What does it mean when a system "virializes"? How does this happen? What does it mean when a gas virializes in a potential? Is it only that it now satisfies the virial theorem? How does it happen?
 A: The virial theorem states that if we have a potential:
$$ U(r) = k r^n $$
for some constant $k$ then the average kinetic and potential energy will be related by:
$$ 2\langle T \rangle = n \langle U \rangle $$
To take a concrete example, for an inverse square potential like gravity $n=-1$ and we get $\langle T \rangle = - \tfrac{1}{2} \langle U \rangle$ and this is exactly what we observe in gravitationally bound systems.
But note that this is an average for an ensemble of interacting particles. If we consider a single body falling into a gravitational potential then clearly its KE and PE do not obey the virial theorem. So if we take an ensemble of such bodies that do not interact they will not obey the virial theorem either.
The virial theorem applies when the bodies involved can interact and exchange energy and momentum with each other, and consequently settle into an equilibrium state. So if we take as an example gas falling into a gravitational potential then initially the average KE and PE of the gas molecules will not obey the virial theorem, but once they have interacted with each other and settled into an equilibrium state they will obey the virial theorem.
And it is this process of interacting and settling into an equilibrium state that is referred to as virialisation.
