Studying the validity of relativistic hydrodynamics with a new exact solution of the Boltzmann equation

Abstract

We present an exact solution to the Boltzmann equation which describes a system undergoing boost-invariant longitudinal and azimuthally symmetric radial expansion for arbitrary shear viscosity to entropy density ratio. This new solution is constructed by considering the conformal map between Minkowski space and the direct product of three dimensional de Sitter space with a line.The resulting solution respects SO(3) q ⊗ SO(1, 1) ⊗ Z 2 symmetry. We compare the exact kinetic solution with exact solutions of the corresponding macroscopic equations that were obtained from the kinetic theory in ideal and second-order viscous hydrodynamic approximations. The macroscopic solutions are obtained in de Sitter space and are subject to the same symmetries used to obtain the exact kinetic solution.

Anisotropic hydrodynamics for conformal Gubser flow

“…(13) and (16). In order to satisfy SO(3) q invariance, the distribution function can only depend onp 2 Ω ≡p 2 θ +p 2 φ /sin 2 θ [53]. As a result, one must haveξ θ =ξ φ .…”

“…(42)- (44) as Refs. [52,53] is that the distribution function was assumed to be isotropic at ρ 0 . As we will show below, if one assumes that the initial distribution function is of spheroidal form in de…”

“…In this paper, we derive the dynamical equations of anisotropic hydrodynamics for a system subject to Gubser flow [50,51] and compare them to recently obtained analytic solutions to the Boltzmann equation in the relaxation-time approximation subject to the same flow [52,53]. Since Gubser flow includes both cylindrically-symmetric transverse and boost-invariant longitudinal (1+1d) expansion, this will allow us to test the efficacy of anisotropic hydrodynamics in a more realistic setting.…”

“…Since Gubser flow includes both cylindrically-symmetric transverse and boost-invariant longitudinal (1+1d) expansion, this will allow us to test the efficacy of anisotropic hydrodynamics in a more realistic setting. We will also compare to recently obtained solutions using the IsraelStewart second-order viscous hydrodynamics framework [54] and a complete second-order Grad 14-moment approximation [53].…”

“…VII we generalize the exact solution of the relaxation-time-approximation Boltzmann equation subject to Gubser flow obtained originally in Refs. [52,53], to allow for arbitrary anisotropy initial conditions and demonstrate that the anisotropic hydrodynamics equations of motion give the exact result in both the ideal and free-streaming limits. In Sec.…”