Hydrodynamic Limit of Brownian Particles Interacting with Short- and Long-Range Forces

We investigate the time evolution of a model system of interacting particles moving in a d-dimensional torus. The microscopic dynamics is first order in time with velocities set equal to the negative gradient of a potential energy term 9 plus independent Brownian motions: 9 is the sum of pair potentials, V(r)+#J(#r); the second term has the form of a Kac potential with inverse range #. Using diffusive hydrodynamic scaling (spatial scale #, temporal scale #) we obtain, in the limit # a 0, a diffusive-type integrodifferential equation describing the time evolution of the macroscopic density profile.