A mathematical model for crowd dynamics: Multiscale analysis, fluctuations and random noise

Abstract

We propose and analyze a multiscale mathematical model that reproduces the predominant features of crowd dynamics by taking into account the distance among pedestrians. The Liouville equation, some ideas borrowed from kinetic theory and Grad limiting procedure are at the basis of the derivation of the model. Fluctuations and random noise (e.g. Brownian motion) are also considered. The asymptotic analysis shows  that the probability distribution function of the crowd model, when converging,  leads to standard kinetic models. Mono-kinetic descriptions are also investigated. Finally a momentum balance equation is readily stated and the macroscopic average velocity  is obtained by averaging the mesoscopic  description. The methodology used in this paper is based on particle, kinetic and hydrodynamic descriptions.