  We consider the dynamic large deviation behaviour of Kac's collisional
process for a range of initial conditions including equilibrium. We prove an
upper bound with a rate function of the type which has previously been found
for kinetic large deviation problems, and a matching lower bound restricted to
a class of sufficiently good paths. However, we are able to show by an explicit
counterexample that the predicted rate function does not extend to a global
lower bound: even though the particle system almost surely conserves energy,
large deviation behaviour includes solutions to the Boltzmann equation which do
not conserve energy, as found by Lu and Wennberg, and these occur strictly more
rarely than predicted by the proposed rate function. At the level of the
particle system, this occurs because a macroscopic proportion of energy can
concentrate in $\mathfrak{o}(N)$ particles with probability
$e^{-\mathcal{O}(N)}$.
