The canonical ensemble is investigated for classical self-gravitating matter in a finite containerΛ [d]⊂ℝ d ,d=3 and 2. Starting with modified gravitational interactions (smoothed-out singularity), it is proven by explicit construction that, in thew *-topology, the canonical equilibrium measure converges to a superposition of Dirac measures when the limit of exact Newtonian gravitational interactions between classical point particles is taken. The consequences of this result for more realistic classical systems are evaluated, and the existence of a gravitational phase transition is proven. The results are discussed with view toward applications in astrophysics and space science. Some attention is paid also to the problem of founding thermodynamics by means of statistical mechanics.
Read full abstract