# Homework Help: Elements of plasma kinetic theory, Bittencourt

Tags:
1. Sep 23, 2016

### Fernando Mourao

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. Sep 23, 2016

### 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.

3. Sep 23, 2016

### Fernando Mourao

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.

Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted