Transport coefficients for bulk viscous evolution in the relaxation-time approximation

We derive the form of the viscous corrections to the phase-space distribution function due to bulk viscous pressure and shear stress tensor using the iterative Chapman-Enskog method. We then calculate the transport coefficients necessary for the second-order hydrodynamic evolution of the bulk viscous pressure and the shear stress tensor. We demonstrate that the transport coefficients obtained using the Chapman-Enskog method are different than those obtained previously using the 14-moment approximation for a finite particle mass. Specializing to the case of boost-invariant and transversally homogeneous longitudinal expansion, we show that the transport coefficients obtained using the Chapman-Enskog method result in better agreement with the exact solution of the Boltzmann equation in the relaxation-time approximation compared to results obtained in the 14-moment approximation. Finally, we explicitly confirm that the time evolution of the bulk viscous pressure is significantly affected by its coupling to the shear stress tensor.

Figure 1

Comparison of the exact transport coefficients obtained herein using the Chapman-Enskog method (blue dashed line) with those calculated using the 14-moment approximation (brown dotted line). The two panels correspond to the transport coefficients which enter (a) the bulk viscous pressure and (b) the shear stress tensor evolution equations, as a function of the ratio of mass and temperature. The inset in panel (a) shows the $m/T$ dependence of the transport coefficients ${\delta }_{\Pi \Pi }$ and ${\lambda }_{\Pi \pi }$ obtained by using the two methods on a linear scale. Here $P_{\left(0\right)}$ is the pressure at vanishing mass; i.e., $P_{\left(0\right)}\equiv P(m=0,T)$.

Figure 2

Time evolution of the pressure anisotropy ${\mathcal{P}}_L/{\mathcal{P}}_T$ (top) and the bulk viscous pressure times $\tau$ (bottom) for three different calculations: the exact solution of the RTA Boltzmann equation [24] (red solid line), second-order viscous hydrodynamics using the 14-moment method [12] (brown dotted line), and the Chapman-Enskog method used herein (blue dashed line). For both panels we use $T_0=600$ MeV at ${\tau }_0=0.5$ $\mathrm{fm}/c,m=300$ MeV, and ${\tau }_{\mathrm{eq}}={\tau }_{\pi }={\tau }_{\Pi }=0.5$ $\mathrm{fm}/c$. The initial spheroidal anisotropy in the distribution function, ${\xi }_0=0$, corresponds to isotropic initial pressures with ${\pi }_0=0$ and ${\Pi }_0=0$.

Figure 3

Same as Fig. 2 except here we take ${\xi }_0=100$ corresponding to ${\pi }_0=51.11$ ${\mathrm{GeV}/\mathrm{fm}}^3$ and ${\Pi }_0=0.85$ ${\mathrm{GeV}/\mathrm{fm}}^3$.

Figure 4

Same as Fig. 2 except here we take $m=1$ GeV.

Figure 5

Same as Fig. 3 except here we take $m=1$ GeV, which for ${\xi }_0=100$ implies ${\pi }_0=35.12$ ${\mathrm{GeV}/\mathrm{fm}}^3$ and ${\Pi }_0=3.08$ ${\mathrm{GeV}/\mathrm{fm}}^3$.

Figure 6

Proper time evolution of the second-order terms scaled by the first-order term in the evolution equation for bulk viscous pressure, Eq. (58). For both panels we use $T_0=600$ MeV at ${\tau }_0=0.5$ $\mathrm{fm}/c$ and ${\tau }_{\mathrm{eq}}={\tau }_{\pi }={\tau }_{\Pi }=0.5$ $\mathrm{fm}/c$. The initial spheroidal anisotropy in the distribution function, ${\xi }_0=0$, corresponds to an isotropic pressure configuration ${\pi }_0=0$ and ${\Pi }_0=0$. For the top panel, we show results for $m=300$ MeV whereas the bottom panel corresponds to $m=1$ GeV.