  The subject of this article is the Kac equation without cutoff. We first show
that in the asymptotic of grazing collisions, the Kac equation can be
approximated by a Fokker-Planck equation. The convergence is uniform in time
and we give an explicit rate of convergence. Next, we replace the small
collisions by a small diffusion term in order to approximate the solution of
the Kac equation and study the resulting error. We finally build a system of
stochastic particles undergoing collisions and diffusion, that we can easily
simulate, which approximates the solution of the Kac equation without cutoff.
We give some estimates on the rate of convergence.
