Abstract
Kinetic equations for quarks and gluons are solved numerically in the relaxation time approximation for the case of one-dimensional boost-invariant geometry. Quarks are massive and described by the Fermi-Dirac statistics, while gluons are massless and obey Bose-Einstein statistics. The conservation laws for the baryon number, energy, and momentum lead to two Landau matching conditions which specify the coupling between the quark and gluon sectors and determine the proper-time dependence of the effective temperature and baryon chemical potential of the system. The numerical results illustrate how a non-equlibrium mixture of quarks and gluons approaches hydrodynamic regime described by the Navier-Stokes equations with appropriate forms of the kinetic coefficients. The shear viscosity of a mixture is the sum of the shear viscosities of quark and gluon components, while the bulk viscosity is given by the formula known for a gas of quarks, however, with the thermodynamic variables characterising the mixture. Thus, we find that massless gluons contribute in a non-trivial way to the bulk viscosity of a mixture, provided quarks are massive. We further observe the hydrodynamization effect which takes place earlier in the shear sector than in the bulk one. The numerical studies of the ratio of the longitudinal and transverse pressures show, to a good approximation, that it depends on the ratio of the relaxation and proper times only. This behaviour is connected with the existence of an attractor solution for conformal systems.
Highlights
Comparisons between predictions of hydrodynamic models and exact kinetic-theory results have become an important method to verify the validity of hydrodynamic frameworks [1,2,3,4,5,6,7,8,9,10,11,12], which are our basic tools to interpret the processes of heavy-ion collisions studied experimentally at RHIC and the LHC [13,14,15,16,17]
We present explicit expressions for various physical quantities such as the particle and energy densities or the transverse and longitudinal pressures. These expressions are obtained with the use of different distribution functions, which do not necessarily correspond to local equilibrium
We have compared the results of the numerical calculations with the first-order hydrodynamic calculations to demonstrate the hydrodynamization process
Summary
Comparisons between predictions of hydrodynamic models and exact kinetic-theory results have become an important method to verify the validity of hydrodynamic frameworks [1,2,3,4,5,6,7,8,9,10,11,12], which are our basic tools to interpret the processes of heavy-ion collisions studied experimentally at RHIC and the LHC [13,14,15,16,17]. Our studies of the time evolution of the ratio of the longitudinal and transverse pressures indicate that, to a very good approximation, it depends on the ratio of the relaxation and proper times only This behavior is related to the presence of an attractor, which was found and discussed. II and III we introduce the system of kinetic equations for the fermion-boson mixture and study their momentum moments. IV we discuss an algebraic method useful for dealing with tensors describing our main observables Appendixes A–C contain: details of the calculations of the generalized thermodynamic functions, discussion of the Navier-Stokes equations, and the explicit calculation of the shear and bulk viscosities for a fermion-boson mixture, respectively.
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