K. Ohnishi (TIT)

K. Tsumura (Kyoto Univ.)T. Kunihiro (YITP, Kyoto)

Derivation of Relativistic Dissipative Hydrodynamic equationsby means of the Renormalization Group method

July 8, 2006 @ Riken

Introduction

Renormalization Group method

Derivation of Hydrodynamic equation

Summary

1.1.

Outline

4.4.

2.2.

3.3.

１． Introduction

QGP = Perfect Fluid Relativistic Hydrodynamical simulationwithout dissipation

Hadronic corona ・・・ dissipative hydrodynamic or kinetic description

QGP phase is also dissipative for Initial Condition based on Color Glass Condensate

Dissipative Relativistic Hydrodynamical analysis

Just Started (Muronga & Rischke(2004), Heinz et al (2005) )

「 RHIC Serves the Perfect Liquid. 」 April 18, 2005 April 18, 2005

cf) Asakawa, Bass and Muller, hep-ph/0603092

T. Hirano et al: Phys. Lett. B636 (2006) 299See also, Nonaka & Bass, nucl-th/0510038

Occurrence of instability due to lack of causality

Israel & Stewart’s regularization (1979)by introducing Relaxation time

Relativistic Hydrodynamic eq. with Dissipation　　　　 Not yet established

Hydro dynamical frame: choice of frame or flow

Landau frame(1959)Landau frame(1959)Eckart frame(1940)Eckart frame(1940) vs.

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Ambiguity in Hydrodynamic eq.Fluid dynamics = a system of balance equations

If dissipative, there arises an ambiguity

Eckart frameEckart frame

Landau frameLandau frame

no dissipation in the number flow

no dissipation in energy flow

Describing the matter flow.

Describing the energy flow.

: Energy-momentum : Number

nuN

XXuXuTXpuuT

2)~~

()(

Xp

nTnuN

XXpuuT

2)(

:transport coefficients ,,

cf. Non-relativistic case　　 Boltzmann eq. Navier-Stokes eq. Hatta & Kunihiro: Ann.Phys.298(2002)24 Kunihiro & Tsumura: J.Phys.A: Math.Gen.39(2006)8089

Purpose of this work: Unified understanding of the frame dependence

Derive the fluid dynamics by performing the dynamical reductionof the relativistic Boltzmann equation

By means of the Renormalization Group method as a reduction theory

Chen, Goldenfeld & Oono: PRL72(1995)376, PRE54(1996)376Kunihiro: PTP94(1995)503, 95(1997)179Ei, Fujii & Kunihiro: Ann Phys.280(2000)236

We will obtain a unified scheme such that the Eckart and Landau framesare included as special cases.

Fluid dynamics as long-wavelength (or slow) limit of the relativisticBoltzmann equation

2. Review of Renormalization Group method

2.1 　 General argument of dynamical reduction

RG method is a framework which can perform the dynamical reduction

(Kuramoto: 1989)

Invariant manifold

Evolution Eq. n-dim vector

m-dim vector

Reduced Eq.

2.2 RG eq. as an Envelope eq. 　 (Kunihiro: PTP94(1995)503)

RG eq can be used to solve a differential equation (Chen et al (1995))

Local solutions (a family of curves) ：))(,;( 00 tCttxx

0))(,;(d

d

0

000

tt

tCttxt

： RG eq.

)(E tx

)(tCDifferential eq. for Reduced dynamical eq.

Envelope ： ))(,;()( 00E ttCtttxtx

Suppose we have only locally valid solution to the differential eq (by some reason)Suppose we have only locally valid solution to the differential eq (by some reason)

Globally valid solution can be obtained by smoothening the local solutions.

　　　　　　 Construction of envelope

Global solution

2.3 Simple example --- Damped Oscillator ---

Damping slowly 　　　　　 Emergence of slow mode 　　　　　 Extraction of Slow dynamics

Perturbative analysis

Approximate solution

： Integral constants

Appearance of secular terms due to the existence of Slow mode

Local solution valid only near

Substitution into Initial valueSubstitution into Initial value

RG (Envelope) eq:

Equation of motion describing the Slow dynamics (Reduction of dynamics)

Envelope (Global solution):

Exact solution:

Well reproduced!

Resummation is performed

Relativistic Boltzmann eq.

Collision termCollision term

Arrangement to the expression convenient for RG method

3. Derivation of Relativistic Hydrodynamic eqTsumura, Kunihiro & K.O.: in preparation

Relativistic Boltzmann eq.Macro Flow vector :

Coordinate changesCoordinate changes

will be specified later

“time” derivative“time” derivative “spatial” derivative“spatial” derivative

perturbation termperturbation term

Order-by-order analysis

0th0th

0th Invariant manifold ：

Static solutionStatic solution

Five Integral consts. ：m = 5m = 5

Juettner distributioncf. Maxwell distribution (N.R.)

1st1st

Order-by-order analysis

Evolution Op.: Inhomogeneousterm:

Spectroscopy of the modified evolution op.

Collision operatorCollision operator

1.1.

Inner productInner product

Self-adjointSelf-adjoint

2.2. Non-positiveNon-positive

3.3.

has 5 zero modes, and other eigenvalues are negative

Order-by-order analysisProjection Op.Projection Op.

metricmetric

Eq. of 1st order :

Fast motionFast motion1st Initial value1st Initial value

1st Invariant manifold ：

5 zero modes ：

2nd2nd

Order-by-order analysis

Fast motionFast motion2nd Initial value2nd Initial value

2nd Invariant manifold ：

Inhomogeneousterm:

Collecting 0th, 1st and 2nd terms, we have;Collecting 0th, 1st and 2nd terms, we have;

RG (Envelope) equation

Expression of Invariant manifoldExpression of Invariant manifold

Approximate solution (Local solution)Approximate solution (Local solution)

RG equation ：

Coarse-Graining Conditions

1.1.

2.2. Choice of : e.g.new

RG equation ：

under

Equation for the Integral consts: , ,Does it reproduce the fluid dynamics of Eckart or Landau frames

by choosing the macro flow vector ?

RG (Envelope) equation

Dissipative Relativistic Hydrodynamic eq.

Landau frameLandau frame

Reproduce perfectly the Landau frame !

Eckart-like frameEckart-like frame

Eckart equation up to the volumeViscosity term

Dissipative Relativistic Hydrodynamic eq.

Stewart frameStewart frame

4. 　 SummaryCovariant dissipative hydrodynamic equationas a reduction theory of Boltzmann equation.

Macro Flow vector plays a role which generateshydrodynamic equations of various frames.

Successful for reproduction of Landau theory.

Stewart theory rather than Eckart for the framewithout particle flow dissipation.

Extension to Mixture (multi-component system)for Landau frame (in preparation)

Israel & Stewart’s regularization can be also derivedin this scheme by the extension of P-space. (Tsumura and Kunihiro: in preparation)

1.1.

2.2.

3.3.

4.4.