1. K. Ohnishi (TIT) K. Tsumura (Kyoto Univ.) T. Kunihiro (YITP, Kyoto) Derivation of Relativistic Dissipative Hydrodynamic equations by means of the Renormalization Group method July 8, 2006 @ Riken

2. Introduction Renormalization Group method Derivation of Hydrodynamic equation Summary 1. Outline 4. 2. 3.

3. １． Introduction QGP = Perfect Fluid Relativistic Hydrodynamical simulation without 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 cf) Asakawa, Bass and Muller, hep-ph/0603092 T. Hirano et al: Phys. Lett. B636 (2006) 299 See also, Nonaka & Bass, nucl-th/0510038

4. 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) Eckart frame(1940) vs. Next page

5. Ambiguity in Hydrodynamic eq. Fluid dynamics = a system of balance equations If dissipative, there arises an ambiguity Eckart frame Landau frame no dissipation in the number flow no dissipation in energy flow Describing the matter flow. Describing the energy flow. : Energy-momentum : Number :transport coefficients

6. 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 reduction 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 We will obtain a unified scheme such that the Eckart and Landau frames are included as special cases. Fluid dynamics as long-wavelength (or slow) limit of the relativistic Boltzmann equation

7. 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.

8. 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) ： ： RG eq. Differential eq. for Reduced dynamical eq. Envelope ： 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

9. 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

10. Substitution 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

11. Relativistic Boltzmann eq. Collision term Arrangement to the expression convenient for RG method 3. Derivation of Relativistic Hydrodynamic eq Tsumura, Kunihiro & K.O.: in preparation

12. Relativistic Boltzmann eq. Macro Flow vector : Coordinate changes will be specified later “time” derivative “spatial” derivative perturbation term

13. Order-by-order analysis 0th 0th Invariant manifold ： Static solution Five Integral consts. ： m = 5 Juettner distribution cf. Maxwell distribution (N.R.)

14. 1st Order-by-order analysis Evolution Op.: Inhomogeneous term: Spectroscopy of the modified evolution op. Collision operator 1. Inner product Self-adjoint 2. Non-positive 3. has 5 zero modes, and other eigenvalues are negative

15. Order-by-order analysis Projection Op. metric Eq. of 1st order : Fast motion 1st Initial value 1st Invariant manifold ： 5 zero modes ：

16. 2nd Order-by-order analysis Fast motion 2nd Initial value 2nd Invariant manifold ： Inhomogeneous term:

17. Collecting 0th, 1st and 2nd terms, we have; RG (Envelope) equation Expression of Invariant manifold Approximate solution (Local solution) RG equation ： Coarse-Graining Conditions 1. 2. Choice of : e.g. new

18. 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

19. Dissipative Relativistic Hydrodynamic eq. Landau frame Reproduce perfectly the Landau frame !

20. Eckart-like frame Eckart equation up to the volume Viscosity term Dissipative Relativistic Hydrodynamic eq. Stewart frame

21. 4. Summary Covariant dissipative hydrodynamic equation as a reduction theory of Boltzmann equation. Macro Flow vector plays a role which generates hydrodynamic 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 & Stewart’s regularization can be also derived in this scheme by the extension of P-space. (Tsumura and Kunihiro: in preparation) 1. 2. 3. 4.