3
In the previous section, we examined a model in which there were an
infinite number of roots 
of either sign. The numerical work
indicates that the root 
controls the
asymptotic exponential behaviour downstream from the source, and that
controls the behaviour upstream. In this section, we discuss the
time dependence analytically, and show how the steady state solution is
established. It will be seen that the positive and negative branches of
the roots control the swarm behaviour upstream and downstream of the source
respectively.
Let us initially model the situation with a one dimensional time dependent diffusion equation with a constant source switched on at time t=0:
The solution can be found by integrating the shifted Gaussian
solution (
) with respect to time [Abramowitz and
Stegun
[1965] 7.4.33]
:
The error functions are nearly constant over most of the real
line, but change sharply from one value to another near the
origin. The effect is of wave fronts in the form of error
functions propagating at velocity 
leaving exponential
functions in their wake, as shown in figure .
Figure: Sketch of the density with the important features labelled
This model gives us the dynamical picture. We now must turn to
a full Boltzmann equation theory to determine how
non-hydrodynamic modes enter this picture.
We start with the spatially one dimensional time dependent Boltzmann equation
for a steady source 
switched on at t=0.
This may be formally solved by means of the assumption of a discrete
spectrum [eq. ()] to give
where
We assume that the source has been chosen in such a way that the integral
over k in () is well defined. For example, with the
Klein-Kramers model we may choose a Gaussian source located at z=0:
Upon substituting () and () into
(), we find
To get 
to vanish fast enough as 
for
() to be convergent, we must choose 
.
Assuming 
and 
are analytic functions of k, the integrand in
() is analytic. Also, 
as 
, and so the contour of integration in ()
may be translated by an arbitrary amount. In particular, we may
move the
contour so that it passes through the saddle point 
of
, which in the Klein-Kramers case is -ai. We may then
use the method of steepest descent [Jeffreys (1961)] to evaluate the time
dependent portion of the integral at large times:
The behaviour of f in time will depend critically upon the signs of
. If 
is positive for any n, then the
time dependent part will grow exponentially, and the system will not
approach a steady state. On the other hand, if 
is
negative for all n, then the time dependent term is exponentially
damped, and a steady state is reached. In the Klein-Kramers case,
If 
, then there is no steady state approached (figure
),
otherwise the system does approach a steady state (figs
to ).
Let us consider a system satisfying 
for all n. The
steady state term is given by the integral in (). Since 
and
are analytic, the only singularities of the integrand occur at
the zeros of 
. In the Klein-Kramers
case, there are only two singularities as shown in fig .
The contour of
integration must lie between the poles 
and 
for the time
dependent term to approach zero according to (). As we shall
see, this leads to the term proportional to 
not contributing
to the distribution at positive z and similarly the 
term not
contributing to the distribution at negative z.
Figure:
Contour used for the large z asymptotic argument.
Since in general, as in the Klein-Kramers model,
grows much faster than any exponential as
, it is not possible to evaluate the integral in
() by completing the contour around the positive imaginary half
plane for positive z, and around the negative half plane for negative z.
Instead, we must use a large z asymptotic
argument that is similar to the method described in section 2.6 of
Jeffreys (1961). In this, we complete the contour in the fashion shown in
fig , with 
an arbitrarily large positive
but finite value. We may
now apply Cauchy's residue theorem to obtain
But the absolute value of the second term is
Since 
may be chosen arbitrarily large, the second term must
vanish faster than any exponential as a function of z, and so
By taking 
negative, one can similarly show that
In general, we may state the selection principle thus: the contour passing
through the saddle point of Re
divides the complex plane;
those roots of 
that lie above this contour contribute to the
asymptotic behaviour of 
downstream of the source, and those that
lie below contribute to the asymptotic behaviour upstream of the source.
