Transport Coefficients of Degenerate Plasma

An improved nonrelativistic calculation, based on the quantum Lenard-Balescu transport equation, is performed for the thermal and electrical conductivity of a plasma of highly degenerate, weakly coupled electrons and nondegenerate, weakly coupled ions. Dynamic shielding in the random-phase approximation is treated correctly, and both electron-ion ($\mathrm{ei}$) and electron-electron ($\mathrm{ee}$) collisions are included in the thermal conductivity. The argument that $\mathrm{ee}$ collisions are negligible, because the Pauli exclusion principle limits their effect to ${\left(\frac{\mathrm{kT}}{E_F}\right)}^2$ less than that of $\mathrm{ei}$ collisions, is refuted in the case of the thermal conductivity. For temperatures of about ${10}^8$°K and densities of ${10}^{26}$ to ${10}^{30}$ electrons/cc, appropriate to red-giant stellar cores, $\mathrm{ee}$ collisions reduce the thermal conductivity by 25 to 50%. However, $\mathrm{ee}$ collisions are insignificant in terrestrial solids. The thermal conductivity $\kappa$ is given by $\frac{1}{\kappa }=\frac{1}{{\kappa }_{\mathrm{ei}}}+\frac{1}{{\kappa }_{\mathrm{ee}}}$, where ${\kappa }_{\mathrm{ei}}$ and ${\kappa }_{\mathrm{ee}}$ are conductivities determined by $\mathrm{ei}$ and $\mathrm{ee}$ collisions. ${\kappa }_{\mathrm{ei}}\propto \frac{\mathrm{Tn}}{\left[\ln \right(\frac{1}{{\lambda }_i})+C_{\mathrm{ei}}]}$, where ${{\lambda }_i}^2\ll 1$ is the ion weak-coupling parameter, and the correction $C_{\mathrm{ei}}$ involves dynamic shielding. If ${\lambda }^2\ll 1$ is the electron weak-coupling parameter, and $\gamma \equiv \frac{4\lambda E_F}{\mathrm{kT}}\gg 1$, then ${\kappa }_{\mathrm{ee}}\propto {\lambda }^3{n}^{\frac{7}{3}}{T}^{-1\mathrm{}}$, instead of the usual logarithmic form. If $\gamma \ll 1$, then ${\kappa }_{\mathrm{ee}}\propto \frac{T^2{n}^{\frac{1}{3}}}{\left[\ln \right(\frac{1}{\gamma })+C_{\mathrm{ee}}]}$, with the temperature dependence contrary to Fermi-liquid theory.