Vlasov-Fokker-Plank(VFP) equation from Boltzmann equation

[Pedro de la Torre]

Dear community,
I am studying some equations related to the acceleration of cosmic rays(CRs) in magnetized plasma and I have seen a couple the equations I am not able to understand.
First, I see that it is used as time-dependent Boltzmann equation for the CRs

∂ƒ/∂t + (v_x + u)∂ƒ/∂x - ∂u/∂xp_x∂ƒ/∂ p_x + F_L/a ∂ƒ/∂p = C(f)
​

, where u is the background plasma velocity, F_L is the Lorentz force and C(f) is the collision term.
The problem is that I have always seen this equation as ∂ƒ/∂t + (v_x + u)∂ƒ/∂x + F_L/a ∂ƒ/∂p = C(f), this is, without the term - ∂u/∂x p_x ∂ƒ/∂p_x that I am not able toderive.

Then, this equation is linearized supposing that the distribution function, f = f₀ + f₁ p/p to obtain the VFP equation:

u∂ƒ₀/∂x + c/3 ∂ƒ/∂x -1/3 ∂u/∂x p ∂ƒ₀/∂p_x
​

How do we arrive to this expression? It is clear for me that the steady state solution is taken, so ∂ƒ/∂t = 0, but I do not even understand why the factor f_x appears there (f_x is the x component of the vector f₁).

Could someone explain me the first "modified" Boltzmann eq. and how to reach the expression of the VFP eq?

Thank you in advance.

 

[jim mcnamara]

It might help if you tell us where you read this, which would help with context, e.g., a paper or textbook. Thanks.

 

[Pedro de la Torre]

It might help if you tell us where you read this, which would help with context, e.g., a paper or textbook. Thanks.

The paper in which I have seen that is: A. R. Bell, Cosmic ray acceleration (https://www.sciencedirect.com/science/article/pii/S0927650512001272)
And what I have called "modified" Boltzmann equation is the equation 1 of the paper.

 

[vanhees71]

I think that term comes from the strange fact that they refer to $f(x,\vec{p})$ being defined in the local fluid rest frame, i.e., the CR momenta $\vec{p}$ are taken in the fluid rest frame. That's unusual in relativistic transport theory, where one usually uses one "computational frame", and $f(x,\vec{p})$ is a Lorentz scalar function anyway (by definition!). For an intro to relativistic transport theory, see

https://th.physik.uni-frankfurt.de/~hees/publ/kolkata.pdf