Hamiltonian and Brownian systems with long-range interactions: I Statistical equilibrium states and correlation functions

We discuss the equilibrium statistical mechanics of systems with long-range interactions. We contrast the microcanonical description of an isolated Hamiltonian system from the canonical description of a stochastically forced Brownian system. We show that the mean-field approximation is exact in a proper thermodynamic limit N→+∞. The one-point equilibrium distribution function is solution of an integrodifferential equation obtained from a static BBGKY-like hierarchy. It also optimizes a thermodynamical potential (entropy or free energy) under appropriate constraints. In the case of attractive potentials of interaction, we show the existence of a critical temperature Tc separating a homogeneous phase (T Tc) from a clustered phase (T Tc). The homogeneous phase becomes unstable for T