Closing the equations of motion of anisotropic fluid dynamics by a judicious choice of a moment of the Boltzmann equation

In Molnár et al. Phys. Rev. D 93, 114025 (2016) the equations of anisotropic dissipative fluid dynamics were obtained from the moments of the Boltzmann equation based on an expansion around an arbitrary anisotropic single-particle distribution function. In this paper we make a particular choice for this distribution function and consider the boost-invariant expansion of a fluid in one dimension. In order to close the conservation equations, we need to choose an additional moment of the Boltzmann equation. We discuss the influence of the choice of this moment on the time evolution of fluid-dynamical variables and identify the moment that provides the best match of anisotropic fluid dynamics to the solution of the Boltzmann equation in the relaxation-time approximation.

Figure 1

From top to bottom: the evolution of the anisotropy parameter $\xi$, temperature $T$, fugacity $\lambda$, and the ratio of longitudinal and transverse pressure components ${\widehat{P}}_l/{\widehat{P}}_{\perp }$ as a function of proper time $\tau$. The thin green line in the figures in the second row represents the temperature evolution for an ideal fluid. The other lines are the solution of the conservation equations (56) and (57) closed by different moment equations: the dotted blue line (${\widehat{I}}_{000}^{RS}$) corresponds to Eq. (64), the dash-dotted blue line (${\widehat{I}}_{300}^{RS}$) to Eq. (62), the full black line (${\widehat{P}}_l$) to Eq. (58), the dash-dotted black line (${\widehat{I}}_{320}^{RS}$) to Eq. (63), the dashed red line (${\widehat{I}}_{440}^{RS}$) to Eq. (65), and the dotted red line (${\widehat{I}}_{540}^{RS}$) to Eq. (66), respectively.

Figure 2

Similar to Fig. 1, but for the case without particle-number conservation, such that always $\lambda (\tau )=1$ (and thus not explicitly shown). The only difference to Fig. 1 is that now Eq. (61) is available to close the energy-conservation equation. The respective dashed blue line is labeled $\widehat{n}$.

Figure 3

The evolution of temperature $T$, fugacity $\lambda$, and the ratio of longitudinal and transverse pressure components ${\widehat{P}}_l/{\widehat{P}}_{\perp }$ as a function of proper time $\tau$. The full black, the dashed blue, the dashed-dotted green and the dotted red lines are the solution of the conservation equations closed by the relaxation equation for ${\widehat{P}}_l$ for ${\tau }_{\text{eq}}=1\mathrm{fm}$ and for ${\tau }_{\text{eq}}$ from Eq. (55) with the three different choices for $\eta /s$. The large dots show the corresponding solution of the Boltzmann equation.

Figure 4

Similar to Fig. 3, but for the case without particle-number conservation [such that $\lambda (\tau )=1$ and thus not shown explicitly].

Figure 5

The ratio of the exact solution $F_{320}(\tau )$ to the corresponding fluid-dynamical solution ${\widehat{I}}_{320}^{RS}(\tau )$, obtained from Eqs. (56)–(58). The choice of moment to close the equations is indicated in brackets, $[{\widehat{P}}_l]$, behind the label on the ordinate. The various lines are similar to Figs. 3 and 4.

Figure 6

Similar to Fig. 5, but without particle-number conservation. Here $n(\tau )\equiv F_{100}(\tau )$ represents the solution of the Boltzmann equation while $\widehat{n}(\tau )\equiv {\widehat{I}}_{100}^{RS}(\tau )$ is computed from the solution of Eqs. (57) and (58). The choice of moment to close the equations is indicated in brackets, $[{\widehat{P}}_l]$, behind the label on the ordinate.

Figure 7

The ratio of the exact solution $F_{300}(\tau )$ to ${\widehat{I}}_{300}^{RS}({\alpha }_{220},{\beta }_{220},{\xi }_{220};\tau )$, where ${\alpha }_{220}$, ${\beta }_{220}$, and ${\xi }_{220}$ were obtained by matching to $F_{220}(\tau )$. See the text for more details.

Figure 8

Similar to Fig. 7. The ratio of the exact solution $F_{220}(\tau )$ to ${\widehat{I}}_{220}^{RS}({\alpha }_{300},{\beta }_{300},{\xi }_{300};\tau )$, where ${\alpha }_{300}$, ${\beta }_{300}$, and ${\xi }_{300}$ were obtained by matching to $F_{300}(\tau )$. See the text for more details.