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In this work, we generalize M. Kac's original many-particle binary stochasticmodel to derive a space homogeneous Boltzmann equation that includes a linearcombination of higher-order collisional terms. First, we prove an abstracttheorem about convergence from a finite hierarchy to an infinite hierarchy ofcoupled equations. We apply this convergence theorem on hierarchies formarginals corresponding to the generalized Kac model mentioned above. As acorollary, we prove propagation of chaos for the marginals associated to thegeneralized Kac model. In particular, the first marginal converges towards thesolution of a Boltzmann equation including interactions up to a finite order,and whose collision kernel is of Maxwell-type with cut-off.

Derivation of a Boltzmann equation with higher-order collisions from a generalized Kac model