Electrostatic plasma wave ($\delta \mathbf{E}=-i\mathbf{k}\varphi$)

A plasma wave dominated by a longitudinal electric field. The leading dispersion relation is $\mathbf{k}\cdot \mathbf{ϵ}\cdot \mathbf{k}=0$. In a cold magnetized plasma, this becomes $k_{\perp }^{2}S+k_{\parallel }^{2}P=0$. An exactly longitudinal finite-wavenumber solution must also satisfy the full Maxwell equations; the oblique electrostatic condition is usually an asymptotic large-refractive index approximation.

For an isotropic completely degenerate nonrelativistic Fermi gas in classical-trajectory collisionless kinetic theory, the parallel marginal distribution is $F(u)=3n(v_F^2-u^2)1_{|u|<v_F}/(4v_F^3)$. Integrating $F^{\prime }(u)/(u-\omega /k)$ yields the electron dielectric function in the title for fixed ions. Its real roots outside the occupied interval have no Landau damping. Quantum recoil, collisions and finite-temperature tails require an enlarged model.

The short-wavelength root of the degenerate-plasma dielectric response, approaching the edge of the occupied parallel velocity interval from outside. Writing $s=1+\delta$ makes its logarithmic singularity give the exponentially small positive correction in the title. It is the high-wavenumber limit of the charged Langmuir wave branch in this approximation; neutral zero sound has a different interaction and long-wavelength limit.

Rapid parallel thermal electron streaming can approximately balance electrostatic and isothermal pressure forces, giving $n_e\propto e^{e\varphi /T_e}$ and its linear form in the title. This assumes a suitable connection along the magnetic field, $|\omega |\ll k_{\parallel }{v}_{\mathrm{th},e}$, and an approximately electrostatic branch. A spatially constant potential on a closed line requires its own particle-number constraint.