One-Dimensional Short-Range Nearest-Neighbor Interaction and Its Nonlinear Diffusion Limit

Abstract

SIAM Journal on Applied Mathematics, Volume 84, Issue 1, Page 1-18, February 2024.
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 [math] as well as their external velocity [math] 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 [math] 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.