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\ndensity function for a system of Langevin processes interacting via a pairwise\npotential. The resulting equation is considerably different from the\nphenomenological equations usually used to describe the dynamics of non\nconserved (Model A) and conserved (Model B) particle systems. The major feature\nis that the spatial white noise for this system appears not additively but\nmultiplicatively. This simply expresses the fact that the density cannot\nfluctuate in regions devoid of particles. The steady state for the density\nfunction may however still be recovered formally as a functional integral over\nthe coursed grained free energy of the system as in Models A and B.\n