Dear community,(adsbygoogle = window.adsbygoogle || []).push({});

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, where u is the background plasma velocity, F_{x}+ u)∂ƒ/∂x - ∂u/∂xp_{x}∂ƒ/∂ p_{x}+F_{L}/a ∂ƒ/∂p= C(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 = f_{0}+f_{1}p/p to obtain the VFP equation:

u∂ƒHow 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_{0}/∂x + c/3 ∂ƒ/∂x -1/3 ∂u/∂x p ∂ƒ_{0}/∂p_{x}

_{x}appears there (f_{x}is the x component of the vectorf)._{1}

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

Thank you in advance.

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

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