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Elements of plasma kinetic theory, Bittencourt

  1. Sep 23, 2016 #1
    1. The problem statement, all variables and given/known data
    Consider the motion of charged particles, in one dimension only, in
    the presence of an electric potential V ( x). Show, by direct substitution,
    that a function of the form

    f=f(1/mv^2 + qV)

    is a solution of the Boltzmann equation under steady-state conditions.

    2. Relevant equations

    ∂f/∂t + v⋅∇f + a[∇][/v]f = 0

    a= dv/dt
    v=dr/dt

    3. The attempt at a solution

    I've been having problems with mathematical demonstrations due to lack of practice in the past 6 years. Any steering into the correct direction would be greatly appreciated.
     
  2. jcsd
  3. Sep 23, 2016 #2

    DrClaude

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    Staff: Mentor

    You simply need to replace the f in the left-hand-side of the Boltzmann equation with the function that was given and show that result is indeed 0. Since this is in 1D only, then ∇ = d/dx.
     
  4. Sep 23, 2016 #3
    Thank you for your reply Dr Claude.

    I understand the process of direct substitution and the fact that, in this case, ∇ = d/dx and ∇v=d/dvx.
    But my problem is the mathematical proof. How to explicitly show that:

    ∂/∂t(f(½mv2+qV)) + v⋅∂/∂x(f(½mv2+qV)) + a⋅∂/∂vx(f(½mv2+qV))=0

    v and a being vectors.
     
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