How to derive fluid equations for electrons from the linearised kinetic equations?

Question

Starting from the linearised kinetic equation for electrons and ignoring perturbations of the ion distribution function completely, how can I work out the fluid equations for electrons (i.e., the evolution equations for the electron density $n_e$ and velocity $u_e$)?

Answer 1

The first thing to start with are the velocity moments (e.g., see https://physics.stackexchange.com/a/218643/59023). Then start taking velocity moments of the Vlasov equation for each particle species to derive things like the continuity equation etc. You most likely only need up to the second velocity moment (i.e., energy flux density). You will have three equations for each particle species (i.e., electrons and ions) now, so six total equations. You will also need an equation of state which will close the velocity moments (i.e., without this, you would need to integrate indefinitely up to infinite velocity moments to properly define each since the lower moments depend upon terms from the higher moments). Typically one assumes some form of an ideal gas for the pressure term at this point.

Starting from the linearised kinetic equation for electrons and ignoring perturbations of the ion distribution function completely, how can I work out the fluid equations for electrons (i.e., the evolution equations for the electron density $n_{e}$ and velocity $u_{e}$?

Now that you have the basic equations of motion, assume all electron terms go as $Q = Q_{o} + \delta Q$, where the $Q_{o}$ terms are the quasi-static, stationary terms and the $\delta Q$ terms are proportional to $e^{i \left( \mathbf{k} \cdot \mathbf{x} - \omega t \right)}$, where $\mathbf{k}$ is some wave vector, $\mathbf{x}$ is a vector position, $\omega$ is a fluctuation angular frequency, and $t$ is a time. You also assume that all ion terms go as $Q = Q_{o}$.

From this you can see that the ion terms will not survive and differentiation, whether partial or total since we assume the $Q_{o}$ terms are the quasi-static in spatial and temporal dimensions. There will be some second order terms (e.g., $\mathbf{u}_{e} \cdot \nabla \mathbf{u}_{e}$) that can be dropped in the linear limit as well but the rest should follow in a straight forward manner.