Second-order $(2+1)$-dimensional anisotropic hydrodynamics

We present a complete formulation of second-order $(2+1)$-dimensional anisotropic hydrodynamics. The resulting framework generalizes leading-order anisotropic hydrodynamics by allowing for deviations of the one-particle distribution function from the spheroidal form assumed at leading order. We derive complete second-order equations of motion for the additional terms in the macroscopic currents generated by these deviations from their kinetic definition using a Grad-Israel-Stewart 14-moment ansatz. The result is a set of coupled partial differential equations for the momentum-space anisotropy parameter, effective temperature, the transverse components of the fluid four-velocity, and the viscous tensor components generated by deviations of the distribution from spheroidal form. We then perform a quantitative test of our approach by applying it to the case of one-dimensional boost-invariant expansion in the relaxation time approximation (RTA) in which case it is possible to numerically solve the Boltzmann equation exactly. We demonstrate that the second-order anisotropic hydrodynamics approach provides an excellent approximation to the exact (0+1)-dimensional RTA solution for both small and large values of the shear viscosity.

Figure 1

Ratio of the longitudinal to transverse pressure for $4\pi \overline{\eta }\in \{1,3,10,100\}$ (rows) and ${\xi }_0\in \{0,10,100\}$ (columns). 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, NLO anisotropic hydrodynamics (vaHydro), LO anisotropic hydrodynamics (aHydro), and third-order viscous hydrodynamics, respectively. The initial conditions in this figure are $T_0=600$ MeV, ${\tilde{\pi }}_0=0$, and ${\tau }_0=0.25$ fm/$c$.

Figure 2

The same results as in Fig. 1, but now plotted as ratios of the approximations to the exact (Boltzmann equation) result. An additional set of purple dotted curves shows results from second-order viscous hydrodynamics [4, 36, 41, 42].

Figure 3

The effective temperature scaled by the value obtained from the exact solution of the Boltzmann equation for the scenarios and approximations shown in Fig. 2.

Figure 4

Particle production measure ${\Delta }_n=\left({\tau }_{f}n\left({\tau }_f\right)\right)/\left({\tau }_{0}n\left({\tau }_0\right)\right)-1$ as a function of $4\pi \eta /\mathcal{S}$. The black points, red dashed line, blue dashed-dotted line, green dashed line, and purple dotted line correspond to the exact solution of the Boltzmann equation, viscous anisotropic hydrodynamics, LO anisotropic hydrodynamics, third-order viscous hydrodynamics, and second-order viscous hydrodynamics, respectively. The initial conditions in this figure are ${\mathrm{T}}_0=600\mathrm{MeV}$, ${\xi }_0=0$, ${\tilde{\pi }}_0=0$, and ${\tau }_0=0.25\mathrm{fm}/\mathrm{c}$. The freeze-out temperature was taken to be $T_f=150$ MeV.