England and Elford [1987] discuss the end-effects in mobility
measurements under five headings; those of contact potentials,
non-hydrodynamic
effects, higher-order diffusion effects, field interpenetration and boundary
effects. The first effect is simply an error in the measured value of the
drift potential, which introduces an uncertainty proportional to 
in the
the electric field. In this chapter, I have examined the second two effects,
and showed that they give rise to corrections in the form of a power series in
of about the same magnitude as observed in experiment.
The final two effects are due to inhomogeneities in the field caused by the
shutters being imperfect, and those due to the selective removal of particles
incident on the boundaries of the apparatus. The theory is formally the same,
and it would appear that either effect is quite difficult to describe.
Formally we would include these effects into the Boltzmann equation by writing
an additional operator 
which repesents the loss of particles to the
boundary or the scattering of particles on a field inhomogeneity. This
operator is localized, and may be idealized as being proportional to a surface
delta function.
If one has the free space Greens function 
(e.g. eq (
)), then one can write the complete Greens function in the
presence of a boundary as a Dyson equation: [Kumar
[1984]]
If we consider an operator 
representing a completely impenetrable
barrier, then the two regions must be causally unconnected, i.e. 
where 
and 
lie on opposite sides of the
barrier. However, this would imply that G is not an analytic function, and
so
that the approximation () in terms of analytic functions is doomed
at best to be slowly convergent. Furthermore, the correction terms to the
free
space Greens function must be of infinite range.
Another method, in which the corrections to the free space solution are localized around the boundary, involves taking linear combinations of free space solutions valid in half spaces on either side of the boundary, and requiring them to satisfy some auxiliary condition at the boundary. A lot of work has been done with this method considering a one-dimensional Klein-Kramers equation with an absorbing or reflecting barrier at the origin. See Selinger and Titulaer [1984] for a review. Even in this simple case, the method is said to be slowly convergent owing to the problems of approximating non-analytic functions. It may be more feasible to consider a leaky barrier, or a field inhomogeneity where presumably this is not a problem, and then to extrapolate the results to an impenetrable barrier. Furthermore, the analytic result of Marshall and Watson [1987] may provide some insight on this particular problem.
As to the original problem of how the boundaries affect swarm measurements, these methods prove to be intractable owing to the complex geometries found inside a typical drift tube. What is needed is a new paradigm in which the form of the boundary effects is independent of the specific details of shutter design etc., in much the same way as thermodynamics is independent of the specific details of molecular motion. Without this, one cannot be satisfied that we completely understand how to correct the swarm data in spite of the successes of the empirical methods used to date.
