Transport coefficients in ultrarelativistic kinetic theory

A spatially periodic longitudinal wave is considered in relativistic dissipative hydrodynamics. At sufficiently small wave amplitudes, an analytic solution is obtained in the linearized limit of the macroscopic conservation equations within the first- and second-order relativistic hydrodynamics formulations. A kinetic solver is used to obtain the numerical solution of the relativistic Boltzmann equation for massless particles in the Anderson-Witting approximation for the collision term. It is found that, at small values of the Anderson-Witting relaxation time $\tau$, the transport coefficients emerging from the relativistic Boltzmann equation agree with those predicted through the Chapman-Enskog procedure, while the relaxation times of the heat flux and shear pressure are equal to $\tau$. These claims are further strengthened by considering a moment-type approximation based on orthogonal polynomials under which the Chapman-Enskog results for the transport coefficients are exactly recovered.

Figure 1

Comparison between the analytic solutions in Eq. (3.20) corresponding to the Chapman-Enskog value for $\eta$ (continuous lines) and numerical results (dotted lines and points) for $\tau =8.3\times {10}^{-3}$. The vertical axes represent the values of (a) $\widetilde{\delta n}$, (b) $\widetilde{\delta P}$, (c) $\tilde{\beta }$, and (d) $\tilde{\mathrm{\Pi }}$, divided by ${\beta }_0$. The coefficients ${\alpha }_{d,\mathrm{CE}}$ and ${\alpha }_{d,\mathrm{G}}$ in the asymptotic dampening lines are obtained from Eq. (3.12) by substituting the Chapman-Enskog (3.3b) and Grad (3.3a) expressions for $\eta$. The system was initialized according to case 1, i.e., $\delta n_0=\delta P_0=0$ and ${\beta }_0={10}^{-3}$.

Figure 2

The time evolution of $\tilde{q}$ for the initializations corresponding to case 2a (i.e., $\delta n_0={\beta }_0=0$ and $\delta P_0={10}^{-3})$ and case 2b (i.e., $\delta P_0={\beta }_0=0$ and $\delta n_0={10}^{-3})$. The dotted lines with points represent the numerical results. The analytic solution (3.21) is represented for the two cases using solid lines when ${\alpha }_{\lambda }$ is computed using the Chapman-Enskog expression for $\lambda$ (3.3b), while the dotted lines correspond to the case when the Grad expression (3.3a) is used. The relaxation time was set to $\tau =0.0083$.

Figure 3

The ratio $\overline{\beta }/{\beta }_0$ for various values of ${\beta }_0$ corresponding to case 1 presented in Sec. 2c. As ${\beta }_0$ increases, the time evolution of $\tilde{\beta }/{\beta }_0$ departs from the solution (3.20), indicating that nonlinear effects become important.

Figure 4

The dependence of (a) ${\alpha }_d$ and (b) ${\alpha }_o$ on ${\beta }_0$; the four curves correspond to the two-parameter nonlinear fits of $\widetilde{\delta n},\widetilde{\delta P},\tilde{\beta }$, and $\tilde{\mathrm{\Pi }}$ in Eq. (3.20) to the corresponding numerical data, as described in Sec. 3e. (c) The dependence of ${\alpha }_{\lambda }$, obtained using a nonlinear fit of Eq. (3.21) to the numerical data, on the amplitudes $\delta P_0$ (case 2a) and $\delta n_0$ (case 2b) of the initial perturbation. The relaxation time was always kept at $\tau =0.0083$.

Figure 5

Graphical representation of the dependence on $\tau$ of (a) ${\alpha }_d$, (b) ${\alpha }_o$, and (c) ${\alpha }_{\lambda }$ obtained using the nonlinear fitting procedure described in Sec. 3e. In panel (d), the ratio ${\alpha }_{\lambda }/{\alpha }_d$ is represented with respect to $\tau$, where ${\alpha }_d$ is obtained using the two-parameter nonlinear fit of $\tilde{\beta }$ (3.20) on the numerical data, while ${\alpha }_{\lambda }$ is obtained using a one-parameter nonlinear fit of $\tilde{q}$ (3.21) for case 2a. The perturbation amplitude is in all cases set to ${10}^{-3}$. On each plot, the analytic curves corresponding to the Chapman-Enskog and Grad methods are displayed.

Figure 6

[(a), (b)] Comparison between the numerical results (dotted lines and points) and the analytic expressions (3.20) and (4.35) obtained in the frame of the first- (dotted lines) and second-order (continuous lines) relativistic hydrodynamics for the evolution of $\tilde{\mathrm{\Pi }}/{\beta }_0\left({\beta }_0={10}^{-3}\right)$ at (a) $\tau =0.056$ and (b) $\tau =0.1$. (c) Evolution of $\tilde{\mathrm{\Pi }}/\tau {\beta }_0$ at various values of $\tau$ (to ease the comparison, the horizontal axis shows $t/\tau )$. (d) Evolution of $\tilde{\mathrm{\Pi }}/\tau {\beta }_0$ at $\tau =0.26$. The fitted curve corresponding to the first-order hydrodynamics is obtained by performing a nonlinear fit of Eq. (3.20) using ${\alpha }_{\eta }$ and ${\alpha }_o$ as free parameters. In the second-order case, the nonlinear fit is performed using Eq. (4.35) by considering ${\alpha }_{\eta ,r},{\alpha }_{\eta ,d},{\alpha }_{\eta ,o}$, and the ratio $\eta /{\tau }_{\mathrm{\Pi }}$ as free parameters. All nonfitted analytic curves are obtained using the Chapman-Enskog value for ${\eta }_0$ (3.3b) and ${\tau }_{\mathrm{\Pi },0}=1$ (4.5).

Figure 7

Comparison with respect to $\tau$ between the analytic prediction (4.20) and the numerical fit for the coefficients (a) ${\alpha }_{\eta ,r}$, (b) ${\alpha }_{\eta ,d}$, and (c) ${\alpha }_{\eta ,o}$. The analytic curves corresponding to the Chapman-Enskog procedure (continuous lines) and Grad moment method (dotted lines) are obtained by using the expressions (3.3b) and (3.3a) for ${\eta }_0$ in Eq. (4.20). The numerical curves (dotted lines and points) are obtained by performing a nonlinear fit on the functional forms (4.29) of $\tilde{\beta },\widetilde{\delta n},\widetilde{\delta P}$, and $\tilde{\mathrm{\Pi }}$. The simulations were initialized according to case 1, i.e., $\delta n_0=\delta P_0=0$ and ${\beta }_0={10}^{-3}$.

Figure 8

Comparison with respect to $\tau$ between the analytic and numerical predictions for (a) the relaxation time ${\tau }_{\mathrm{\Pi }}$; (b) the ratio $\eta /{\tau }_{\mathrm{\Pi }}{P}_0$; (c) the coefficient of shear viscosity $\eta$. The analytic expression for ${\tau }_{\mathrm{\Pi }}$ (continuous line), given in Eq. (4.5), is ${\tau }_{\mathrm{\Pi }}=\tau$, while the analytic expression for $\eta$ is given in Eqs. (3.3b) and (3.3a) for the Chapman-Enskog (continuous line) and Grad (dotted lines) cases. The numerical results (dotted lines and points) are obtained as described in Sec. 4d.

Figure 9

Comparison between the numerical results (dotted lines and points) for $\tilde{\beta }$ (first column), $\widetilde{\delta P}$ (second column), and $\tilde{\mathrm{\Pi }}$ (third column) and the corresponding analytic predictions of the first-order (dotted lines) and second-order (continuous lines) hydrodynamics when ${\tau }_{\mathrm{\Pi }}=\tau$ and the Chapman-Enskog value (3.3b) for $\lambda$ is employed. The initial conditions for the plots on the first line correspond to case 2b, i.e., ${\beta }_0=\delta P_0=0$ and $\delta n_0={10}^{-3}$. On the second line, ${\beta }_0$ is increased to ${10}^{-5}$ and on the third line, ${\beta }_0={10}^{-3}$. All curves are normalized with respect to $\delta n_0={10}^{-3}$ and $\tau =0.0083$ was employed.

Figure 10

(a) Numerical results (dotted lines and points) for the evolution of $\tilde{q}/\tau \delta n_0$ at various values of $\tau$ for $\delta n_0={10}^{-3}$ (to ease the comparison, the horizontal axis shows $t/\tau )$. The analytic curves corresponding to the first-order (3.20) and second-order (4.35) hydrodynamics are shown using dotted and continuous lines, respectively. (b) Time evolution of $\tilde{q}/\tau \delta n_0$ at $\tau =0.1$ and $\delta n_0={10}^{-3}$. The fitted curve corresponding to the first-order hydrodynamics is obtained by performing a nonlinear fit of Eq. (3.21) using ${\alpha }_{\eta }$ and ${\alpha }_o$ as free parameters. In the second-order case, the nonlinear fit is performed on Eqs. (4.37a) and (4.37b) for the overdamped and underdamped cases by considering ${\alpha }_{\lambda ,d}$ and ${\alpha }_{\lambda ,o}({\overline{\alpha }}_{\lambda ,d}$ and ${\overline{\alpha }}_{\lambda ,o})$ as free parameters. All nonfitted analytic curves are obtained using the Chapman-Enskog value for ${\lambda }_0$ (3.3b) and ${\tau }_{q,0}=1$ (4.5).

Figure 11

Analysis with respect to $\tau$ of (a) ${\alpha }_{\lambda ,d}$ and (b) ${\alpha }_{\lambda ,o}$ for the initial conditions of case 2b (i.e., ${\beta }_0=\delta P_0=0$ and $\delta n_0={10}^{-3})$. The analytic curve in panel (a) is ${\alpha }_{\lambda ,d}=1/2\tau$, while in panel (b), the analytic result (4.17) is represented using the Chapman-Enskog (continuous line) and Grad (dashed line) values for $\lambda$, while ${\tau }_q$ was taken equal to $\tau$. The numerical curves shown with dotted lines and filled symbols are obtained by performing a nonlinear fit on the overdamped (OD) solution (4.37a) while considering ${\alpha }_{\lambda ,d}$ and ${\alpha }_{\lambda ,o}$ as free parameters. The dotted lines with hollow symbols are obtained by fitting the values of ${\overline{\alpha }}_{\lambda ,d}$ and ${\overline{\alpha }}_{\lambda ,o}$ to the numerical results using the underdamped solution (4.37b). The numerical curves represented with dashed lines and without points are obtained by piecing together the UD and OD results. The transition from the OD to the UD regime occurs when ${\alpha }_{\lambda ,o}=0$, as indicated by the spikes in panel (b).

Figure 12

Analysis with respect to $\tau$ of (a) ${\tau }_q$ and (b) $\lambda$ for the initial conditions of case 2b (i.e., ${\beta }_0=\delta P_0=0$ and $\delta n_0={10}^{-3})$. The analytic curve shown in panel (a) using a continuous line is ${\tau }_q=\tau$, while in panel (b), the analytic curves correspond to the Chapman-Enskog (continuous line) and Grad (dotted line) values for $\lambda$, given in Eqs. (3.3b) and (3.3a), respectively. The numerical results are obtained as explained in Sec. 4e.

Figure 13

Time evolution of $\tilde{q}/\tau \delta n_0$ at (a) $\tau =0.22$ and (b) $\tau =1.0$ for $\delta n_0={10}^{-3}$. Since for the Chapman-Enskog value of ${\lambda }_0$ (3.3b) and ${\tau }_{q,0}=1$ (4.5), in both cases $\tau >{\tau }_{\lambda ,\lim }\simeq 0.14$ (4.16), such that the curve corresponding to the second-order hydrodynamics theory is given by the underdamped (UD) solution (4.37b). The analytic solution corresponding to the moment method is given in Eq. (5.36). The fitted curves are obtained as explained in Sec. 5d.

Figure 14

Analysis with respect to $\tau$ of (a) the parameter $\tau$, (b) ${\alpha }_{\lambda ,r}$, (c) ${\alpha }_{\lambda ,d}$, and (d) ${\alpha }_{\lambda ,o}$ for the initial conditions of case 2b (i.e., ${\beta }_0=\delta P_0=0$ and $\delta n_0={10}^{-3})$. The analytic curves are given by Eq. (5.24) in panels (b)–(d), while in panel (a), the analytic curve represents $\tau$. The numerical curves shown with dotted lines and symbols are obtained by performing a nonlinear fit, as described in Sec. 5d.

Figure 15

Time evolution of (a) $\widetilde{\delta n}$, (b) $\widetilde{\delta P}$, (c) $\tilde{\beta }$, (d) $\tilde{q}$, and (e) $\tilde{\mathrm{\Pi }}$ in the free-streaming regime, divided by the wave amplitude. The system is initialized with the following initial conditions: Case 1, $\delta n_0=\delta P_0=0$ and the wave amplitude is ${\beta }_0={10}^{-3}$; case 2a, $\delta n_0={\beta }_0=0$ and the wave amplitude is $\delta P_0={10}^{-3}$; and case 2b, $\delta P_0={\beta }_0=0$ and the wave amplitude is $\delta n_0={10}^{-3}$, such that the wave amplitude is always ${10}^{-3}$. The numerical results are represented with dashed lines and points, while the analytic results corresponding to Eq. (6.6) are represented using solid lines. The analytic and numerical curves are indistinguishable.