繰り込み群法による

Derivation of Relativistic Dissipative Hydrodynamic equations
by means of the Renormalization Group method
K. Ohnishi (TIT)
K. Tsumura (Kyoto Univ.)
T. Kunihiro (YITP, Kyoto)
July 8, 2006 @ Riken
Outline
1.
Introduction
2.
Renormalization Group method
3.
Derivation of Hydrodynamic equation
4.
Summary
１．Introduction
「RHIC Serves the Perfect Liquid.」
QGP = Perfect Fluid
April 18, 2005
Relativistic Hydrodynamical simulation
without dissipation
cf) Asakawa, Bass and Muller, hep-ph/0603092
 Hadronic corona ・・・ dissipative hydrodynamic or kinetic description
 QGP phase is also dissipative for Initial Condition based
on Color Glass Condensate
T. Hirano et al: Phys. Lett. B636 (2006) 299
See also, Nonaka & Bass, nucl-th/0510038
Dissipative Relativistic Hydrodynamical analysis
Just Started
(Muronga & Rischke(2004), Heinz et al (2005) )
Relativistic Hydrodynamic eq. with Dissipation
Not yet established
Hydro dynamical frame: choice of frame or flow
Eckart frame(1940)
vs. Landau frame(1959)
Next page
Occurrence of instability due to lack of causality
Israel & Stewart’s regularization (1979)
by introducing Relaxation time
Ambiguity in Hydrodynamic eq.
Fluid dynamics = a system of balance equations
: Energy-momentum
: Number
If dissipative, there arises an ambiguity
no dissipation in the number flow
Eckart frame
T

~
 ~
 u u  ( p  X )  T (u X  u X )  2X 
 
N   nu
Landau frame


Describing the matter flow.
no dissipation in energy flow
T   u  u  ( p  X )  2X 
nT


N  nu  
X
p
 , ,  :transport coefficients
Describing the energy flow.
Purpose of this work: Unified understanding of the frame dependence
Derive the fluid dynamics by performing the dynamical reduction
of the relativistic Boltzmann equation
Fluid dynamics as long-wavelength (or slow) limit of the relativistic
Boltzmann equation
By means of the Renormalization Group method as a reduction theory
Chen, Goldenfeld & Oono: PRL72(1995)376, PRE54(1996)376
Kunihiro: PTP94(1995)503, 95(1997)179
Ei, Fujii & Kunihiro: Ann Phys.280(2000)236
cf. Non-relativistic
Boltzmann eq.
case
Navier-Stokes eq.
Hatta & Kunihiro: Ann.Phys.298(2002)24
Kunihiro & Tsumura: J.Phys.A: Math.Gen.39(2006)8089
We will obtain a unified scheme such that the Eckart and Landau frames
are included as special cases.
2. Review of Renormalization Group method
2.1 General argument of dynamical reduction (Kuramoto: 1989)
Evolution Eq.
n-dim vector
m-dim vector
Invariant manifold
Reduced Eq.
RG method is a framework which can perform the dynamical reduction
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))
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
xE (t )
Local solutions (a family of curves)：
x  x(t; t0 , C(t0 ))
d
x(t ; t0 , C (t0 ))
dt0
t
0
0 t
Differential eq. for C (t )
Envelope：
： RG eq.
Reduced dynamical eq.
xE (t )  x(t; t0  t , C(t0  t ))
Global solution
2.3 Simple example
Damping slowly
--- Damped Oscillator ---
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
RG (Envelope) eq:
Equation of motion describing the Slow dynamics (Reduction of dynamics)
Substitution into Initial value
Envelope (Global solution):
Exact solution:
Well reproduced!
 Resummation is performed
3. Derivation of Relativistic Hydrodynamic eq
Tsumura, Kunihiro & K.O.: in preparation
Relativistic Boltzmann eq.
Collision term
Arrangement to the expression convenient for RG method
Relativistic Boltzmann eq.
Macro Flow vector :
will be specified later
Coordinate changes
“time” derivative
“spatial” derivative
perturbation term
Order-by-order analysis
0th
Static solution
Juettner distribution
cf. Maxwell distribution (N.R.)
Five Integral consts.：
m=5
0th Invariant manifold：
Order-by-order analysis
1st
Evolution Op.:
Inhomogeneous
term:
Collision operator
Spectroscopy of the modified evolution op.
Inner product
1.
Self-adjoint
2.
Non-positive
3.
has 5 zero modes, and other eigenvalues are negative
Order-by-order analysis
Projection Op.
metric
Eq. of 1st order :
1st Initial value
1st Invariant manifold：
5 zero modes：
Fast motion
Order-by-order analysis
Inhomogeneous
term:
2nd
2nd Initial value
2nd Invariant manifold：
Fast motion
RG (Envelope) equation
Collecting 0th, 1st and 2nd terms, we have;
Expression of Invariant manifold
Approximate solution (Local solution)
RG equation：
Coarse-Graining Conditions
1.
new 2.
Choice of
:
e.g.
RG (Envelope) equation
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
?
Dissipative Relativistic Hydrodynamic eq.
Landau frame
Reproduce perfectly the Landau frame !
Dissipative Relativistic Hydrodynamic eq.
Eckart-like frame
Eckart equation up to the volume
Viscosity term
Stewart frame
4. Summary
1.
Covariant dissipative hydrodynamic equation
as a reduction theory of Boltzmann equation.
Macro Flow vector plays a role which generates
hydrodynamic equations of various frames.
2.
Successful for reproduction of Landau theory.
Stewart theory rather than Eckart for the frame
without particle flow dissipation.
3.
Extension to Mixture (multi-component system)
for Landau frame (in preparation)
4.
Israel & Stewart’s regularization can be also derived
in this scheme by the extension of P-space.
(Tsumura and Kunihiro: in preparation)

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