Active Crowds

TL;DR

This chapter reviews mathematical models of active particles in crowded environments, deriving macroscopic equations from microscopic models, and explores boundary effects and applications through numerical simulations.

Contribution

It provides a unified framework for various macroscopic models of active particles, highlighting similarities, differences, and boundary effects in crowded settings.

Findings

Macroscopic models exhibit cross diffusion and degenerate mobilities.

Boundary conditions significantly influence particle behavior.

Numerical simulations illustrate effects of boundary conditions.

Abstract

This chapter focuses on the mathematical modelling of active particles (or agents) in crowded environments. We discuss several microscopic models found in literature and the derivation of the respective macroscopic partial differential equations for the particle density. The macroscopic models share common features, such as cross diffusion or degenerate mobilities. We then take the diversity of macroscopic models to a uniform structure and work out potential similarities and differences. Moreover, we discuss boundary effects and possible applications in life and social sciences. This is complemented by numerical simulations that highlight the effects of different boundary conditions.