I am attempting to learn about Coulomb collisions in the kinetic theory of plasmas. In doing so, I have come up with an "intuitive" derivation of a Coulomb collision operator. However, the collision operator I derive is not the same as the one described in textbooks. Could you help describe why my reasoning is wrong?
One glaring error is that my collision operator would always be zero if the distribution of the background particles being collided against does not depend on position, like a Maxwellian.
A second problem is that the collision operator in some textbooks, such as Helander and Sigmar, is expressed as a Taylor expansion in terms of changes in velocity $(\Delta v)$:
$$C(f_a,f_b) = -\frac{\partial}{\partial v} \left(\frac{\langle \Delta v\rangle }{\Delta t}f_a \right) + \frac{\partial^2}{\partial v^2} \left(\frac{\langle (\Delta v)^2\rangle }{2\Delta t}f_a \right)$$
The derivation I show expresses the result in terms of an acceleration, and does not have the second term altogether.
Derivation
Suppose that we have a plasma with distribution function $f_a(x,v,t)$ under macroscopic electric and magnetic fields, $E$ and $B$. If another distribution of charged particles $f_b$ is present, then the particles from $f_a$ should undergo "microscopic" Coulomb forces caused by particles from $f_b$ coming within a Debye length of particles from $f_a$.The Coulomb force between the two particles is $F_c(x,x') = \frac{q_aq_b(x-x')}{4\pi\epsilon_0||x-x'||^3}$ for particles with positions $x,x'$. If a single particle $b$ from distribution $f_b$ is close enough to particle $a$ from $f_a$, then the total force on particle $a$ is $$F = F_L + F_c$$ where $F_L = q(E + v\times B)$ is the Lorentz Force. The average force experienced by $a$ from interactions with many particles $b$ is $$\bar{F}(x,v) = F_L(x,v) + \bar{F}_c(x)$$ where $\bar{F}_c(x) = \int F_c(x,x')f_b(x',v',t)dx' dv'$. Notice that $\bar{F}_c = 0$ if $f_b$ is a a distribution that does not depend on $x'$, such as a Maxwellian!
To get the collision operator, we revisit the derivation of the Fokker-Planck equation. The Fokker-Planck equation says that the time derivative of the probability mass of $f_a$ inside a phase space volume is the flux of particles out of that volume (ignoring sources/sinks). Under the macroscopic and microscopic forces, the flux in phase-space is $\nabla \cdot \dot{z}f_a(x,v,t)$ where $\dot{z} = (\dot{x},\dot{v}) = (v, \frac{\bar{F}}{m})$ is the time derivative of the phase space coordinate $z = (x,v)$.
By this logic, the Fokker-Planck equation is $$ \frac{\partial f_a}{\partial t} + \nabla \cdot (\dot{z}f_a) = 0$$ which expands to $$ \frac{\partial f_a}{\partial t} + \nabla \cdot f_a\begin{pmatrix}v \\ \frac{1}{m}F_L\end{pmatrix} = -\nabla\cdot f_a \begin{pmatrix}0 \\\frac{1}{m}\bar{F}_c \end{pmatrix}$$ where $m$ is the mass of particle $a$. We can then define collision operator as $C(f_a, f_b) = -\nabla_v\cdot \frac{1}{m}\bar{F}_cf_a$.
Since $\bar{F}_c$ is zero when $f_b$ is a Maxwellian, the Collision operator becomes zero when $f_b$ is Maxwellian. I know this should not be the case, but it is somewhat intuitive since if collisions occur isotropically they should have no effect.
