Fundamentals of Plasma Physics

2.2 Distribution function and Vlasov equation 31

reverse direction is called a passing particle, while a particle confined to a certain region of
phase-space (e.g., a particle with periodic motion) is called a trapped particle.

2.2 Distribution function and Vlasov equation

At any given time, each particle has a specific position and velocity. We can therefore char-
acterize the instantaneous configuration of a large number of particles byspecifying the
density of particles at each pointx,vin phase-space. The function prescribing the instan-
taneous density of particles in phase-space is called thedistribution functionand is denoted
byf(x,v,t).Thus,f(x,v,t)dxdvis the number of particles at timethaving positions in
the range betweenxandx+dxand velocities in the range betweenvandv+dv.As time
progresses, the particle motion and acceleration causes the number of particles in thesex
andvranges to change and sofwill change. This temporal evolution offgives a descrip-
tion of the system more detailed than afluid description, but less detailed than following
the trajectory of each individual particle. Using the evolution offto characterize the sys-
tem does not keep track of the trajectories of individual particles, but rather characterizes
classes of particles having the samex,v.

x

v

dx

dv

Figure 2.2: A box with in phase space having widthdxand heightdv.

Now consider the rate of change of the number of particles inside a small boxin phase-
space such as is shown in Fig.2.2. Defininga(x,v,t)to be the acceleration of a particle,
it is seen that the particleflux in the horizontal direction isfvand the particleflux in the
vertical direction isfa.Thus, the particlefluxes into the four sides of the box are:

1. Flux into left side of box isf(x,v,t)vdv


2. Flux into right side of box is−f(x+ dx,v,t)vdv


3. Flux into bottom of box isf(x,v,t)a(x,v,t)dx


4. Flux into top of box is−f(x,v+ dv,t)a(x,v+ dv,t)dx
   The number of particles in the box isf(x,v,t)dxdvso that the rate of change of parti-
   cles in the box is