Statistical mechanics of classical particles with gravitational interactions: Exactly solvable (for N ≤ ∞) in d = 1 and d > 2

This paper is concerned with a curious gap in a string of exactly solvable models, a gap that is suggestively related to a completely integrable nonlinear PDE in d = 2 known as Liouville's equation. This PDE emerges in a limit N → ∞ from the equilibrium statistical mechanics of classical point particles with gravitational interactions (SMGI) in dimension d = 2 which, accordingly, is an exactly solvable continuum model in this limit. Interestingly, in d = 1 and all d > 2, the SMGI can be, and partly has been, exactly evaluated for all N ≤ ∞. This entitles one to suspect that the SMGI for d = 2 is likewise exactly solvable for N > ∞, but currently this is an unproven hypothesis. If this conjecture can be answered in the affirmative, spin-offs in various areas associated with Liouville's equation, such as vortex gases, superfluidity, random matrices, and string theory, can be expected.