As a way of obtaining insight into how the behaviour of swarms are related to various mathematical features of the Boltzmann equation, it is useful to have an exactly solvable model. The models most often employed in transport theory are briefly reviewed in section 13 of Kumar et al. [1980]. The model that reflects best the theory developed so far is the Rayleigh gas model, which describes Brownian motion.
Brownian motion may be considered to be a limiting case of swarms in which the diffusing particle is very much more massive than the background fluid molecules. This phenomenon is described in stochastic theory by the master equation, which is an integral equation, and an equivalence with a differential form, known as the Kramers-Moyal expansion. This expansion is often truncated at second order, where it is known as the Fokker-Planck equation, which is essentially exact for Brownian motion. A general introduction to this subject can be found by Risken [1984], and in a kinetic theory context by Braglia [1980].
The analogy of Brownian motion and swarm physics suggests that a
differential
form of the collision operator can be found in which the ratio 
plays
the part of a small expansion parameter, where M is the background gas
molecule
mass, and m the charged particle mass. Kumar et al.
[1980] develop this
expansion, which they call the Fokker-Planck expansion, and another
expansion
valid when 
is small.
If we make the approximation of the background gas molecule velocities being much larger that the charged particle velocities, and truncate the Fokker-Planck expansion at second order, we obtain the Klein-Kramers equation:
The reaction rate 
is positive
when ionization occurs. The coefficients 
and 
are related
to
physical quantities by
where 
is Boltzmann's constant and T the background gas
temperature.
The
collision frequency 
is approximately the product of the swarm drift velocity,
the background gas number density, and the collision cross section.
We look for a solution to the Klein-Kramers equation in the form
where the Greens function G satisfies
Taking the Fourier transform of (
) with respect to
position, we find:
The operator 
can be
transformed into a hermitian operator by means of the similarity
transformation 
, where 
.
This has a discrete spectrum with a complete set of eigenfunctions:
The Green's function can then be written using a spectral expansion
Making the change of variables
the eigenvalue equation () for 
is transformed to the harmonic oscillator problem
familiar from quantum mechanics:
This equation is separable in cartesian coordinates 
,
, 
and has solutions in terms of Hermite polynomials
[Abramowitz and Stegun
[1965] 22.6.20]
:
where 
, 
and 
are whole numbers, and n is the
cartesian triple 
. These solutions can also be
expressed in spherical coordinates by means of Burnett functions
[Kumar
[1980]]
. In one dimension, the result is
identical
except that g is now a scalar, and the index set 
is
the set of whole numbers.
The Greens function may now be evaluated by substituting
() and () into ():
The sum over n may be evaluated using a generating function for Hermite polynomials [Erdélyi [1954] 10.13.22] :
Thus we obtain
The inverse Fourier transform of 
is a Gaussian
integral, so we obtain
where 
and
In one dimension, the exponent on the second factor in
() is 
, i.e.
Figure () shows the one dimensional density distribution
plotted at successive values of t, from an initial delta
function pulse, i.e.
In the r-t plane, the path that the centroid traces out is plotted
(dotted
curve) and can be compared with the path expected if the system were
hydrodynamic with a constant drift velocity (dashed curve).
Non-hydrodynamic effects manifest themselves within time 
of the origin, with a residual constant retardation of the
swarm at large times. This was an effect first observed by Harris
[1981],
who noted the ``non-diffusive'' nature of the Klein-Kramers equation at
large
times. Titulaer [1983] explained the phenomenon using a
detailed
analysis of the Klein-Kramers equation. This effect has been discovered a
number of times in different contexts, for example Robson [1975]
noted
the effect in the BGK model. However, it was generally ignored as not
having
physical consequences. I rediscovered this phenomenon
independently, and showed that it should occur under quite general
circumstances and have measurable consequences for swarm experiments.
More will be said on this in chapter .
Figure: Evolution of the one dimensional Klein-Kramers equation showing
non-hydrodynamic effects at large times
