Stochastic process of equilibrium fluctuations of a system with long-range interactions

The relaxation towards equilibrium of systems with long-range interactions is not yet understood. As a step towards such a comprehension, we propose the study of dynamical equilibrium fluctuations in a model system with long-range interaction. We compute analytically, from the microscopic dynamics, the autocorrelation function of the order parameter. From this result, we derive analytically a Fokker-Planck equation which describes the stochastic process of the impulsion of a single particle in an equilibrium bath. The diffusion coefficient is explicitly computed.