We consider a system of N particles interacting via stochastic binary collisions, like the celebrated Kac's model. We discuss the large deviations asymptotic of the empirical measure and flow reviewing some relevant results and open problems. In particular, we show that - with probability exponentially small in N - there are paths that do not conserve the total energy. The structure of the large deviations rate function provides a natural gradient flow formulation of the underlying Boltzmann-Kac equation. As an application of this formulation, we finally discuss - in the context of non-homogeneous linear kinetic equations - the diffusive scaling limit.
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