Diffusion of particles with short-range interactions


We consider methods to obtain the continuum description of a system of interacting particles with short-range repulsive potentials. Nonlinear diffusion equations arise as the continuum description of interacting Brownian particles, and can be used to explain how such particles behave as a collective. But correlations due to interactions between particles are difficult to incorporate correctly into a diffusion equation. Closure approximations such as the mean field or Kirkwood superposition approximations are common. In this paper, we present an alternative approach using matched asymptotic expansions, in which the approximation is entirely systematic. We perform numerical simulations to compare the continuum models obtained by our approach and closure approximations with the stochastic particle system for various potentials. We find that our approach works best for very repulsive short-range potentials, while the mean-field closure is suitable for long-range interactions. The Kirkwood superposition closure provides an accurate description for both short- and long-range potentials, but has limited practical use.

SIAM Journal on Applied Mathematics