^{1,2}

^{3}

^{1}

^{2}

^{3}

A comparison between first-order microscopic and macroscopic differential models of crowd dynamics is established for an increasing number

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 [

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 [

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 [

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

(a) Classical mean field point of view, in which the total mass

This point of view is being introduced also in the context of other systems of interacting particles, for example, vehicular traffic. Quoting from the conclusions of the lecture [

real traffic is microscopic. Ideally, accurate macroscopic models should not focus on the limit

The two types of models which will be considered throughout the paper assume first-order position-dependent pedestrian dynamics, given via the walking velocity. Specifically, in the microscopic case, let

Models (

Models (

Such a measure-based framework features an intrinsic generality; indeed it can describe a discrete crowd distribution when

Once plugged into (

According to the arguments set forth, (

In this section we study and compare the stationary behavior of one-dimensional (

The evaluation of the pedestrian speed in homogeneous crowding conditions is an established experimental practice which leads to the so-called

For the sake of simplicity, we consider the one-dimensional problem on a periodic domain

We assume that the desired velocity is a positive constant

Prototype of interaction kernel

We make the following assumptions

with

with, moreover,

Thus pedestrian interactions decay in the interior of the sensory region as the mutual distance increases and, moreover, pedestrians do not “self-interact.” This forces

Before proceeding with the comparison of asymptotic pedestrian speeds resulting from microscopic and macroscopic dynamics, we ascertain that the spatially homogeneous solutions (

The recent literature about discrete and continuous models of collective motions is quite rich in contributions dealing with the stability of special patterns, for example, flocks, mills, and double mills; see [

The equispaced lattice solution (

Proposition

The spatially homogeneous solution (

For the sake of completeness, we report the proofs of Propositions

We now calculate and compare the speed diagrams corresponding to the stable stationary homogeneous solutions studied in the previous sections, that is, the mappings

Specifically, from Proposition

Notice that both

In order to compare the two speed diagrams we introduce the quantity

From Assumption

For the first integral,

because

For each of the integrals in the sum,

For the last integral,

The considered partition of the domain

It follows that

A numerical evaluation of

Speed diagrams (

One has

We consider one by one the terms at the right-hand side of (

First, the right endpoint of

Second, in view of the smoothness of

Finally,

From Theorem

Additionally, from Theorem

Speed diagrams (

In this section we consider the Cauchy problem

It makes thus sense to consider sequences of discrete and continuous initial conditions

Formally speaking, we will operate in the setting of the

In the following, from Sections

Following the theory developed in [

There exist

Using the expression (

The basic tool for the subsequent analysis is the continuous dependence of the solution to (

Consider the flow map (

For all

Let

Let

Let us now consider two sequences of initial conditions of growing mass, say

An admissible interaction kernel in the family (

An interaction kernel somehow opposite to

Besides the extreme cases in Examples

A scaling of interactions of type (

Solutions to (

Let

For all

Let

The flow maps

The solutions satisfy

(i) By direct calculation we find

(ii) We check that

Next we observe that

(iii) Due to the result in (ii) we have

As a consequence of Proposition

Let

By (

Conditions (

Let

In the remaining part of this section we compute the Wasserstein distance between

Let

Here we use expression (

We pass now to characterize transference plans between

Let us consider, for a transference plan of the form (

Notice that the transference plan

Transference plans (

Thanks to Proposition

Let

From Proposition

It is worth remarking that, because of (

Homogeneous pedestrian distributions can be obtained, for instance, by considering regular lattices.

Let

In dimension

which converges to zero for

In dimension

which converges to zero for

In dimension

which converges to zero for

Discrete pedestrians (dots) and their continuous counterparts (circular regions delimited by dashed lines) in

In this paper we have investigated microscopic and macroscopic differential models of systems of interacting particles, chiefly inspired by human crowds, for an increasing number

In particular, we have proved the solutions to the following two models:

A schematic illustration of the result of the paper. The ODE and PDE solutions approach each other for

First of all, we point out that interactions are modeled in an

Second, our analysis shows the correct parallelism between first-order microscopic and macroscopic models which do not originate from one another but are formulated independently by aprioristic choices of the scales. In this view, the utility of such a parallelism is twofold. On one hand, it accounts for the

First we observe that measure (

To show that

stable if

stable and attractive if

In case (a) the sum in (

In case (b) we make the ansatz

The constant solution (

By linearity we consider one term of the sum at a time; that is, we take

We claim that

First, we observe that

(i) From (

(ii) Again by (

Here we use the expression (

The authors declare that there is no conflict of interests regarding the publication of this paper.

The first author was funded by a “Lagrange” Ph.D. scholarship granted by the CRT Foundation, Torino, Italy, and by Eindhoven University of Technology, Eindhoven, Netherlands.