Ornstein-Uhlenbeck limit for the velocity process of an $N$-particle system interacting stochastically

19 pages ; streamlined notations ; new section on many particles with momentum conservation ; new appendix on Kac systemAn $N$-particle system with stochastic interactions is considered. Interactions are driven by a Brownian noise term and total energy conservation is imposed. The evolution of the system, in velocity space, is a diffusion on a $(3N-1)$-dimensional sphere with radius fixed by the total energy. In the $N\rightarrow\infty$ limit, a finite number of velocity components are shown to evolve independently and according to an Ornstein-Uhlenbeck process