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

Abstract

We derive first-order relativistic dissipative hydrodynamic equations (RDHEs) from relativistic Boltzmann equation (RBE) on the basis of the renormalization-group (RG) method. We introduce a macroscopic-frame vector (MFV) to specify the local rest frame (LRF) on which the macroscopic dynamics is described. The five hydrodynamic modes are identified with the same number of the 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 orthogonal to the zero modes in terms of the inner product: Here, any ansatz's, such as the so-called conditions of fit used in the standard methods in an ad-hoc way, are not necessary. We elucidate that the Burnett term dose not affect the RDHEs owing to the very nature of the hydrodynamic modes as the zero modes. Applying the RG equation, we obtain the RDHE in a generic LRF specified by the MFV, as the coarse-grained and covariant equation. Our generic RDHE reduces to RDHEs in various 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 and Lifshitz, while the derived RDHE in the particle LRF is slightly different from that of Eckart, owing to the presence of the dissipative internal energy. We prove that the Eckart equation can not be compatible with the underlying RBE. The proof is made on the basis of the observation that the orthogonality condition to the zero modes coincides with the ansatz's posed on the dissipative parts of the energy-momentum tensor and the particle current in the phenomenological RDHEs. We also present an analytic proof that all of our RDHEs have a stable equilibrium state owing to the positive definiteness of the inner product.