Anisotropic hydrodynamics for rapidly expanding systems

Abstract

We exactly solve the relaxation-time approximation Boltzmann equation for a system which is transversely homogeneous and undergoing boost-invariant longitudinal expansion. We compare the resulting exact numerical solution with approximate solutions available in the anisotropic hydrodynamics and second order viscous hydrodynamics frameworks. In all cases studied, we find that the anisotropic hydrodynamics framework is a better approximation to the exact solution than traditional viscous hydrodynamical approaches.The application of relativistic viscous hydrodynamics is important in a wide variety of situations including, for example, the dynamics of high energy astrophysical plasmas and the quark-gluon plasma created in relativistic heavy-ion collisions. Since the seminal work of Israel and Stewart [1,2] there have been many papers that have addressed the questions of how to apply and systematically improve relativistic viscous hydrodynamics [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. Recently, a new framework called anisotropic hydrodynamics (aHydro) has emerged for describing the nonequilibrium dynamics of relativistic systems [22][23][24][25][26][27][28][29].

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

“…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.…”

“…In this appendix we briefly review the exact solution [46,47] of the Boltzmann equation in relaxation time approximation [58] for a (0+1)-dimensional boost-invariant system:…”

“…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.…”

Anisotropic hydrodynamics for a mixture of quark and gluon fluids

“…The value of the relaxation time used in this work is constant, τ eq = 0.25 fm/c. A temperature dependent τ eq can be also used, as it is described in [13,14]. However, to check the agreement with the kinetic theory it is enough to use a constant value.…”

Anisotropic hydrodynamics for conformal Gubser flow

“…In the anisotropic hydrodynamics framework, the most important (diagonal) components of the energy-momentum tensor are treated non-perturbatively and non-spheroidal/off-diagonal components are treated perturbatively. This approach has been shown to more accurately describe the evolution of systems subject to boost-invariant and transversely homogeneous (0+1d) flow than traditional viscous hydrodynamics approaches [42,44,46,48,49].…”