Equilibrium statistical mechanics of one-dimensional Hamiltonian systems with long-range force

The system of N identical classical particles on the circle of length L interacting via a pair potential is investigated in the mean field limit (N-->infinity, L fixed). Its physical properties are determined by the Fourier components of the interaction (mean) field. The partition function, the joint distribution of the interaction fields, the local field, and the correlation functions are computed. If the interaction is semidefinite non-negative, field components become independent for N-->infinity and satisfy central limit theorems. If the interaction has negative Fourier components, a phase transition occurs.