Derivation of relativistic hydrodynamic equations consistent with relativistic Boltzmann equation by renormalization-group method

Abstract

Abstract. We review our work on the application of the renormalization-group method to obtain first-and second-order relativistic hydrodynamics from the relativistic Boltzmann equation (RBE) as a dynamical system, with some corrections and new unpublished results. For the first-order equation, we explicitly obtain the distribution function in the asymptotic regime as the invariant manifold of the dynamical system, which turns out to be nothing but the matching condition defining the energy frame, i.e., the Landau-Lifshitz one. It is argued that the frame on which the flow of the relativistic hydrodynamic equation is defined must be the energy frame, if the dynamics should be consistent with the underlying RBE. A sketch is also given for derivation of the second-order hydrodynamic equation, i.e., extended thermodynamics, which is accomplished by extending the invariant manifold so that it is spanned by excited modes as well as the zero modes (hydrodynamic modes) of the linearized collision operator. On the basis of thus constructed resummed distribution function, we propose a novel ansatz for the functional form to be used in Grad moment method; it is shown that our theory gives the same expressions for the transport coefficients as those given in the Chapman-Enskog theory as well as the novel expressions for the relaxation times and lengths allowing natural interpretation.

Derivation of second-order relativistic hydrodynamics for reactive multicomponent systems

“…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].…”

“…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.…”

“…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. Their numerical values are indeed different from those by the moment method [32].…”

Analytical and numerical Gubser solutions of the second-order hydrodynamics

“…On general grounds, one expects that the relativistic Navier-Stokes equation is pathological, and indeed this solution [19,20] shows unphysical behaviors such as a negative temperature at early time. Though attempts have been made to cure this problem by solving, semi-analytically and numerically, the IsraelStewart equation [21] and the microscopic Boltzmann equation in the relaxation time approximation [22][23][24], it is important to search for more complete and consistent second-order relativistic hydrodynamic equations [25][26][27][28][29][30][31][32] and their solutions.…”

Mesoscopic dynamics of fermionic cold atoms — Quantitative analysis of transport coefficients and relaxation times

“…The aim of this article is to give basic analyses of nonequilibrium properties of cold atomic gasses in a quantitative way based on the kinetic theory [28,29,30,31]; a focus is put on a quantitative and comprehensive analysis of the viscous relaxation times based on the novel microscopic formulas of them [28,29,30,31]. Such an analysis is of fundamental importance in this field because the quantitative extraction of the relaxation times is indispensable to elucidate the dynamics of cold atomic gases in a precise way.…”