Derivation of a Boltzmann Equation with Higher-Order Collisions from a Generalized Kac Model

SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 5409-5444, August 2024.
Abstract. In this work, we generalize Kac’s original many-particle binary stochastic model to derive a space homogeneous Boltzmann equation that includes a linear combination of higher-order collisional terms. First, we prove an abstract theorem about convergence from a finite hierarchy to an infinite hierarchy of coupled equations. We apply this convergence theorem on hierarchies for marginals corresponding to the generalized Kac model mentioned above. As a corollary, we prove propagation of chaos for the marginals associated to the generalized Kac model. In particular, the first marginal converges towards the solution of a Boltzmann equation including interactions up to a finite order and whose collision kernel is of Maxwell type with cut-off.