数理解析研究所講究録
第 1576 巻 2008 年 82-83

82

Construction of Invariant Manifold byRenormalization-group Method: reduction ofdynamical systems and its applications
Teiji Kunihiro
Yukawa Institute for Theoretical Physics, Kyoto University,
Sakyo-ku, Kyoto 606-8502, Japan
We have first given a comprehensive review of the renormalization group(RG) method for
global and asymp-totic analysis on the basis of the following articles[l, 2, 3, 4, 5, 6]: Anemphasis is put on the relevance to the classical theory of envelopes and the existencethat an oesentialof invariant manifolds of the dynamics under consideration. We
point of the method is to convert the problem from solving differential equations to obtaining suitable initial (or boundary) conditions: The RG equation determines the slow
motion of the would-be integral constants in the unperturbative solution on the invariant$cla\dot{n}6^{r}$manifold.
The RG method is applied to derive the relativistic Navier-Stokes equation from the Boltzmann equation $[7, 8]$ , as an example of the reduction of dynamics; the non-relativistc case
was already treated successfully in $[1, 6]$ . It tums out that the derived equation in the particle frame is a stable dissipative relativistic hydrodynamic equation [8]. We indicate thatafterthe usual constraint on the dissipative part of the energy-momentum tensor
Eckart in the particle frame is not compatible with the underlying relativistic Boltzmann
equation. We demonstrate that the solution around the thermal equilibrium state obtained in the new equations in the particle frame is stable, performing the linear stability
analysis with the use of the equation of state and the transport coefficients for a rarefied
gas[8]. It is worth emphasizing that the establishment of a stable relativistic hydrodynamic equation in the particle frame is significant since the so called causal equations suchas Israel-Stwert are usually constructed in the particle frame with the constraint of Eckart.
$\delta T^{\mu\nu}$This work supported by a $Grant-in$-Aid for Scientific Research by Monbu-Kagakusyoof Japan (No. 17540250) and for the 21st Century COE “Center for Diversity and Uni$is$versality in Physics” of Kyoto University and by the Yukawa Intemational Program forQuark-hadron Sciences.

References[1] T. Kunihiro and K. Tsumura,“Application of the renormalization-group method to the reduction of transport equations”
J.Phys.A39:8089-8104,2006; hep-th/0512108.
[2] S.-I. Ei, K. IFUjii and T. Kunihiro,“Renormalization Group Method for Reduction of Evolution Equations: Invariant

83

Manifold and Envelopes”
Ann. Phys. 280 (2000), 236.
[3] T. Kunihiro,“A Geometrical Formulation of the Renormalization Group Method for Global Analysis”
Prog.Theor.Phys. 94, 503(1995); 95, 835(1996)(E).

[4] T. Kunihiro,“A Geometrical Formulation of the Renormalization Group Method for Global Analysis II: Partial Di.erential Equations”
Jpn. J. $Ind$ . Appl. Math. 14, 51 (1997).
[5] T. Kunihiro,“The Renormalization-Group Method Applied to Asymptotic Analysis of Vector
Fields”
Prog. Theor. Phys. 97, 179 (1997).
[6] Y. Hatta and T. Kunihiro,“Renormalization Group Method Applied to Kinetic Equations: roles of initial values
and time”
Ann. Phys. 298 (2002) 24.

[7] K. Tsumura, T. Kunihiro and K. Ohnishi,“Derivation of covariant dissipative fluid dynamics in the renormalization-groupmethod”
Phys. Lett. $B646,134- 140(2007);hep- ph/0609056$ .
[8] K. Tsumura and T. Kunihiro,“Stable First-order $Particle\cdot hame$ Relativistic Hydrodynamics for Dissipative System ,$arXiv:0709.3645$ [nucl-th].
$s$