Nonconformal viscous anisotropic hydrodynamics

We generalize the derivation of viscous anisotropic hydrodynamics from kinetic theory to allow for nonzero particle masses. The macroscopic theory is obtained by taking moments of the Boltzmann equation after expanding the distribution function around a spheroidally deformed local momentum distribution whose form has been generalized by the addition of a scalar field that accounts nonperturbatively (i.e., already at leading order) for bulk viscous effects. Hydrodynamic equations for the parameters of the leading-order distribution function and for the residual (next-to-leading order) dissipative flows are obtained from the three lowest moments of the Boltzmann equation. The approach is tested for a system undergoing ($0+1$)-dimensional boost-invariant expansion for which the exact solution of the Boltzmann equation in the relaxation time approximation is known. Nonconformal viscous anisotropic hydrodynamics is shown to approximate this exact solution more accurately than any other known hydrodynamic approximation.

Figure 1

Ratio of the longitudinal to transverse pressure ${\mathcal{P}}_{\perp }/{\mathcal{P}}_L$ (left column) and the bulk viscous pressure $\mathrm{\Pi }$ (right column). The top panels correspond to an initial anisotropy parameter ${\xi }_0=0$ whereas the bottom panels are for ${\xi }_0=100$. The black solid, red short-dashed, blue dashed-dotted, and green long-dashed lines are the results obtained from the exact solution of the Boltzmann equation, viscous anisotropic hydrodynamcs (vaHydro), LO anisotropic hydrodynamics (aHydro), and second-order viscous hydrodynamics [7, 8, 9], respectively. The initial conditions in this figure are $T_0=600\mathrm{MeV}, m=1\mathrm{GeV}, {\tilde{\mathrm{\Pi }}}_0=0, {\tilde{\pi }}_0=0, {\tau }_{\mathrm{eq}}=0.5\mathrm{fm}/c$, and ${\tau }_0=0.5\mathrm{fm}/c$.

Figure 2

Similar to Fig. 1, but for a ten-times-smaller mass $m=0.1\mathrm{GeV}$.

Figure 3

Similar to Fig. 1, but for ${\tau }_{\mathrm{eq}}=5\mathrm{fm}/c$. An additional set of purple dotted curves shows the effect of setting the bulk-shear coupling terms in the evolution equations for $\tilde{\mathrm{\Pi }}$ and $\tilde{\pi }$ to zero.