Solving More Difficult Partial Differential Equations (Pt. 4): General Solution of the 1D Vlasov Equation

 In Plasma Physics' comprehensive PH D exams, one 's first problem generally starts with asking for the solution of the 1D  Vlasov-Boltzmann equation:

¶ f/ ¶ t + v  · Ñ x f +  v'  Ñ v f = 0

A time-independent solution is generally required and I show here how to proceed. Assuming constants of the motion, then df/dt =0, so we have these conditions for the zero order Vlasov equation:

i) f =   f _o  + f ₁

ii) E =   E _o  + E ₁

_{iii) B=   Bo  + B1}

 (¶ f/ ¶ t)_c     =  0

Then we can write:

v Ñ x   f _o + q/ m (E _o + v x B_o ) · Ñ v  f _o  =  0

The total derivative  of   f _o following the zero order or unperturbed orbit is:

(d f _o / dt) _o  = 0

For the first order equation we have:

¶ f ₁ / ¶ t  +  v Ñ x  f _o + q/ m ( E _o + v x B_o ) · Ñ v  f ₁  

= - q/ m ( E ₁ + v x B₁ ) · Ñ v f _o

Þ   

( d f ₁  / dt ) _o   =   - q/ m ( E ₁ + v x B₁ ) · Ñ v f _o

We now need to specify the zero order orbit in phase space, viz.

X' (t) = V (t)

V (t) =    q/ m [F ₀ (X, t) + V (t)  x B _0..(X, t) ]

Then:   X(t') =  x(t) +  ò ^t1  _t  X(t'') dt''

 V(t') =  v(t) +  ò ^t' _t  v(t'') dt''

We need to follow the prescription to solve for  f ₁  by integrating both sides of the earlier boxed equation, e.g.

 ò ^t_-¥    ( df ₁ / dt ) _o   =  ò ^t_-¥   RHS dt'   (zero order orbit)

This leads to the general solution of the linear Vlasov equation:

 f ₁  (x, v, t) =  f ₁ [X(t'), V(t'), t' =  (-∞)] =

- q/m ò ^t_-¥ dt{E ₁ [X(t'), t'] +  V(t') x  B ₁[X(t'), t'] · Ñ v f ₀

Suggested Problem:

Show that:

Ñ x · v = 0   and:  Ñ v · (E  + v x B )  = 0

And write the resulting partial differential equation.