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Prog. Theor. Phys. Vol. 126 No. 5 (2011) pp. 761-809

[ Full Text PDF : FREE ACCESS (357K) ]

Invited Papers

First-Principles Derivation of Stable First-Order Generic-Frame Relativistic Dissipative Hydrodynamic Equations from Kinetic Theory by Renormalization-Group Method

Kyosuke Tsumura1,* and Teiji Kunihiro2,**

1Analysis Technology Center, Fujifilm Corporation, Minami-Ashigara 250-0193, Japan
2Department of Physics, Kyoto University, Kyoto 606-8502, Japan

(Received August 9, 2011)

Abstract:

We derive first-order relativistic dissipative hydrodynamic equations from the relativistic Boltzmann equation by the renormalization-group (RG) method. We introduce a macroscopic-frame vector, which does not necessarily coincide with the flow velocity, to specify the local rest frame on which the macroscopic dynamics is described. The five hydrodynamic modes are naturally identified with the same number of zero modes of the linearized collision operator, i.e., the collision invariants. After defining the inner product in the function space spanned by the distribution function, the higher-order terms, which give rise to the dissipative effects, are constructed so that they are precisely orthogonal to the zero modes in terms of the inner product: Here, no ansatzs, such as the so-called conditions of fit used in the standard methods in an ad hoc way, are necessary. We elucidate that the Burnett term does not affect the hydrodynamic equations owing to the very nature of the hydrodynamic modes as the zero modes. Then, applying the RG equation, we obtain the hydrodynamic equation in a generic frame specified by the macroscopic-frame vector, as the coarse-grained and covariant equation. Our generic hydrodynamic equation reduces to hydrodynamic equations in various local rest frames, including the energy and particle frames with a choice of the macroscopic-frame vector. We find that our equation in the energy frame coincides with that of Landau and Lifshitz, while the derived equation in the particle frame is slightly different from that of Eckart, owing to the presence of the dissipative internal energy. We prove that the Eckart equation is not compatible with the underlying relativistic Boltzmann equation. The proof is made on the basis of the observation that the orthogonality condition to the zero modes coincides with the ansatzs posed on the dissipative parts of the energy-momentum tensor and the particle current in the phenomenological equations. We also present an analytic proof that all our equations ensure the stability of steady states including the thermal equilibrium state owing to the positive definiteness of the inner product.

Subject Index : 250
URL : http://ptp.ipap.jp/link?PTP/126/761/
DOI : 10.1143/PTP.126.761


*E-mail: kyousuke_tsumura@fujifilm.co.jp
**E-mail: kunihiro@ruby.scphys.kyoto-u.ac.jp

[ Full Text PDF : FREE ACCESS (357K) ] Citation:

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Citing Article(s) :

  1. Progress of Theoretical Physics Supplement No.195 (2012) pp. 19-28 :
    New Forms of Non-Relativistic and Relativistic Hydrodynamic Equations as Derived by the Renormalization-Group Method
    Kyosuke Tsumura and Teiji Kunihiro

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