K. Ohnishi (TIT)

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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 _at_ Riken
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Outline
Introduction Renormalization Group
method Derivation of Hydrodynamic
equation Summary
1.
2.
3.
4.
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1.Introduction
?RHIC Serves the Perfect Liquid.?
April 18, 2005
Relativistic Hydrodynamical simulation without
dissipation
QGP Perfect Fluid
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) )
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Relativistic Hydrodynamic eq. with Dissipation
Not yet established
Hydro dynamical frame choice of frame or flow
Landau frame(1959)
Eckart frame(1940)
vs.
Next page
Occurrence of instability due to lack of causality
Israel Stewarts regularization (1979) by
introducing Relaxation time
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Ambiguity in Hydrodynamic eq.
Fluid dynamics a system of balance equations
Energy-momentum
Number
If dissipative, there arises an ambiguity
Eckart frame
no dissipation in the number flow
Describing the matter flow.
Landau frame
no dissipation in energy flow
Describing the energy flow.
transport coefficients
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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 case Boltzmann eq.
Navier-Stokes eq. Hatta
Kunihiro Ann.Phys.298(2002)24
Kunihiro Tsumura J.Phys.A Math.Gen.39(2006)808
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We will obtain a unified scheme such that the
Eckart and Landau frames are included as special
cases.
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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
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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
Local solutions (a family of curves)
RG eq.
Differential eq. for
Reduced dynamical eq.
Envelope
Global solution
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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
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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

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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
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Relativistic Boltzmann eq.
Macro Flow vector
will be specified later
Coordinate changes
spatial derivative
time derivative
perturbation term
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Order-by-order analysis
0th
Static solution
Juettner distribution cf. Maxwell distribution
(N.R.)
Five Integral consts.
m 5
0th Invariant manifold
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Order-by-order analysis
1st
Evolution Op.
Inhomogeneous term
Collision operator
Spectroscopy of the modified evolution op.
Inner product
Self-adjoint
1.
2.
Non-positive
3.
has 5 zero modes, and other eigenvalues are
negative
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Order-by-order analysis
Projection Op.
metric
Eq. of 1st order
Fast motion
1st Initial value
1st Invariant manifold
5 zero modes
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Order-by-order analysis
2nd
Inhomogeneous term
Fast motion
2nd Initial value
2nd Invariant manifold
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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
Choice of e.g.
2.
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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
?
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Dissipative Relativistic Hydrodynamic eq.

Landau frame
Reproduce perfectly the Landau frame !
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Dissipative Relativistic Hydrodynamic eq.
Eckart-like frame
Eckart equation up to the volume Viscosity term
Stewart frame
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4. Summary
Covariant dissipative hydrodynamic equation as a
reduction theory of Boltzmann equation. Macro
Flow vector plays a role which generates hydrodyna
mic equations of various frames. Successful for
reproduction of Landau theory. Stewart theory
rather than Eckart for the frame without particle
flow dissipation. Extension to Mixture
(multi-component system) for Landau frame (in
preparation) Israel Stewarts regularization
can be also derived in this scheme by the
extension of P-space. (Tsumura and Kunihiro in
preparation)
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