In this chapter, a non-hydrodynamic theory of the parallel plane
steady state Townsend experiment is developed relating the
asymptotic properties of the steady state solution to the distribution of
zeros of the eigenvalues 
of the inhomogeneous Boltzmann
operator 
. It was found that
non-hydrodynamic effects are clustered around the source, and decay
exponentially away from the source.
The spectrum of 
was taken
to be discrete and the eigenvalues and eigenfunctions taken to be analytic
functions of k. These assumptions were chosen to make the
theory simple, and are sufficient for understanding the simple
model used in this work. However, in general, the situation may
be more complex. Consider what would happen if 
from
equation (
) is not analytic everywhere, but has a singularity at
say 
. If this singularity is a pole, then the effect is of an additional
term that behaves like exp
in the steady state solution. If,
however, the singularity is part of a branch cut, then there is an
additional term whose form is not generally exponential, but will be
bounded asymptotically by exp
.
A similar situation arises if the spectrum contains a continuous portion.
Here we might expect that the sum over n is replaced by the integration
over a continuous parameter 
:
If Re
is always less than some value Q, then 
will be bounded by an exponential of the form exp
. Needless to
say, the formal theory of continuous spectra is beyond the
scope of this work. The purpose of mentioning it here it to point out how
this work might be generalized to handle these cases.