Coupled kinetic equations for fermions and bosons in the relaxation-time approximation

Kinetic equations for fermions and bosons are solved numerically in the relaxation-time approximation for the case of one-dimensional boost-invariant geometry. Fermions are massive and carry baryon number, while bosons are massless. The conservation laws for the baryon number, energy, and momentum lead to two Landau matching conditions, which specify the coupling between the fermionic and bosonic sectors and determine the proper-time dependence of the effective temperature and baryon chemical potential of the system. The numerical results illustrate how a nonequilibrium mixture of fermions and bosons 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 fermion and boson components, while the bulk viscosity is given by the formula known for a gas of fermions, however, with the thermodynamic variables characterising the mixture. Thus, we find that massless bosons contribute in a nontrivial way to the bulk viscosity of a mixture, provided fermions 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 behavior is connected with the existence of an attractor solution for conformal systems.

Figure 1

Effective temperature $T$ (top panels) and $\mu /T$ ratio (bottom panels), shown as functions of the proper time $\tau$ and normalized to unity at the initial proper time $\tau ={\tau }_0$. Results correspond to the initial oblate-oblate configuration with the anisotropy parameters given in the figure. Three different types of lines correspond to three different choices of the statistics and the quark mass (the label “cs” denotes classical statistics used for both quarks and gluons, while the label “qs” denotes the use of Fermi-Dirac and Bose-Einstein statistics for quarks and gluons, respectively). Other parameters of the calculations are shown in the figure and discussed in the text.

Figure 2

Same as Fig. 1 but for the initial prolate-oblate configuration with the parameters given in the figure.

Figure 3

Same as Figs. 1 and 2 but for the initial prolate-prolate configuration with the parameters given in the figure.

Figure 4

Proper-time dependence of the ratios (a) $T/T_0$ and (b) $\mu T_0/\left({\mu }_{0}T\right)$ obtained in the range ${\tau }_0<\tau <10$ fm from: kinetic theory (red solid lines), perfect-fluid hydrodynamics (green dot-dashed lines), and Navier-Stokes hydrodynamics (navy blue dashed lines). All results are normalized to the initial values $T_0$ and ${\mu }_0$ used in the kinetic theory. The initial values of temperature and chemical potential in the hydrodynamic calculations are chosen in such a way that the final values of $T$ and $\mu$ agree with the values found in the kinetic-theory calculation. The calculations are done for the oblate-oblate initial conditions with a finite quark mass of 300 MeV, quantum statistics, and ${\mathcal{B}}_0=1{\mathrm{fm}}^{-3}$.

Figure 5

Effective shear viscosity ${\eta }_{\mathrm{eff}}$ defined by Eq. (84) (red solid line) and the shear viscosity coefficients $\eta$ calculated using Eq. (85) for the two $T\left(\tau \right)$ and $\mu \left(\tau \right)$ profiles, found from the perfect-fluid hydrodynamics (green dot-dashed line) and from the Navier-Stokes equations (navy blue dashed line). The effective shear viscosity agrees well with the standard definition of $\eta$ for $\tau >0.5$ fm. The initial conditions are the same as in Fig. 4.

Figure 6

(a) Proper-time dependence of the shear viscosity of the mixture (red solid line), of the quark component (navy blue dashed line), and of the gluon component (green dot-dashed line), see Eqs. (85, 86, 87). The initial conditions are the same as in Fig. 4. (b) The ratio ${\eta }_Q/{\eta }_G$ as a function of $m/T$ and $\mu /T$. The colored line represents the system evolution trajectory with the parameters corresponding to (a).

Figure 7

Effective bulk viscosity ${\zeta }_{\mathrm{eff}}$ defined by Eq. (88) (red line) and the bulk viscosity coefficient $\zeta$ calculated with the help of Eq. (90) for the two $T\left(\tau \right)$ and $\mu \left(\tau \right)$ profiles, found from the perfect-fluid hydrodynamics (green dot-dashed line) and from the Navier-Stokes equations (blue dashed line). We find that the effective bulk viscosity agrees with the standard definition of $\zeta$ for $\tau >2$ fm. The initial conditions are the same as in Fig. 4.

Figure 8

Bulk viscosity coefficient $\zeta$ calculated with the help of Eq. (90) with the thermodynamic coefficients ${\kappa }_1$ and ${\kappa }_2$ determined for the whole quark-gluon system (red solid line) and the coefficient ${\zeta }_0$ obtained from Eq. (90) with ${\kappa }_1$ and ${\kappa }_2$ determined only for the quark component (blue dashed line). The initial conditions are the same as in Fig. 4.

Figure 9

${\mathcal{P}}_T/\mathcal{E}$ (top panels) and ${\mathcal{P}}_L/{\mathcal{P}}_T$ (bottom panels) for initially oblate-oblate system. Green solid lines correspond to massless quarks and classical distribution functions, black dashed lines to the massive quarks and classical distribution functions, while red solid lines are for massive quarks and quantum distributions. Left (right) panels describe the results for ${\mathcal{B}}_0=0.001{\mathrm{fm}}^{-3}\left({\mathcal{B}}_0=1{\mathrm{fm}}^{-3}\right)$.

Figure 10

Same as Fig. 9 but for initially prolate-oblate system.

Figure 11

Same as Fig. 9 but for initially prolate-prolate system.

Figure 12

${\mathcal{P}}_T/\mathcal{E}$ (top panels) and ${\mathcal{P}}_L/{\mathcal{P}}_T$ (bottom panels) for massless quarks and classical distribution functions. Green dot-dashed, navy blue dashed, and red solid lines describe the results for the oblate-oblate, prolate-oblate, and prolate-prolate initial conditions. The blue dotted line describes ${({\mathcal{P}}_L/{\mathcal{P}}_T)}_{\mathrm{NS}}$ obtained from the Navier-Stokes hydrodynamics.

Figure 13

Same as Fig. 12 but for massive quarks and classical statistics.

Figure 14

Same as Fig. 12 but for massive quarks and quantum statistics.

Figure 15

Contour plots of ${({\mathcal{P}}_L/{\mathcal{P}}_T)}_{\mathrm{NS}}$ obtained for the Navier-Stokes hydrodynamics for (a) $\mu =0$ and (b) $\mu /T=2$. In the two cases quarks and gluons are described by quantum statistics. The solid black lines together with the contour shading represent the classical baryon-free system.

Figure 16

The quantity $A_1$ plotted as a function of $\mathrm{\Gamma }$ for three different initial anisotropies, finite quark mass, and quantum statistics.

Figure 17

Same as Fig. 16 but for $A_2$ vs $\mathrm{\Gamma }$.