The pseudo-Maxwell molecules model is defined through its Boltzmann-Enskog equation, and not in terms of a true particle dynamics. This equation reads:
The prefactor
accounts for various terms, e.g. it
can be taken proportional to
if the derivation
of Bobylev et al. is followed (see paragraph 5.1.5). In this
case the Enskog correction
is the pair correlation
function at distance
and density
, which averagely takes
into account correlations between the positions (scattering angle) of
colliding particles.
The assumption of homogeneity changes the above equation in the following way:
When the velocities are scalar and the dependence (e.g. the
temperature) is absorbed by a time reparametrization
, the
equation simplifies:
with
.
This equation is the master equation for the scalar Ulam model
discussed at the end of the paragraph 5.1.5: at each step a
pair of velocities and
from a set of
velocities is
chosen and transformed according to the usual rule of one-dimensional
inelastic collisions. It should be noted that the time
is
proportional to the number of collisions per particles.