  We study the Boltzmann equation with elastic point-like scalar interactions
in two different versions of the the classical approximation. Although solving
numerically the Boltzmann equation with the unapproximated collision term poses
no problem, this allows one to study the effect of the ultraviolet cutoff in
these approximations. This cutoff dependence in the classical approximations of
the Boltzmann equation is closely related to the non-renormalizability of the
classical statistical approximation of the underlying quantum field theory. The
kinetic theory setup that we consider here allows one to study in a much
simpler way the dependence on the ultraviolet cutoff, since one has also access
to the non-approximated result for comparison.
