Mean Field Limit for Congestion Dynamics in One Dimension

TL;DR

This paper demonstrates that a continuum PDE model for congested transport in one dimension can be rigorously derived as the mean-field limit of a particle system with finite-distance constraints, connecting microscopic and macroscopic descriptions.

Contribution

It provides a rigorous derivation of the macroscopic congestion PDE from a particle system in one dimension, using Eulerian and Lagrangian approaches.

Findings

The continuum PDE is the mean-field limit of the particle system.

The derivation uses two different approximations for density and pressure.

The approach bridges microscopic particle dynamics and macroscopic congestion models.

Abstract

This paper addresses congested transport, which can be described, at macroscopic scales, by a continuity equation with a pressure variable generated from the hard-congestion constraint (maximum value of the density). The main goal of the paper is to show that, in one spatial dimension, this continuum PDE can be derived as the mean-field limit of a system of ordinary differential equations that describes the motion of a large number of particles constrained to stay at some finite distance from each others. To show that these two models describe the same dynamics at different scale, we will rely on both the Eulerian and Lagrangian points of view and use two different approximations for the density and pressure variables in the continuum limit.