Langevin Equation for the Density of a System of Interacting Langevin Processes

Abstract

We present a simple derivation of the stochastic equation obeyed by the density function for a system of Langevin processes interacting via a pairwise potential. The resulting equation is considerably different from the phenomenological equations usually used to describe the dynamics of non conserved (Model A) and conserved (Model B) particle systems. The major feature is that the spatial white noise for this system appears not additively but multiplicatively. This simply expresses the fact that the density cannot fluctuate in regions devoid of particles. The steady state for the density function may however still be recovered formally as a functional integral over the coursed grained free energy of the system as in Models A and B.