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

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 + (vx + u)∂ƒ/∂x - ∂u/∂xpx∂ƒ/∂ px + FL/a ∂ƒ/∂p = C(f)
, where u is the background plasma velocity, FL is the Lorentz force and C(f) is the collision term.
The problem is that I have always seen this equation as ∂ƒ/∂t + (vx + u)∂ƒ/∂x + FL/a ∂ƒ/∂p = C(f), this is, without the term - ∂u/∂x px ∂ƒ/∂px 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/∂px
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 fx appears there (fx 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?

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

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
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