In 1959 Mark Kac introduced a simple model for the evolution of
a gas of hard spheres undergoing elastic collisions. The main
simplification consisted in replacing deterministic collisions
with random Poisson distributed collisions.
It is possible to obtain many interesting results for this
simplified dynamics, like estimates on the rate of convergence
to equilibrium and validity of the Boltzmann equation.
The price paid is that this system has no space structure.
I will review some classical results on the Kac model and
report on an attempt to reintroduce some form of space
structure and non-equilibrium evolution in a way that
preserve the mathematical tractability of the system.