Higher order and anisotropic hydrodynamics for Bjorken and Gubser flows

We study the evolution of hydrodynamic and nonhydrodynamic moments of the distribution function using anisotropic and third-order Chapman-Enskog hydrodynamics for systems undergoing Bjorken and Gubser flows. The hydrodynamic results are compared with the exact solution of the Boltzmann equation with a collision term in relaxation time approximation. While the evolution of the hydrodynamic moments of the distribution function (i.e., of the energy momentum tensor) can be described with high accuracy by both hydrodynamic approximation schemes, their description of the evolution of the entropy of the system is much less precise. We attribute this to large contributions from nonhydrodynamic modes coupling into the entropy evolution, which are not well captured by the hydrodynamic approximations. The differences between the exact solution and the hydrodynamic approximations are larger for the third-order Chapman-Enskog hydrodynamics than for anisotropic hydrodynamics, which effectively resums some of the dissipative effects from anisotropic expansion to all orders in the anisotropy, and are larger for Gubser flow than for Bjorken flow. Overall, anisotropic hydrodynamics provides the most precise macroscopic description for these highly anisotropically expanding systems.

Figure 1

Proper time evolution of (a) the normalized shear stress $\pi /(\epsilon +P)$, (b) the pressure anisotropy $P_L/P_T$, (c) the entropy density per unit rapidity and transverse area $s\tau$, and (d) the normalized entropy density $s/s_{\mathrm{eq}}$, for Chapman-Enskog third-order hydrodynamics (dotted red lines), anisotropic hydrodynamics (dashed blue lines), and the exact solution of the RTA Boltzmann equation (solid green lines). For each theory, three sets of curves are shown, corresponding to three different values for the specific shear viscosity, $4\pi \eta /s=1,3,$ and 10. All curves assume an initial equilibrium state (i.e., ${\pi }_0=0$) with temperature $T_0=300\mathrm{MeV}$ at ${\tau }_0=0.25\mathrm{fm}/c$.

Figure 2

The same quantities as shown in Fig. 1, but now plotted as a function of the scaling variable $\tilde{w}\equiv \tau T/(4\pi \eta /s)$. See text for discussion.

Figure 3

de Sitter time evolution of the (normalized) temperature $\widehat{T}$ for Gubser flow, in absolute terms (a) and relative to the temperature corresponding to the energy density associated with the exact solution of the RTA Boltzmann equation (b). The results for anisotropic hydrodynamics (blue) and the third-order Chapman-Enskog approach (red) are compared with the exact solution (green). Panel (a) shows results for $\eta /s=10/\left(4\pi \right)$ only whereas in panel (b) results are compared for three different values of the specific shear viscosity, $4\pi \eta /s=1$, 3, and 10.

Figure 4

de Sitter time evolution of (a) the normalized shear stress $\widehat{\pi }/(\widehat{\epsilon }+\widehat{P})$, (b) the pressure anisotropy ${\widehat{P}}_L/{\widehat{P}}_T$, (c) the entropy content $\widehat{s}{cosh}^2\left(\rho \right)$, and (d) the normalized entropy density $\widehat{s}/{\widehat{s}}_{\mathrm{eq}}$, for Chapman-Enskog third-order hydrodynamics (red), anisotropic hydrodynamics (blue), and the exact solution of the RTA Boltzmann equation (green). For each theory, three sets of curves are shown, corresponding to three different values for the specific shear viscosity, $4\pi \eta /s=1$ (solid), 3 (dashed), and 10 (dash-dotted). Thermal equilibrium initial conditions ($\widehat{\pi }=0$) with initial temperature ${\widehat{T}}_0=0.002$ were implemented at ${\rho }_0=-10$.

Figure 5

The same quantities as shown in Fig. 4, but now plotted as a function of the scaling variable $\tilde{w}\equiv (4\pi \eta /s)(2tanh\rho )/\widehat{T}$. See text for discussion.