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 Psi plus independent Brownian motions: Psi is the sum of pair potentials, V(I) + gamma(d)J(gamma r); the second term has the form of a Kac potential with inverse range gamma. Using diffusive hydrodynamic scaling (spatial scale gamma(-1), temporal scale gamma(-2)) we obtain, in the limit gamma down arrow 0, a diffusive-type integrodifferential equation describing the time evolution of the macroscopic density profile.