HYDRODYNAMICAL LIMIT FOR A HAMILTONIAN SYSTEM WITH WEAK NOISE

Starting from a general Hamiltonian system with superstable pairwise p otential, we construct a stochastic dynamics by adding a noise term wh ich exchanges the momenta of nearby particles. We prove that, in the s caling limit, the time conserved quantities, energy, momenta and densi ty, satisfy the Euler equation of conservation laws up to a fixed time t provided that the Euler equation has a smooth solution with a given initial data up to time t. The strength of the noise term is chosen t o be very small (but nonvanishing) so that it disappears in the scalin g limit.