Testing viscous and anisotropic hydrodynamics in an exactly solvable case

TL;DR

This paper provides an exact solution to the one-dimensional Boltzmann equation with arbitrary shear viscosity and compares it with viscous and anisotropic hydrodynamics, demonstrating improved agreement of recent formulations.

Contribution

It offers the first exact kinetic-theory solution for this case and evaluates the accuracy of different hydrodynamic approximations against it.

Findings

Recent second-order viscous hydrodynamics formulations align better with the exact solution.

Anisotropic hydrodynamics closely approximates the exact kinetic solution when properly connected.

Standard Israel-Stewart approach shows less agreement with the exact solution.

Abstract

We exactly solve the one-dimensional boost-invariant Boltzmann equation in the relaxation time approximation for arbitrary shear viscosity. The results are compared with the predictions of viscous and anisotropic hydrodynamics. Studying different non-equilibrium cases and comparing the exact kinetic-theory results to the second-order viscous hydrodynamics results we find that recent formulations of second-order viscous hydrodynamics agree better with the exact solution than the standard Israel-Stewart approach. Additionally, we find that, given the appropriate connection between the kinetic and anisotropic hydrodynamics relaxation times, anisotropic hydrodynamics provides a very good approximation to the exact relaxation time approximation solution.