Post on 14-Jan-2016
transcript
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.
Next page
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.