Elements of plasma kinetic theory, Bittencourt

[Fernando Mourao]

Homework Statement

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.

Homework Equations

∂f/∂t + v⋅∇f + a[∇][/v]f = 0

a= dv/dt
v=dr/dt

The Attempt at a Solution

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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.

 

[DrClaude]

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.

 

[Fernando Mourao]

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.

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/dv_x.
But my problem is the mathematical proof. How to explicitly show that:

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

v and a being vectors.