First-Principles 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 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.

Global flow structure and exact formal transseries of the Gubser flow in kinetic theory

“…Since a reduced description of hydrodynamics only needs parametrization of microscopic variables projected as macroscopic fields v, p, ε, a "lifting procedure" is required to acquire knowledge about a complete microscopic picture. This procedure is well-defined on a "slow" invariant manifold -or simply slow manifold, 5 where the hydrodynamic flow meets the kinetic motion of a fluid as the kinetics becomes well-behaved at a small Knudsen number and hydrodynamic field becomes bounded close to the local equilibrium [41][42][43]. If the slow manifold exists, a complete understanding of hydrodynamics can be obtained based on these macroscopic fields.…”

Relativistic causal hydrodynamics derived from Boltzmann equation: A novel reduction theoretical approach

“…because ∂ n K(s)/∂s n | s=0 does not vanish for any n; see Eq. (76). Admittedly the existence of such an infinite number of terms would be undesirable for the construction of the (closed) mesoscopic dynamics.…”

Landau and Eckart frames for relativistic fluids in nuclear collisions

“…There are two distinctive ways of defining the local rest frame for the flow: the Landau (or energy) frame [45] and the Eckart (or conserved charge/particle) frame [46]. There have been decades of debate over the eligibility of the two definitions of the local rest frame [20][21][22][23][24][47][48][49][50][51][52]. Most of the numerical analyses of hydrodynamic models for relativistic nuclear collisions so far do not give explicit consideration to the frame because the diffusion or the dissipation current is neglected, but the Landau frame is often considered to be a preferred choice when there is a theoretical need.…”