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? Thank you in advance.