r ,.
t + dt
Fig. 4 In the absence of collisions the particles within the volume element d r d3v about (r,v), at an instant t, will occupy after a time interval dt a new volume element d3r' d3v', about (r',V).
r'{t + dt) =r{t) + v dt
v'(t + dt) - v(t) + adt d3r' dV = | J| <f3r d3v
Show that: \J \ = 1
5.6 Show that the Vlasov equation for a homogeneous plasma under the influence of a uniform external magnetostatic field Bo, in the equilibrium state, is satisfied by any homogeneous distribution function, f{vt,v±), which is cylindrically symmetric with respect to the magnetostatic field.
5.7 The entropy of a system can be expressed, in terms of the distribution function, as
S=-kJ j / ln(/) d:!r d\'
Show that, for a system that obeys the collisionless Boltzmann equation, the total time derivative of the entropy vanishes.
5.2 Consider the following two-dimensional Maxwellian distribution function:
2kT
(a) Verify that no represents correctly the particle number density, that is, the number of particles per unit area.
(b) Sketch, in a three-dimensionsal perspective view, the surface for this distribution function, plotting f{vx,vy) in terms of vx and vy. Draw, on this surface, curves of constant i/z, curves of constant vy, and curves of constant /.
7.9 Consider the particles in the Earth's atmosphere under equilibrium conditions in the presence of the Earth s gravitationl field. Assume a horizontally stratified (x,y plane) atmosphere with constant temperature T and consider a constant value g = -gz for the acceleration due to gravity. Derive an expression for the number density na(z) as a function of height 2, for the type a species, in terms of the number density na(zo) at a base level zq and of the scale height HQ = kT/mag. How is the expression for nQ(z) modified, when T and g vary with height?