From microscopic to macroscopic: the large number dynamics of agents and cells, possibly interacting with a chemical background

TL;DR

This paper reviews recent advances in understanding how large systems of agents and cells transition from microscopic to macroscopic behavior, using mean-field methods and numerical simulations across various fields.

Contribution

It introduces a mean-field approach based on marginals for analyzing large agent and cell systems, highlighting new theoretical insights and numerical findings.

Findings

Numerical results reveal unexpected features in large system dynamics.

The mean-field approach using marginals offers a new perspective.

Applications span kinetic theory, biology, and flocking phenomena.

Abstract

In this paper we review some recent results dealing with the transition between microscopic and macroscopic scales in different fields, including kinetic theory, cells movement in biology, chemotaxis, flocking phenomena and agent systems. The methodology of the mean-field approach of this study uses the concept of marginals instead of the empirical measure paradigm. Numerical computations showing some theoretically unexpected features are presented at the end of the paper.