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We derive first-order relativistic dissipative hydrodynamic equations (RDHEs)from relativistic Boltzmann equation (RBE) on the basis of therenormalization-group (RG) method. We introduce a macroscopic-frame vector(MFV) to specify the local rest frame (LRF) on which the macroscopic dynamicsis described. The five hydrodynamic modes are identified with the same numberof the zero modes of the linearized collision operator, i.e., the collisioninvariants. After defining the inner product in the function space spanned bythe distribution function, the higher-order terms, which give rise to thedissipative effects, are constructed so that they are orthogonal to the zeromodes in terms of the inner product: Here, any ansatz's, such as the so-calledconditions of fit used in the standard methods in an ad-hoc way, are notnecessary. We elucidate that the Burnett term dose not affect the RDHEs owingto the very nature of the hydrodynamic modes as the zero modes. Applying the RGequation, we obtain the RDHE in a generic LRF specified by the MFV, as thecoarse-grained and covariant equation. Our generic RDHE reduces to RDHEs invarious LRFs, including the energy and particle LRFs with a choice of the MFV.We find that our RDHE in the energy LRF coincides with that of Landau andLifshitz, while the derived RDHE in the particle LRF is slightly different fromthat of Eckart, owing to the presence of the dissipative internal energy. Weprove that the Eckart equation can not be compatible with the underlying RBE.The proof is made on the basis of the observation that the orthogonalitycondition to the zero modes coincides with the ansatz's posed on thedissipative parts of the energy-momentum tensor and the particle current in thephenomenological RDHEs. We also present an analytic proof that all of our RDHEshave a stable equilibrium state owing to the positive definiteness of the innerproduct.

progress of theoretical physics

First-principle Derivation of Stable First-order Generic-frame  Relativistic Dissipative Hydrodynamic Equations from Kinetic Theory by  Renormalization-group Method