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

1. 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

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

3. 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))

4. Relativistic Hydrodynamic eq. with Dissipation Not yet established Hydro dynamical frame: choice of frame or flow Eckart frame(1940) Landau frame(1959) vs. Next page Occurrence of instability due to lack of causality Israel & Stewart’s regularization (1979) by introducing Relaxation time

5. 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

6. 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)8089 We will obtain a unified scheme such that the Eckart and Landau frames are included as specialcases.

7. 2. Review of Renormalization Group method 2.1General 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

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)) 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

9. 2.3 Simple example --- Damped Oscillator --- Damping slowlyEmergence of slow modeExtraction 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. 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

11. 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

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

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

14. Order-by-order analysis 1st Evolution Op.: Inhomogeneous term: Collision operator Spectroscopy of the modified evolution op. Inner product Self-adjoint 1. Non-positive 2. 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. Order-by-order analysis 2nd Inhomogeneous term: Fast motion 2nd Initial value 2nd Invariant manifold：

17. 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.

18. 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 ?

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

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