Testing viscous and anisotropic hydrodynamics in an exactly solvable case

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.

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

“…VII of Ref. [47] where this result is obtained without moment expansion). All approaches will be compared at the same value ofη, using…”

“…We additionally include the corresponding approximate result obtained by using the second-order viscous hydrodynamic equations of vaHydro is seen to yield the best overall approximation in all situations, with third-order hydrodynamics a close second for sufficiently small specific shear viscosities. We also point out that, among the approximations explored here, the second-order viscous hydrodynamic equations of Denicol et al which were shown in [36,46,47] to work better than Israel-Stewart theory, provide the poorest approximation to the exact solution, in all cases studied.…”

“…For transversely homogeneous systems undergoing boost-invariant longitudinal expansion, the Boltzmann equation (5) with an RTA collision kernel (96) can be solved exactly [46,47,57], and this can be used to determine the efficacy of various approximation schemes. The transport coefficients to be used in the different hydrodynamic approximations can be related to the RTA relaxation rate Γ by matching to the exact solution at asymptotically late times when the system approaches local momentum isotropy (ξ → 0).…”

“…3 In Refs. [46,47] the non-perturbative equations of aHydro were compared to various second-order (perturbative) hydrodynamic approaches for (0+1)-dimensional expansion in which case there exists an exact solution to the Boltzmann equation in the relaxation time approximation. In all cases tested, anisotropic hydrodynamics most accurately approximated the exact solution when compared to various second-order viscous hydrodynamical approaches, showing the power of this non-perturbative approach.…”

Viscous hydrodynamics for strongly anisotropic expansion

“…For (0+1)-d longitudinally boost-invariant expansion of a transversally homogeneous system, the Boltzmann equation can be solved exactly in RTA [9], and the solution can be used to test the various macroscopic hydrodynamic approximation schemes. Setting homogeneous initial conditions in r and space-time rapidity η s and zero transverse flow, π µν reduces to a single non-vanishing componentπ:π µν = diag(0, −π/2, −π/2,π) at z = 0.…”