Advances in Mathematical Physics

Volume 2016, Article ID 6902086, 17 pages

http://dx.doi.org/10.1155/2016/6902086

## Comparing First-Order Microscopic and Macroscopic Crowd Models for an Increasing Number of Massive Agents

^{1}Department of Structural and Geotechnical Engineering, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy^{2}Department of Mathematics and Computer Science, Eindhoven University of Technology, 5600 MB Eindhoven, Netherlands^{3}Istituto per le Applicazioni del Calcolo “M. Picone”, Consiglio Nazionale delle Ricerche, Via dei Taurini 19, 00185 Roma, Italy

Received 27 October 2015; Accepted 11 February 2016

Academic Editor: Prabir Daripa

Copyright © 2016 Alessandro Corbetta and Andrea Tosin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A comparison between first-order microscopic and macroscopic differential models of crowd dynamics is established for an increasing number of pedestrians. The novelty is the fact of considering massive agents, namely, particles whose individual mass does not become infinitesimal when grows. This implies that the total mass of the system is not constant but grows with . The main result is that the two types of models approach one another in the limit , provided the strength and/or the domain of pedestrian interactions are properly modulated by at either scale. This is consistent with the idea that pedestrians may adapt their interpersonal attitudes according to the overall level of congestion.

#### 1. Introduction

Pedestrians walking in crowds exhibit rich and complex dynamics, which in the last years generated problems of great interest for different scientific communities including, for instance, applied mathematicians, physicist, and engineers (see [1, Chapter ] and [2, 3] for recent surveys). This led to the derivation of numerous mathematical models providing qualitative and possibly also quantitative descriptions of the system [4–6].

When deducing a mathematical model for pedestrian dynamics different observation scales can be considered. Two extensively used options are the microscopic and the macroscopic scales. Microscopic models describe the time evolution of the position of each single pedestrian, addressed as a discrete particle [7–10]. Conversely, macroscopic models deal with a spatially averaged representation of the pedestrian distribution, which is treated as a continuum in terms of the pedestrian density [11–15]. Furthermore, crowds have been also represented at the mesoscopic scale [16–18] or via discrete systems such as Cellular Automata [19, 20].

Different observation scales serve different purposes: the microscopic scale is more informative when considering very localized dynamics, in which the action of single individuals is relevant; conversely, the macroscopic scale is appropriate when insights into the ensemble (collective) dynamics are required or when high densities are considered. In addition to this, spatially discrete and continuous scales may provide a dual representation of a crowd useful to formalize aspects such as pedestrian perception and the interplay between individualities and collectivity [1, 21, 22]. Selecting the most adequate representation may present difficulties, because different outcomes at different scales are likely to be observed. Nevertheless, independently of the scale, models are often deduced out of common phenomenological assumptions; hence they are expected to reproduce analogous phenomena. The question then arises when and how they are comparable to each other.

These arguments provide the motivation for this paper, in which a comparison of microscopic and macroscopic crowd models is carried out for a growing number of pedestrians. It is well known that the statistical behavior of microscopic systems of interacting particles can be described, for , by means of a Vlasov-type kinetic equation derived in the* mean field limit* under the assumption that the strength of pairwise interactions is scaled as (*weak coupling scaling*); see [23, 24] and references therein. If the total mass of the system is , where is the mass of each particle, this corresponds to assuming that particles generate an interaction potential in space proportional to their mass (like, e.g., in gravitational interactions). The mean field limit requires the assumption of a constant total mass of the system, say , which implies that the mass of each particle becomes infinitesimal as grows (cf. Figure 1(a)). On the contrary, considering continuous models* per se*, parallelly to discrete ones, allows one to keep the mass of each individual constant, say ; thus holds (cf. Figure 1(b)). In this perspective, a comparison with discrete models based on the role of acquires a renewed interest.