If the timescale of one's interest is of comparable size to the
mean free time, or if the length scale is of the order of the
mean free path, then the assumptions involved in the Chapman-
Enskog procedure break down. Examples of where this is the case
include the behaviour of the fluid near the boundary, or the
short-time evolution of the fluid in an arbitrary initial
configuration. Even if the timescale of interest is large in
comparison with the mean free time, then the hydrodynamic
trajectory that the system follows is not identical to the
hydrodynamic trajectory that passes through the system's initial
point. As we shall see in the case of swarms in chapter
, this difference has a long range persistence,
and can
be quite significant on hydrodynamic timescales.
A further example where the assumptions of the Chapman-Enskog method break down is where the mean free time of the system becomes infinite. This might occur if the collision frequency diminishes sufficiently rapidly with increasing energy, leading to a runaway effect of the fluid becoming increasingly hotter in time [see Waldman and Mason [1981] for a discussion] . This effect has been observed experimentally in swarms [Howorka et al. [1979], Morruzzi and Kondo [1980]] , and has been reviewed by Kumar [1984].
The operator 
introduced earlier controls the decay of
the velocity distribution to its hydrodynamic distribution 
. In this thesis, 
is assumed to have a
discrete spectrum, so we can identify the ground eigenstate of
with the hydrodynamic distribution 
. The presence of field inhomogeneities and boundary
processes in which particles are being absorbed, (and perhaps re-
emitted with different energies,) may be represented by
an operator 
, which is localized in position, in the
Boltzmann equation. This operator is unlikely to commute with
, so it will have the effect of mixing the eigenstates
in this region, producing a non-hydrodynamic velocity
distribution.