32 Chapter 2. Derivation offluid equations: Vlasov, 2-fluid, MHD
∂f(x,v,t)
∂tdxdv = −f(x+ dx,v,t)vdv+f(x,v,t)vdv
−f(x,v+ dv,t)a(x,v+ dv,t)dx
+f(x,v,t)a(x,v,t)dx(2.1)
or, on Taylor expanding the quantities on the right hand side, we obtain the one dimensional
Vlasov equation,
∂f
∂t+v
∂f
∂x+
∂
∂v(af) = 0. (2.2)It is straightforward to generalize Eq.(2.2) to three dimensions and soobtain the three-
dimensional Vlasov equation
∂f
∂t+v·
∂f
∂x+
∂
∂v·(af) = 0. (2.3)Becausex,vare independent quantities in phase-space, the spatial derivative term has the
commutation property:
v·∂f
∂x=
∂
∂x·(vf). (2.4)The particle acceleration is given by the Lorentz force
a=
q
m(E+v×B). (2.5)Because(v×B)i=vjBk−vkBjis independent ofvi, the term∂(v×B)i/∂vivanishes
so that even though the accelerationais velocity-dependent, it nevertheless commutes with
the vector velocity derivative as
a·
∂f
∂v=
∂
∂v·(af). (2.6)Because of this commutation property the Vlasov equation can also be written as
∂f
∂t
+v·∂f
∂x
+a·∂f
∂v= 0. (2.7)
If we “sit on top of” a particle moving in phase-space with trajectory x=x(t),v=v(t)
and measure the distribution function as we are carried along by the particle, the ob-
served rate of change of the distribution function will bedf(x(t),v(t),t)/dtwhere the
d/dtmeans that the derivative is measured in the moving frame.Becausedx/dt=vand
dv/dt=a, this observed rate of change is
(
df(x(t),v(t),t)
dt)
orbit=∂f
∂t+v·∂f
∂x+a·∂f
∂v= 0. (2.8)
Thus, the distribution function as measured when moving along a particletrajectory (orbit)
is aconstant.This gives a powerful method for finding solutions to the Vlasov equation.
Since the distribution function is a constant when measured in the frame following an orbit,
we can choose it to depend onany quantitythat is constant along the orbit (Jeans 1915,
Watson 1956).