Shear-bulk coupling in nonconformal hydrodynamics

We compute the temporal evolution of the pressure anisotropy and bulk pressure of a massive gas using second-order viscous hydrodynamics and anisotropic hydrodynamics. We then compare our results with an exact solution of the Boltzmann equation for a massive gas in the relaxation time approximation. We demonstrate that, within second-order viscous hydrodynamics, the inclusion of the full set of kinetic coefficients, particularly the shear-bulk couplings, is necessary to properly describe the time evolution of the bulk pressure. We also compare the results of second-order hydrodynamics with those obtained using the anisotropic hydrodynamics approach. We find that anisotropic hydrodynamics and second-order viscous hydrodynamics including the shear-bulk couplings are both able to reproduce the exact evolution with comparable accuracy.

Figure 1

Time evolution of the pressure anisotropy ${\mathcal{P}}_L/{\mathcal{P}}_T$ (a) and the bulk pressure (b). Three lines describe three different results: the exact solution of the Boltzmann equation [52] (black solid line), the result of the full second-order viscous hydrodynamics [32] including the shear-bulk couplings ${\lambda }_{\Pi \pi }$ and ${\lambda }_{\pi \Pi }$ (red dashed line), and the result of the second-order hydrodynamics with ${\lambda }_{\Pi \pi }={\lambda }_{\pi \Pi }=0$ (blue dot-dashed line). For both panels we use $m=30$ MeV, ${\tau }_0=0.5$ fm/$c$, ${\tau }_{\mathrm{eq}}={\tau }_{\pi }={\tau }_{\Pi }=0.5$ fm/$c$, and $T_0=600$ MeV. The initial spheroidal anisotropy parameter fixing the initial distribution function equals ${\xi }_0=0$, correspondingly, we use ${\pi }_0=0$ and ${\Pi }_0$ = 0.

Figure 2

Same as Fig. 1 except here we take $m=300\mathrm{MeV}$ [(a) and (b)] and $m=1\mathrm{GeV}$ [(c) and (d)].

Figure 3

Same as Fig. 1 except here we take ${\xi }_0=100$.

Figure 4

Same as Fig. 2 except here we take ${\xi }_0=100$.

Figure 5

Proper-time evolution of pressure anisotropy ${\mathcal{P}}_L/{\mathcal{P}}_T$ (a) and bulk pressure (b). The three lines correspond to the exact solution of the Boltzmann equation [52] (black solid line), the full second-order viscous equations including the shear-bulk couplings ${\lambda }_{\Pi \pi }$ and ${\lambda }_{\pi \Pi }$ [32] (blue dot-dashed line), and anisotropic hydrodynamics [53] (red dashed line). For both figures we used $m=30\mathrm{MeV}$, ${\tau }_0$ = 0.5 fm/$c$, ${\tau }_{\mathrm{eq}}={\tau }_{\pi }={\tau }_{\Pi }$ = 0.5 fm/$c$, and $T_0$ = 600 MeV. The initial spheroidal anisotropy parameter for initial distribution function of the exact solution of Boltzmann equation is taken to be ${\xi }_0=0$; in consequence ${\pi }_0=0$ and ${\Pi }_0$ = 0.

Figure 6

Proper-time evolution of ${\mathcal{P}}_L/{\mathcal{P}}_T$ [(a) and (c)] and bulk pressure [(b) and (d)]. Parameters and descriptions are the same as in Fig. 5 except here we take $m=300\mathrm{MeV}$ [(a) and (b)] and $m=1\mathrm{GeV}$ [(c) and (d)].

Figure 7

Proper-time evolution of ${\mathcal{P}}_L/{\mathcal{P}}_T$ (a) and bulk pressure (b). Parameters and descriptions are the same as in Fig. 5 except here we take ${\xi }_0=100$.

Figure 8

Proper-time evolution of ${\mathcal{P}}_L/{\mathcal{P}}_T$ [(a) and (c)] and bulk pressure [(b) and (d)]. Parameters and descriptions are the same as in Fig. 6 except here we take ${\xi }_0=100$.