Boltzmann electron response ($\delta n_e/n_{0e}=e\varphi /T_e$)

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.