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

Abstract

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.

arXiv comment37 pages, in TeX (compile twice), to appear on J. Stat. Phys., e-mail addresses: [email protected], [email protected]