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 are 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, $V(r)+\gamma^d J(\gamma r)$, the second term has the form of a Kac potential with inverse range $\gamma$. Using diffusive hydrodynamical scaling (spatial scale $\gamma^{-1}$, temporal scale $\gamma^{-2}$) we obtain, in the limit $\gamma\downarrow 0$, a diffusive type integro-differential equation describing the time evolution of the macroscopic density profile.