Bulk viscous evolution within anisotropic hydrodynamics

We derive a system of moment-based dynamical equations that describe the $1+1d$ space-time evolution of a cylindrically symmetric massive gas undergoing boost-invariant longitudinal expansion. Extending previous work, we introduce an explicit degree of freedom associated with the bulk pressure of the system. The resulting form generalizes the ellipsoidal one-particle distribution function appropriate for massless particles to massive particles. Using this generalized form, we obtain a system of partial differential equations that can be solved numerically. In order to assess the performance of this scheme, we compare the resulting anisotropic hydrodynamics solutions with the exact solution of the $0+1d$ Boltzmann equation in the relaxation-time approximation. We find that the inclusion of the bulk degree of freedom improves the agreement between anisotropic hydrodynamics and the exact solution for a massive gas.

Figure 1

Proper-time evolution of ${\mathcal{P}}_L/{\mathcal{P}}_T$. The three lines correspond to the exact solution of the Boltzmann equation [53] (solid black line), the full aHydro equations including the bulk degree of freedom [dashed (red) line], and the aHydro equations with the ellipsoidal bulk degree of freedom set to 0 [dot-dashed (blue) line]. For both panels we used $m=1$ GeV, ${\tau }_0=0.5$ fm/$c$, ${\tau }_{\mathrm{eq}}=0.5$ fm/$c$, and $T_0=600$ MeV. In the top panel we fixed the initial spheroidal anisotropy parameter ${\xi }_0=0$ and in the bottom panel we chose ${\xi }_0=100$.

Figure 2

Proper-time evolution of the bulk pressure. Parameters and descriptions are the same as in Fig. 1.

Figure 3

Proper-time evolution of ${\mathcal{P}}_L/{\mathcal{P}}_T$. Parameters and descriptions are the same as in Fig. 1 except here we take $m=300$ MeV.

Figure 4

Proper-time evolution of the bulk pressure. Parameters and descriptions are the same as in Fig. 3.

Figure 5

Bulk pressure and pressure anisotropy as a function of the proper time. All curves correspond to the aHydro evolution including the bulk degree of freedom. We took ${\xi }_0=0$, with all other initial conditions and parameters the same as in previous figures.