( 3+1 )-dimensional anisotropic fluid dynamics with a lattice QCD equation of state

Abstract

Anisotropic hydrodynamics improves upon standard dissipative fluid dynamics by treating certain large dissipative corrections nonperturbatively. Relativistic heavy-ion collisions feature two such large dissipative effects: (i) Strongly anisotropic expansion generates a large shear stress component which manifests itself in very different longitudinal and transverse pressures, especially at early times. (ii) Critical fluctuations near the quarkhadron phase transition lead to a large bulk viscous pressure on the conversion surface between hydrodynamics and a microscopic hadronic cascade description of the final collision stage. We present a new dissipative hydrodynamic formulation for nonconformal fluids where both of these effects are treated nonperturbatively. The evolution equations are derived from the Boltzmann equation in the 14-moment approximation, using an expansion around an anisotropic leading-order distribution function with two momentum-space deformation parameters, accounting for the longitudinal and transverse pressures. To obtain their evolution we impose generalized Landau matching conditions for the longitudinal and transverse pressures. We describe an approximate anisotropic equation of state that relates the anisotropy parameters with the macroscopic pressures. Residual shear stresses are smaller and are treated perturbatively, as in standard second-order dissipative fluid dynamics. The resulting optimized viscous anisotropic hydrodynamic evolution equations are derived in 3 + 1 dimensions and tested in a (0 + 1)-dimensional Bjorken expansion, using a state-of-the-art lattice equation of state. Comparisons with other viscous hydrodynamical frameworks are presented.

Hydrodynamics from free-streaming to thermalization and back again

“…Pursuing this program has led to the development of a variety of so-called second-and third-order causal dissipative hydrodynamic theories [29,30,[33][34][35][36], including an anisotropic hydrodynamic framework that addresses the particularly large discrepancy between longitudinal and transverse flow gradients during the early stages of relativistic heavy-ion collisions [32,[37][38][39][40] and effectively resums certain classes of gradients to infinite order [41,42]. In a highly symmetric limit, Bjorken flow [43], which applies to the earliest expansion stage of the medium created in such collisions, it was found that the diverging gradient series can be resummed in its entirety using Borel resummation [8], resulting in a time evolution that agrees with the above-mentioned low-order causal hydrodynamic approaches at late times, when the system approaches local thermal equilibrium, but remains well-behaved even at very early times when the system is very far away from local thermal equilibrium [11,44].…”

Exact solutions and attractors of higher-order viscous fluid dynamics for Bjorken flow

“…In light of footnote 2 this should perhaps not be too surprising 12. aHydro, a second-order approach that is based on an expansion around a self-consistently adjusted ellipsoidally deformed local momentum distribution[38,74,76,77,79,80], performs even better than the third-order theory[42].…”

Viscous hydrodynamics for nonconformal anisotropic fluids

“…Hydrodynamics being an effective field theory for the long-distance dynamics of multiparticle systems [3], the structure of these equations is universal, with specific medium properties encoded in the EoS and a set of transport coefficients. In [1] the Boltzmann equation in the relaxation-time approximation for a system of weakly interacting quasiparticles with a medium dependent mass m(T ) is used to derive these relaxation equations, including one for a mean field B needed for thermodynamic consistency [4]. We expand the distribution function as…”

“…After using the generalized Landau matching conditions to eliminate all derivatives of the microscopic anisotropy parameters Λ, α ⊥ , α L and the mean field B in terms of the macroscopic hydrodynamic variables E, P ⊥ and P L , a set of purely macroscopic evolution equations is found, with source terms that describe the generation of dissipative flows in terms of hydrodynamic gradients multiplied by transport coefficients [1]. Ref.…”

“…Only the smaller residual shear stresses are treated perturbatively. For systems without conserved charges a full account of the formalism is presented in [1]; here we only summarize its essential features but add a discussion of how to initialize the anisotropic parameters that are necessary for computing the initial values of the anisotropic transport coefficients. A brief comparison of the evolution of the pressure anisotropy P L /P ⊥ in (0+1)-dimensional Bjorken expansion between our new approach and standard 2 nd order viscous hydrodynamics concludes these proceedings.…”