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We study the Boltzmann equation with elastic pointlike scalar interactions in two different versions of 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 nonrenormalizability 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 also has access to the nonapproximated result for comparison. [ABSTRACT FROM AUTHOR]