Perturbative approaches in relativistic kinetic theory and the emergence of first-order hydrodynamics

Hydrodynamics can be formulated in terms of a perturbative series in derivatives of the temperature, chemical potential, and flow velocity around an equilibrium state. Different formulations for this series have been proposed over the years, which consequently led to the development of various hydrodynamic theories. In this work, we discuss the relativistic generalizations of the perturbative expansions put forward by Chapman and Enskog, and Hilbert, using general matching conditions in kinetic theory. This allows us to describe, in a comprehensive way, how different out-of-equilibrium definitions for the hydrodynamic fields affect the development of the hydrodynamic perturbative series. We provide a perturbative method for systematically deriving the hydrodynamic formulation recently proposed by Bemfica, Disconzi, Noronha, and Kovtun (BDNK) from relativistic kinetic theory. The various transport coefficients that appear in BDNK (at first-order) are explicitly computed using a new formulation of the relaxation time approximation for the Boltzmann equation. Assuming Bjorken flow, we also determine the hydrodynamic attractors of BDNK theory and compare the overall hydrodynamic evolution obtained using this formulation with that generated by the Israel-Stewart equations of motion and also kinetic theory.

Figure 1

Comparison between various solutions of BDNK theory, Eq. (117), and the attractor solutions. (a) The black curves represent solution (119) with $a=1,2,4,6,8,10,14,18,22,26,30,36,42$, 48, 54 and are shown in comparison with the hydrodynamic attractor (120). (b) The black curves represent the solutions of (119) with $a=1,1/2,1/3,\dots ,1/10$ shown in comparison with the early time attractor (121).

Figure 2

Comparison of hydrodynamic attractor solutions for different matching conditions, with matching parameter $s=3$ (a) and $s=4$ (b).

Figure 3

Evolution of the nonequilibrium fraction of the energy density (right) and the normalized shear-stress tensor, $\pi /({ϵ}_0+P_0+\delta ϵ+\mathrm{\Pi })=3\pi /[4({ϵ}_0+\delta ϵ)]$ (left), for the Boltzmann equation (RTA), Israel-Stewart (IS) and BDNK for type I exotic Eckart and $s=3$ and equilibrium initial conditions.

Figure 4

Evolution of the nonequilibrium fraction of the energy density (left) and the normalized shear-stress tensor, $\pi /({ϵ}_0+P_0+\delta ϵ+\mathrm{\Pi })=3\pi /[4({ϵ}_0+\delta ϵ)]$ (right), for the Boltzmann equation (RTA), Israel-Stewart (IS), and BDNK for type I exotic Eckart and $s=4$ and equilibrium initial conditions.

Figure 5

Comparison between the time evolution (for various initial conditions) of the nonequilibrium correction for the energy density found by solving the RTA Boltzmann equation, Israel-Stewart (IS), and BDNK using type I exotic Eckart for $s=3$ (left) and $s=4$ (right). The initial conditions are so that $\delta ϵ({\widehat{\tau }}_0)/[({ϵ}_0+\delta ϵ)({\widehat{\tau }}_0)]=0$, 0.2, 0.4, and, when applicable, $\pi ({\widehat{\tau }}_0)=0$.

Figure 6

Evolution of the nonequilibrium fraction of the particle density (left) and the normalized shear-stress tensor, $\pi /({ϵ}_0+P_0)=3\pi /(4{ϵ}_0)$ (right), found by solving the RTA Boltzmann equation, Israel-Stewart (IS), and BDNK for type II exotic Eckart with $s=3$ and equilibrium initial conditions.

Figure 7

Evolution of the nonequilibrium fraction of the particle density (left) and the normalized shear-stress tensor, $\pi /({ϵ}_0+P_0)=3\pi /(4{ϵ}_0)$ (right), found by solving the RTA Boltzmann equation, Israel-Stewart (IS), and BDNK for type II exotic Eckart with $s=4$ and equilibrium initial conditions.

Figure 8

Comparison between time evolution of the nonequilibrium correction to the particle density found by solving the RTA Boltzmann equation, Israel-Stewart (IS), and BDNK for type II exotic Eckart with $s=3$ (left) and $s=4$ (right) and various initial conditions. The initial conditions are such that $\delta n({\widehat{\tau }}_0)/[(n+\delta n)({\widehat{\tau }}_0)]=0.0$, 0.2, 0.4 and $\pi ({\widehat{\tau }}_0)=0$, when applicable.