Liouville field theory and the partition function for two-dimensional Newtonian gravity

It is shown that the grand partition function for a system of particles interacting via gravitational force can be related to the vacuum-to-vacuum amplitude of the Liouville field theory. Finite action solutions to the Liouville field equation can be expressed in closed form in two dimensions. This solution is used to construct the mean field approximation for the grand partition function. The accuracy of the mean field approximation is verified by comparing it with the exact partition function for a system with two particles.