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hydrodynamics from the relativistic Boltzmann equation (RBE) as a dynamical system, with some corrections and new unpublished results.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.
Highlights
The dynamical evolution of the hot and/or dense QCD matter produced in the Relativistic Heavy Ion Collider (RHIC) at the Brookhaven National Laboratory can be well described by relativistic hydrodynamic simulations [1, 2]
We note that such an approach is important for a systematic analysis of RHIC/Large Hadron Collider (LHC) data, because the proper dynamics for the description may change from hydrodynamics to kinetic one and vice versa, as mentioned above
We have reported our attempts to derive first-order and second-order relativistic hydrodynamic equations from relativistic Boltzmann equation which has a manifest Lorentz invariance and does not show any pathological behavior such as the instability and acausality seen in existing hydrodynamic equations
Summary
The dynamical evolution of the hot and/or dense QCD matter produced in the Relativistic Heavy Ion Collider (RHIC) at the Brookhaven National Laboratory can be well described by relativistic hydrodynamic simulations [1, 2]. Taking the relativistic Boltzmann equation (RBE) [21, 22] as a typical kinetic equation, we have been exploring the basic problems with the relativistic hydrodynamics [25,26,27] We note that such an approach is important for a systematic analysis of RHIC/LHC data, because the proper dynamics for the description may change from hydrodynamics to kinetic one and vice versa, as mentioned above. As the system is further relaxed, the time evolution will be described in terms of the hydrodynamic quantities, i.e., the flow velocity, particle-number density, and local temperature In this sense, the hydrodynamics is the far-infrared asymptotic dynamics of the kinetic equation. The distribution function which is explicitly constructed in our theory provides a new ansatz for the functional form of the distribution function in the Grad theory [49]
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