In classical transport theory, the transport coefficients have a simple dependence upon the pressure of the neutral gas if the ratio of the electric field to neutral gas density is kept constant. The presence of end effects introduces anomalous pressure dependencies in the experimental data. It is therefore of interest to calculate the pressure dependence of the two end effects discussed here. This is done by dimensional analysis on the Boltzmann equation.
Introduce a parameter 
which scales proportionally with the
neutral gas density, and, since 
is constant, the
field. As the collision operator 
is proportional to
the neutral gas density, the scaled operator is 
and
has eigenvectors 
with eigenvalues 
. Upon identifying the
transport coefficients with the multipole coefficients of the
lowest eigenvalue, we see
that 
.
The non-hydrodynamic part of the density is given explicitly by
(
).
For a delta function initial pulse, the initial
Fourier transformed phase space distribution does not depend on
, and so 
or 
. Substituting these relations into
equation (), we can obtain the scaling for the spatial
density function 
and similarly, the collector current scaling
.
If we differentiate this expression with respect to t, and solve for 
,
we find
and so the end effects scale as 
. This
property is obeyed by equation () and () for 
and 
.
The experimental data of England and Elford [1987] for
(called 
in their paper) seem to indicate that
is almost independent of pressure. This discrepancy is
surprising in view of the clear nature of the pressure
dependence derived in this section. One must conclude that
other effects must play a significant rôle in the total
end-effect. One such effect might be an error in the initial
position of the swarm. For example, a delta function pulse of
the form 
, where 
is the
true centroid of the initial pulse, will give rise to a pressure
independent component of 
.