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
Ψ
plus independent Brownian motions:
Ψ
is the sum of pair potentials,
V(r)+
γ
d
J(γr)
, the second term has the form of a Kac potential with inverse range
γ
. Using diffusive hydrodynamical scaling (spatial scale
γ
−1
, temporal scale
γ
−2
) we obtain, in the limit
γ↓0
, a diffusive type integro-differential equation describing the time evolution of the macroscopic density profile.