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

  • #1
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.
 

Answers and Replies

  • #2
jim mcnamara
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It might help if you tell us where you read this, which would help with context, e.g., a paper or textbook. Thanks.
 
  • #4
vanhees71
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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
 
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