Anisotropic fluid dynamics for Gubser flow

Exploring a variety of closing schemes to the infinite hierarchy of momentum moments of the exactly solvable Boltzmann equation for systems undergoing Gubser flow, we study the precision with which the resulting hydrodynamic equations reproduce the exact evolution of hydrodynamic moments of the distribution function. We find that anisotropic hydrodynamics, obtained by expanding the distribution function around a dynamically evolving locally anisotropic background whose evolution is matched to exactly reproduce the macroscopic pressure anisotropy caused by the different longitudinal and transverse expansion rates in Gubser flow, provides the most accurate macroscopic description of the microscopic kinetic evolution. This confirms a similar earlier finding for Bjorken flow [Molnár et al., Phys. Rev. D 94, 125003 (2016)]. We explain the physics behind this optimal matching procedure and show that one can efficiently correct for a nonoptimized matching choice by adding a residual shear stress to the energy-momentum tensor whose evolution is again determined by the Boltzmann equation.

Figure 1

de Sitter time evolution of the temperature $\widehat{T}$ and the normalized shear stress $\widehat{\overline{\pi }}$ for the exact solution of the RTA Boltzmann equation (black solid lines) and four different hydrodynamic approximations: second-order viscous hydrodynamics (DNMR theory, short-dashed magenta lines), anisotropic hydrodynamics with ${\widehat{\mathcal{P}}}_L$-matching (dotted red lines), leading-order anisotropic hydrodynamics in the NRS scheme (dash-dotted green lines), and next-to-leading-order anisotropic hydrodynamics in the NRS scheme amended by residual viscous corrections (long-dashed blue lines). For the initial momentum distribution, we here assumed isotropy, i.e., ${\xi }_0=0$. The top, middle, and bottom rows of panels correspond to specific shear viscosity $4\pi \eta /s=1$, 3, and 10, respectively. The four columns of plots show, from left to right, the $\rho$ evolution of the temperature $\widehat{T}$, of the ratio between its hydrodynamic and exact kinetic evolution $\widehat{T}/{\widehat{T}}_{\mathrm{exact}}$, of the normalized shear stress $\widehat{\overline{\pi }}$, and of the difference between its hydrodynamic and exact kinetic evolution $\widehat{\overline{\pi }}-{\widehat{\overline{\pi }}}_{\mathrm{exact}}$.

Figure 2

Same as Fig. 1, but for an initially highly oblate momentum distribution, ${\xi }_0=100$.

Figure 3

Same as Fig. 1, but for an initially highly prolate momentum distribution, ${\xi }_0=-0.9$.

Figure 4

Comparison of the de Sitter time evolution of the momentum anisotropy parameter $\xi$ corresponding to the exact solution of the RTA Boltzmann equation and the various hydrodynamic approximations discussed in this work, for an initial value ${\xi }_0=100$ and a specific shear viscosity $4\pi \eta /s=3$. The different approaches are plotted using the same line styles as in Figs. 1, 2, 3. See text for discussion.