  We introduce a Kac's type walk whose rate of binary collisions preserves the
total momentum but not the kinetic energy. In the limit of large number of
particles we describe the dynamics in terms of empirical measure and flow,
proving the corresponding large deviation principle. The associated rate
function has an explicit expression. As a byproduct of this analysis, we
provide a gradient flow formulation of the Boltzmann-Kac equation.
