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

TL;DR

This paper derives a stochastic density equation for interacting Langevin processes, revealing multiplicative noise and providing a new perspective on the dynamics of particle systems with conserved density.

Contribution

It introduces a novel derivation of the density evolution equation with multiplicative noise, differing from traditional phenomenological models for particle systems.

Findings

The derived equation features multiplicative spatial white noise.

Steady state density can be expressed as a functional integral over free energy.

The approach differs from standard models like Model A and B.

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.