  A pathwise large deviation result is proved for the pure jump models of
$k$-nary interacting particle system introduced by Kolokoltsov that generalize
classical Boltzmann's collision model, Smoluchovski's coagulation model and
many others. The upper bound is obtained by following the standard methods of
using a process "perturbed" by a regular function. To show the lower bound, we
propose a family of orthogonal martingale measures and prove a coupling for the
general perturbations. The rate function is studied based on the idea of
L\'eonard with a simplification by considering the conjugation of integral
functionals on a subspace of $L^{\infty}$. General "gelling" solutions in the
domain of the rate function are also discussed.
