A real gas with gravitation-like interaction Consider a system (a gas) of point-like particles with a gravitation-like interaction (potential) $V(r) \sim \frac{1}{r}$ between pairs of them.
One can rule out statistically that two particles will approach each other exactly along their line, so two particles will never collide directly - thus giving rise to infinite (kinetic) energies. So the particles will always whirl around each other in one way or the other.
What can be said about such a system in the framework of statistical mechanics? 
How does its partition function look like? 
What are its thermodynamical equilibrium states? 
What happens when cooling such a system down (e.g. from very high temperatures)?
 A: A (3d) gas of particles with a gravitational interaction is an example of a system with long range interactions, where the energy is not additive and thus many basic results of classical statistical mechanics are not valid, including the equivalence of the microcanonical, canonical and grand-canonical ensembles. For a general introduction to the subject see these lecture notes by Mukamel, and for an introduction to the stat. mech. of gravitational systems see these lecture notes by Padmanabhan. Because ensembles are not equivalent, one must be careful when discussing issues such as the paritition function of the system. 
I am not sure why you say that particle collisions can be rules out. As far as I understand, a short length-scale cutoff (i.e., a minimum distance between particles) must be introduced to avoid a collapse of particles into the same point. Even with such a cutoff, I believe that particles will phase-separate into what is called a core and a halo: the core is a compact and rather still aggregate of particles; because its potential energy is very low (i.e., large and negative), it enables the existance of a "hot" halo with very high kinetic energy. If I am not mistaken, this halo will spread out towards infinity, and  thus to have a sensible equilibrium state one must also introduce a large distance cutoff (Padmanabhan's lecture notes discuss this issue, but I've only briefly refreshsed my memory now so my answer may be slightly inaccurate). In the discussion above I've assumed a microcanonical system. 
