# Deriving Drude Theory from Plasma Fluid Equations

Does anyone have experience in looking at Drude theory from the perspective of plasma physics instead of the standard, condensed-matter, "electrons in a metal" sort of thing and can point the way? I'm a grad student working on this topic for my masters thesis and I've spent the last hour or so reading the "Plasma Fluid Theory" section of this website, but the math is going over my head and I feel lost. The Ph.D student who is helping me has said I should keep three things in mind:

1. The plasma is cold, $$P = 0$$
2. Density deformations are small, $$\dfrac{dn}{dt} = 0$$
3. The ions are stationary, microwave frequency >> ion plasma frequency

It seems like there are so many different ways to formulate a fluid theory of plasmas and I can't get grasp what the right starting point is.

• Is this along the lines of what you wanted? aanda.org/articles/aa/full_html/2013/06/aa20738-12/… Jun 22, 2020 at 11:41
• I think the Holy Grail I'm looking for is a textbook that uses plasma physics to develop the theory of metals starting with Drude theory. Unfortunately, the article you linked seems to be full of cross-sections flying everywhere so it does not appear to be what I am looking for. I'm still focused on the basics so I'm looking more for derivations at this point. Jun 22, 2020 at 19:45
• But I think your disconnect may be in thinking that plasmas behave like metals. In plasmas, you cannot ignore long-range interactions and the ions can move. Jun 22, 2020 at 20:52
• I'm trying to approach from the other direction though, the one where I examine how metals behave like plasmas Jun 22, 2020 at 21:47

I once derived something similar starting from the momentum equation of a multi-fluid model: $$\begin{equation} m_{\alpha}n_{\alpha}\left[\partial_{t}\mathbf{v}_{\alpha}+\left(\mathbf{v}_{\alpha}\cdot\nabla\right)\mathbf{v}_{\alpha}\right]=n_{\alpha}q_{\alpha}\left(\mathbf{E}+\mathbf{v}_{\alpha}\times\mathbf{B}\right)-\nabla\cdot\boldsymbol{P}_{\alpha}+\sum_{\beta}\mathbf{R}_{\alpha\beta}\label{eq:2fluid_1} \end{equation}$$ Where $$m$$ and $$q$$ denote mass and charge of the fluid elements respectively. This equation decribes the conservation of momentum for each particle species $$\alpha$$ (i.e. Newton's 2nd law). On the left hand side we have inertial forces due to temporal or convective changes. On the right hand side we have forces acting on the fluids. From left to right these are the Lorentz force, internal forces described by the pressure tensor (pressure or viscosity) and a term which allows the momentum transfer from species $$\alpha$$ to species $$\beta$$.

Depending on your exact situation you can now cross terms out if they are not that important, like in your case the pressure. Since I am working on magnetic confinement in fusion devices it was save for me to neglect the whole left side since it is way smaller in these conditions than the vxB term on the right. Depending on your situation you may do the same.

If we assume that the plasma is made of electrons and one species of ions, the term respecting transfer of momentum to electrons by electron-ion collisions can be written as

$$\begin{equation} \mathbf{R}_{ei}=-m_{e}n_{e}\nu_{ei}\left(\mathbf{v}_{e}-\mathbf{v}_{i}\right), \end{equation}$$

where $$\nu_{ei} = \tau_{ei}^{-1}$$ is the collisionality or inverse collision time. This is called friction force and already looks similar to a current $$\mathbf{J}=-n_{e}e\left(\mathbf{v}_{e}-\mathbf{v}_{i}\right)$$. With only electrons and ions you will have 2 version of the momentum equation above, one for each species. This is usually called 2-fluid model.

I think you may use this as a starting point since there is now a current and the electric field in the equations and the combination of constants will make up a conductivity

$$\begin{equation} \sigma =\frac{n_{e}e^{2}}{m_{e}\nu_{ei}}, \end{equation}$$

in the end.

• What would the momentum transfer look like between electrons and neutrals? Jun 21, 2020 at 0:15
• I am not 100% sure but I think it does not really make sense for such a momentum transfer. Since the validity for fluid theory is restricted to highly collisional plasmas, neutrals would immediately strip off their electrons and become ions. Usually the plasma neutral interactions are described with plasma kinetic theory which is more complicated but not restricted to high collisionality. Jun 24, 2020 at 6:50