0M

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 RG/E equation00/ t tt u 0

gives the envelope, which is

The global solution (the invariant manifod):

globally valid:

[ ]W C0

0[ ]P W C

M

0M

( ) C

u

0[ ]W C ( ) C

c.f. Polchinski theorem in renormalization theory In QFT.

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

Extension;

Aa) 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

Pert. Theory:

with RG equation!

A foundation of the RG method a la FRG. T.K. (1998)

Let

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

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.)

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

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.Navier-Stokes　eq.

Hydrodynamics is the effective dynamics of the kinetic (Boltzmann) equation in the infrared refime.

Geometrical image of reductionof dynamics

nR

t X

M

dim M m n

dim X n

( )ts

O dim ms

Invariant and attractive manifold

( )d

dt

XF X

( )d

dtsG s

M={ ( )}X X X s

( , )fX r p ; distribution function in the phase space (infinite dimensions)

{ , , }u T ns ; the hydrodinamic quantities (5 dimensions), conserved quantities.

eg.

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

The entropy current:

Conservation of entropy

i.e., the local equilibrium distribution fn;

(Maxwell-Juettner dist. fn.)

Local equilibrium distribution

( )pf x

Owing to the energy-momentum conservation, the collision integral also vanishes for the local equilibrium distribution fn.;

Remark:

[ ]( ) 0.eqpC f x

Typical　hydrodynamic　equations　for　a　viscous　fluid　

Fluid dynamics = a system of balance equations

Eckart　eq.Eckart　eq.

energy-momentum ：energy-momentum ：

number ：number ：

Landau-Lifshits　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 ---

0,T u

0u N

No dissipativeenergy-densitynor energy-flow.No dissipativeparticle density

Ambiguities of the definition of the flow and the LRFIn 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 quitedifferent 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)

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

trivial

0,T u

Landau

c.f.

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 systematicallyderive dissipative rel. hydrodynamics at once!

Cf. Chapman-Enskog method to derive Landau and Eckart eq.’s; see, eg, de Groot et al (‘80)

perturbationperturbation

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 dynamicsK. 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 which and are obtained.

Chen-Goldenfeld-Oono(’95),T.K.(’95), S.-I. Ei, K. Fujii and T.K. (2000)

may not be u

Solution　by　the　perturbation　theory0th0th

0th invariant manifold

“slow”“slow”

Five conserved quantities

m　=　5m　=　5

Local equilibrium

reduced degrees of freedom

written in terms of the hydrodynamic variables.Asymptotically, the solution can be written solelyin terms of the hydrodynamic variables.

1st1st

　Evolution op. ：inhomogeneous：

The lin. op. has good properties:

Collision operatorCollision operator

1.1. Self-adjointSelf-adjoint

2.2. Semi-negative　definiteSemi-negative　definite

3.3.

has 5 zero modes 、 other eigenvalues are negative.

Def. inner product:

1. Proof of self-adjointness

2. Semi-negativeness of the L

3.Zero modes

1 2 3p p p p

Collision invariants!or conserved quantities.

en-mom.

Particle #

metricmetric

fast　motionto　be　avoidedfast　motionto　be　avoided The initial value yet not

determined　 The initial value yet not

determined　

Modification　of　the　manifold ：

Def. Projection operators:

eliminated by the choice

fast　motionfast　motionThe initial value not yet determinedThe initial value not yet determined

Second order solutions

with

Modification of the invariant manifold in the 2nd order;

eliminated by the choice

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.

The distribution function;

produce the dissipative terms!

Notice that the distribution function as the solution is representedsolely by the hydrodynamic quantities!

A　generic　form　of　the　flow　vector

： a　parameter ： a　parameter

1 2

2 3P

P P P

Projection op. onto space-like traceless second-rank tensor;

bulk pressurebulk pressure

heat flowheat flow

The　microscopic　representation　of　　the　dissipative　flow:The　microscopic　representation　of　　the　dissipative　flow:

stress tensorstress tensor

de　Groot　et　alde　Groot　et　al

Evaluation of the dissipative term thermal forces

All quantities are functions of the hydrodynamic variables, ,T and u

Reduced enthalpy

/P Vc c Ratio of specificheats

Thermal forces

Volume change

Shear flow

Temperature and pressuregradient

/P Vc c

Landau　frameand　Landau　eq.!Landau　frameand　Landau　eq.!

Examples

T

satisfies the Landau constraints

0, 0u u T u T

0u N

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!

-independentpc.f.

( )p pa In a Kubo-type form;

with the microscopic expressions for the transport coefficients;

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:

Collision　operatorCollision　operator

has　5　zero　modes:

Preliminaries:

The dissipative part; =

with

where

due to the Q operator.

Conditions of fit v.s. orthogonality condition

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, 3Constraint 1

Contradiction!

Matching condition!

p u p u

See next page.

(C)

Constraints 2, 3

Constraint 1

Contradiction!

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)

(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:

Du

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 isproved 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))

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)

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.

Du