# Diffusion of particles with short-range interactions

## TL;DR

This paper derives a nonlinear diffusion equation for interacting Brownian particles with short-range repulsive potentials, comparing its accuracy with mean-field, Kirkwood, and Monte Carlo methods.

## Contribution

It systematically derives a continuum model for short-range interactions using matched asymptotic expansions and evaluates its performance against other approximation methods.

## Key findings

- The continuum model performs best for very repulsive short-range potentials.
- Mean-field approximation is suitable for long-range interactions.
- Kirkwood superposition approximation is accurate for both short- and long-range potentials, but computationally intensive.

## Abstract

A system of interacting Brownian particles subject to short-range repulsive potentials is considered. A continuum description in the form of a nonlinear diffusion equation is derived systematically in the dilute limit using the method of matched asymptotic expansions. Numerical simulations are performed to compare the results of the model with those of the commonly used mean-field and Kirkwood-superposition approximations, as well as with Monte Carlo simulation of the stochastic particle system, for various interaction potentials. Our approach works best for very repulsive short-range potentials, while the mean-field approximation is suitable for long-range interactions. The Kirkwood superposition approximation provides an accurate description for both short- and long-range potentials, but is considerably more computationally intensive.