Relativistic third-order viscous corrections to the entropy four-current from kinetic theory

By employing a Chapman-Enskog like iterative solution of the Boltzmann equation in relaxation-time approximation, we derive a new expression for the entropy four-current up to third order in gradient expansion. We show that unlike second-order and third-order entropy four-current obtained using Grad's method, there is a nonvanishing entropy flux in the present third-order expression. We further quantify the effect of the higher-order entropy density in the case of boost-invariant one-dimensional longitudinal expansion of a system. We demonstrate that the results obtained using the third-order evolution equation for the shear stress tensor, derived by employing the method of Chapman-Enskog expansion, show better agreement with the exact solution of the Boltzmann equation as well as with the parton cascade bamps, as compared to those obtained using the third-order equations from the method of Grad's 14-moment approximation.

Figure 1

Time evolution of $s/s_0$ obtained using ideal hydrodynamics (black dotted lines), second-order Grad's approximation (maroon dashed-dotted lines), second-order Chapman-Enskog method (green dashed-dotted-dotted lines), third-order Grad's approximation (red dashed lines), and third-order Chapman-Enskog method (blue solid lines) for initial temperature $T_0=300$ MeV at initial time ${\tau }_0=0.25$ fm/$c$, various $\eta /s$, and for (a) isotropic initial pressure configuration ${\xi }_0=0$ and (b) anisotropic pressure configuration ${\xi }_0=10$.

Figure 2

Same as Fig. 1 except here we take $T_0=600$ MeV.

Figure 3

Temperature dependence of $s/s_0$ obtained using exact solution of the Boltzmann equation (black dotted lines), third-order evolution equations from Grad's approximation (red dashed lines), and third-order equations from the Chapman-Enskog method (blue solid lines) for various $\eta /s$ and for (a) initial temperature $T_0=300$ MeV and (b) $T_0=600$ MeV, at initial time ${\tau }_0=0.25$ fm/$c$ and isotropic initial pressure configuration ${\xi }_0=0$.

Figure 4

Temperature dependence of the shear relaxation time, ${\tau }_{\pi }$, obtained using exact solution of the Boltzmann equation (black dotted lines), third-order evolution equations from Grad's approximation (red dashed lines), and third-order equations from the Chapman-Enskog method (blue solid lines) for initial temperature $T_0=300$ MeV at initial time ${\tau }_0=0.25$ fm/$c$, various $\eta /s$, and for (a) isotropic initial pressure configuration ${\xi }_0=0$ and (b) anisotropic pressure configuration ${\xi }_0=10$.

Figure 5

Same as Fig. 4 except here we take $T_0=600$ MeV.

Figure 6

Time evolution of $P_L/P_T$ obtained using exact solution of the Boltzmann equation (black dotted lines), third-order evolution equations from Grad's approximation (red dashed lines), and third-order equations from the Chapman-Enskog method (blue solid lines) for initial temperature $T_0=300$ MeV at initial time ${\tau }_0=0.25$ fm/$c$, various $\eta /s$, and for (a) isotropic initial pressure configuration ${\xi }_0=0$ and (b) anisotropic pressure configuration ${\xi }_0=10$.

Figure 7

Same as Fig. 6 except here we take $T_0=600$ MeV.

Figure 8

Time evolution of $P_L/P_T$ obtained using bamps (black dots), third-order evolution equations from Grad's approximation (red dashed lines), and third-order equations from the Chapman-Enskog method (blue solid lines) for initial temperature $T_0=500$ MeV at initial time ${\tau }_0=0.4$ fm/$c$, isotropic initial pressure configuration ${\xi }_0=0$, and various $\eta /s$.