Transport coefficients of a relativistic plasma

In this work, a self-consistent transport theory for a relativistic plasma is developed. Using the notation of Braginskii [S. I. Braginskii, in Reviews of Plasma Physics, edited by M. A. Leontovich (Consultants Bureau, New York, 1965), Vol. 1, p. 174], we provide semianalytical forms of the electrical resistivity, thermoelectric, and thermal conductivity tensors for a Lorentzian plasma in a magnetic field. This treatment is then generalized to plasmas with arbitrary atomic number by numerically solving the linearized Boltzmann equation. The corresponding transport coefficients are fitted by rational functions in order to make them suitable for use in radiation-hydrodynamic simulations and transport calculations. Within the confines of linear transport theory and on the assumption that the plasma is optically thin, our results are valid for temperatures up to a few MeV. By contrast, classical transport theory begins to incur significant errors above $k_{B}T\sim 10$ keV, e.g., the parallel thermal conductivity is suppressed by 15% at $k_{B}T=20$ keV due to relativistic effects.

Figure 1

The geometry used in our description of transport, as per Eq. (18). The tensorial transport coefficients are described by their components in the $\mathbf{b}$ $\left({}_{\parallel }\right)$, $\mathbf{b}\times (\mathbf{s}\times \mathbf{b})$ $\left({}_{\perp }\right)$ and $\mathbf{b}\times \mathbf{s}$ $\left({}_{\wedge }\right)$ directions.

Figure 2

The relativistic transport coefficients as a function of ${\omega }^{☆}{\tau }^{☆}$ in the case of a Lorentzian plasma for $\mathrm{\Theta }=0$ (black, solid), $\mathrm{\Theta }=0.01$ (black, dash), $\mathrm{\Theta }=0.1$ (black, dot-dash), $\mathrm{\Theta }=1$ (blue, dot-dash), $\mathrm{\Theta }=10$ (blue, dash), and $\mathrm{\Theta }=\infty$ (blue, solid).

Figure 3

The relativistic transport coefficients as a function of ${\omega }^{☆}{\tau }^{☆}$ in the case of a $Z=1$ plasma for $\mathrm{\Theta }=0$ (black, solid), $\mathrm{\Theta }=0.01$ (black, dash), $\mathrm{\Theta }=0.1$ (black, dot-dash), $\mathrm{\Theta }=1$ (blue, dot-dash), $\mathrm{\Theta }=10$ (blue, dash), and $\mathrm{\Theta }=\infty$ (blue, solid).

Figure 4

Plot of fractional error in fits to (a) ${\alpha }_{\perp }$, (b) ${\alpha }_{\wedge }$, (c) ${\beta }_{\perp }$, (d) ${\beta }_{\wedge }$, (e) ${\kappa }_{\perp }$, (f) ${\kappa }_{\wedge }$ as a function of ${\omega }^{☆}{\tau }^{☆}$ in the case of a Lorentzian plasma $\left(Z\to \infty \right)$ for $\mathrm{\Theta }=0$ (black, solid), $\mathrm{\Theta }=0.01$ (black, dash), $\mathrm{\Theta }=0.1$ (black, dot-dash), $\mathrm{\Theta }=1$ (blue, dot-dash), $\mathrm{\Theta }=10$ (blue, dash), and $\mathrm{\Theta }=\infty$ (blue, solid).

Figure 5

Relativistic correction factors in the case of (a) ${\alpha }_{\perp }$, (b) ${\alpha }_{\wedge }$, (c) ${\beta }_{\perp }$, (d) ${\beta }_{\wedge }$, (e) ${\kappa }_{\perp }$, (f) ${\kappa }_{\wedge }$ as a function of the nonrelativistic Hall parameter $\psi =\omega \tau$ and reduced temperature $\mathrm{\Theta }$ for a $Z=1$ plasma.