Causal relativistic hydrodynamics of conformal Fermi-Dirac gases

In this paper we address the derivation of causal relativistic hydrodynamics, formulated within the framework of divergence type theories (DTTs), from kinetic theory for spinless particles obeying Fermi-Dirac statistics. The approach leads to expressions for the particle current and energy momentum tensor that are formally divergent, but may be given meaning through a process of regularization and renormalization. We demonstrate the procedure through an analysis of the stability of an homogeneous anisotropic configuration. In the DTT framework, as in kinetic theory, these configurations are stable. By contrast, hydrodynamics as derived from the Grad approximation would predict that highly anisotropic configurations are unstable.

Figure 1

A graphic of the $p{p}_z$-plane showing the singularities at ${\theta }_0$ in dashed lines as well as a transverse cut of the surface defined by the equation $p-p^2(1-3{\cos }^2\theta )=0$ as an example.

Figure 2

Comparison between $F_1$ (full line) and its Gaussian approximation ${\mathcal{F}}_1$ (dashed line) for different ${\zeta }_0$ values.

Figure 3

Integral of ${\mathcal{F}}_1$ over [0, 1]. ${\zeta }_0$ axis in logarithmic scale.

Figure 4

Comparison between $F_2$ (full line) and its Gaussian approximation ${\mathcal{F}}_2$ (dashed line) for different ${\zeta }_0$ values.

Figure 5

Integral of ${\mathcal{F}}_2$ over [0, 1]. ${\zeta }_0$ axis in logarithmic scale.

Figure 6

Left: pressure anisotropy asymptotic behavior. ${\zeta }_0$ axis in logarithmic scale. Right: zoom at the region of interest. Vertical line at ${\zeta }_0=1/2$ and horizontal line at ${\delta }_p=1/7$. DTT as a dashed line and Grad’s approximation as a dot-dash-dotted line.

Figure 7

$\tau k_{\max }$ as a function of ${\zeta }_0={\xi }_0$. It shows kinetic theory as a full line, DTT as a dashed line and Grad’s approximation as a dot-dash-dotted line. $\tau k_{\max }$ axis in logarithmic scale.

Figure 8

Dispersion relation for ${\zeta }_0={\xi }_0=0$. It shows kinetic theory as a full line, DTT as a dashed line and Grad’s approximation as a dot-dash-dotted line.

Figure 9

Dispersion relation for different ${\zeta }_0={\xi }_0$ values. It shows kinetic theory as a full line, DTT as a dashed line and Grad’s approximation as a dot-dash-dotted line.

Figure 10

Dispersion relation in a neighborhood of Grad’s instability. It shows kinetic theory as a full line, DTT as a dashed line and Grad’s approximation (where ${\xi }_0={\zeta }_0$) as a dot-dash-dotted line.

Figure 11

$\mathrm{\Omega }$ for both Grad and DTT. Logarithmic ${\zeta }_0$ axis on the left and zoom on the sign change interval on the right.