Uniqueness of Landau-Lifshitz energy frame in relativistic dissipative hydrodynamics

Abstract

We show that the relativistic dissipative hydrodynamic equation derived from the relativistic Boltzmann equation by the renormalization-group method uniquely leads to the one in the energy frame proposed by Landau and Lifshitz, provided that the macroscopic-frame vector, which defines the local rest frame of the fluid velocity, is independent of the momenta of constituent particles, as it should. We argue that the relativistic hydrodynamic equations for viscous fluids must be defined on the energy frame if it is consistent with the underlying relativistic kinetic equation.

Derivation of second-order relativistic hydrodynamics for reactive multicomponent systems

“…The present choice of the coordinate system leads to the relativistic hydrodynamics in the Landau-Lifshitz frame or the energy frame [56]. From now on, we suppress the variable σ.…”

“…It is also understood that, in the context of the reduction theory of dynamical systems [47,48], the hydrodynamical variables constitute the natural coordinates of the invariant/attractive manifold in the functional space spanned by the distribution function. Then the renormalization group equation [33,34] or envelope equation [35,36] gives the evolution equation of the hydrodynamical variables, i.e., the hydrodynamic equation, after an averaging of the distribution function [29,31,32,43,45,46]. It has been shown that the resultant second-order relativistic hydrodynamic equation is causal and stable [29][30][31][32], the microscopic expressions for the transport coefficients coincide with those obtained by Chapman-Enskog method, and those of the relaxation times are different from any other previous ones but allow physically natural interpretations.…”

“…Recently, to eliminate the ambiguity in the derivation of hydrodynamics and perform systematic formulation, the renormalization group (RG) method [33][34][35][36][37][38][39][40][41][42][43][44][45] has been applied to derive the second-order hydrodynamic equation for the singlecomponent system both in the non-relativistic and relativistic cases [29][30][31][32] as an extension of the first-order case [46]: In this method, the Boltzmann equation is faithfully solved to obtain the distribution function around the local equilibrium state without imposing any ansatz in a perturbation theory and the secular terms are resummed away into the would-be integral constants, which constitute the slow variables, i.e., the hydrodynamical variables. It is also understood that, in the context of the reduction theory of dynamical systems [47,48], the hydrodynamical variables constitute the natural coordinates of the invariant/attractive manifold in the functional space spanned by the distribution function.…”

“…In the latter part, more microscopic dynamics should be incorporated to the hydrodynamics since the viscous effect is too large to apply the naive viscous hydrodynamics [6]. The second-order hydrodynamics actually include the mesoscopic dynamics and useful to analyze the hydrodynamic behavior of the spatially inhomogeneous matter [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32].…”

Derivation of Hydrodynamics from the Hamiltonian Description of Particle Systems

“…The simple method presented in this Letter is expected to be useful for the derivation of hydrodynamic equations in other systems such as relativistic systems [28,29], an-isotropic molecular systems [30], visco-elastic systems [31][32][33], dissipative particles [34], and active matter [35]. The formulation may also be developed so as to describe more complicated behavior near boundaries.…”