# One-dimensional short-range nearest-neighbor interaction and its nonlinear diffusion limit

## TL;DR

This paper models one-dimensional particle interactions with finite-range repulsion, rigorously derives their macroscopic nonlinear diffusion limits, and validates findings through numerical simulations, connecting microscopic behavior to well-known PDEs like the porous medium equation.

## Contribution

It provides a rigorous derivation of nonlinear diffusion equations from microscopic particle models with finite-range interactions in one dimension.

## Key findings

- Derivation of macroscopic nonlinear diffusion equations from microscopic models.
- Identification of the porous medium equation as a limit case.
- Numerical simulations confirming analytical results.

## Abstract

Repulsion between individuals within a finite radius is encountered in numerous applications, including cell exclusion, i.e. avoidance of overlapping cells, bird flocks, or microscopic pedestrian models. We define such individual based particle dynamics in one spatial dimension with minimal assumptions of the repulsion force f as well as their external velocity v and prove their characteristic properties. Moreover, we are able to perform a rigorous limit from the microscopic to the macroscopic scale, where we could recover the finite interaction radius as a density threshold. Specific choices for the repulsion force f lead to well known nonlinear diffusion equations on the macroscopic scale, as e.g. the porous medium equation. At both scaling levels numerical simulations are presented and compared to underline the analytical results.