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

Authors: Alessandro Corbetta; Andrea Tosin
Journal: Advances in Mathematical Physics
 (2016)

---

## Abstract

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


---

## Body

## 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 4] 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 numberN of pedestrians. It is well known that the statistical behavior of microscopic systems of interacting particles can be described, for N
→
∞, by means of a Vlasov-type kinetic equation derived in themean field limit under the assumption that the strength of pairwise interactions is scaled as N
-
1 (weak coupling scaling); see [23, 24] and references therein. If the total mass of the system is M
=
m
N, where m is the mass of each particle, this corresponds to assuming that particles generate an interaction potential in space proportional to their mass m
=
M
/
N (like, e.g., in gravitational interactions). The mean field limit requires the assumption of a constant total mass of the system, say M
=
1, which implies that the mass of each particle becomes infinitesimal as N grows (cf. Figure 1(a)). On the contrary, considering continuous modelsper se, parallelly to discrete ones, allows one to keep the mass m of each individual constant, say m
=
1; thus M
=
N holds (cf. Figure 1(b)). In this perspective, a comparison with discrete models based on the role of N acquires a renewed interest.Figure 1
(a) Classical mean field point of view, in which the total massM of the system is constant while the mass m of each particle becomes infinitesimal as N increases. (b) The point of view pursued in this paper, in which m is constant (massive particles) while M grows when N increases.
(a)
(b)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 [25],real traffic is microscopic. Ideally, accurate macroscopic models should not focus on the limitN
→
∞ but represent the solution with the true number of vehicles N.Pedestrian crowds are microscopic as well; hence macroscopic crowd models should be built consistently with the phenomenology of afinite number of microscopicmassive pedestrians. Of course, we cannot expect the microscopic and macroscopic solutions to be the same for all numbers of pedestrians; however we can ascertain if the two types of models are actually “the same model” at least in some asymptotic regime. In this sense we address the limit N
→
∞.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, letX
t
1
,
X
t
2
,
…
,
X
t
N
∈
R
d be the positions of N pedestrians at a time t. Their evolution satisfies(1)
X
˙
t
i
=
v
d
X
t
i
-
∑
j
=
1
N
K
X
t
j
-
X
t
j
,
i
=
1
,
…
,
N
.Conversely, in the macroscopic case, let ρ
t
(
x
) be the density of pedestrians in the point x
∈
R
d at time t, such that ∫
R
d
ρ
t
(
x
)
d
x
=
N for all t
≥
0. In some analogy with (1), its evolution is given by the conservation law(2)
∂
t
ρ
t
+
div
⁡
ρ
t
v
d
-
∫
R
d
K
y
-
·
ρ
t
y
d
y
=
0
.In both cases, pedestrian velocity is modeled as a sum of two terms: adesired velocity
v
d, which walkers would keep in the absence of others, plusrepulsive (whence the minus sign)interactions, which perturb v
d for collision avoidance purposes. By assumption, interactions depend on the relative distance between pairs of interacting pedestrians via aninteraction kernel
K.Models (1) and (2) can be recognized as particular instances of a scale-agnostic measure-valued conservation law. This abstract conservation law will play the role of a pivot in the comparison performed here. The comparison and the paper are organized as follows: in Section 2 we introduce and briefly discuss the scale-agnostic modeling framework. In Section 3 we give a first comparison result of discrete and continuous dynamics in the one-dimensional stationary case, which leads to the computation of the so-calledspeed diagrams. The main result is that the asymptotic pedestrian speeds predicted by models (1), (2) are not the same and that they cannot even be expected to match one another for large numbers of pedestrians. In Section 4 we then give a second more complete comparison result in a general d-dimensional time-evolutionary setting. We consider sequences of pairs of discrete and continuous models of the forms (1), (2) indexed by the total number N of pedestrians and, as main contributions, we establish the following: (i) for fixed N, a stability estimate relating the distance between those pairs of models at a generic instant t
>
0 to the one at the initial time t
=
0; (ii) for fixed t, a family of scalings of the interactions, comprising the aforementioned mean field as a particular case, which control the amplification of such a distance when N grows; (iii) a procedure to construct discrete and continuous initial configurations of the crowd giving rise to mutually convergent sequences of discrete and continuous models at all times when N grows. Finally, in Section 5 we discuss the implications of the obtained results on the modeling of crowd dynamics and other particle systems in a multiscale perspective.


## 2. A Scale-Agnostic Modeling Framework

Models (1) and (2) are particular cases of the measure-valued conservation law in R
d
(3)
∂
t
μ
t
+
div
⁡
μ
t
v
μ
t
=
0
,where t is the time variable, μ
t is a time-dependent spatial measure of the crowding, and v
[
μ
t
] is a measure-dependent velocity field to be prescribed (see below). More specifically, μ
t is a positive locally finite Radon measure defined on B
(
R
d
), the Borel σ-algebra on R
d, which satisfies(4)
μ
t
R
d
=
N
,
∀
t
≥
0
,where, as we recall, N
∈
N is the (conserved) number of pedestrians. The cases d
=
1,2 are commonly considered to model, respectively, scenarios in which pedestrians are aligned, for example, along a walkway (d
=
1) or can walk in a given planar area (d
=
2). Equation (3) has to be understood in the proper weak formulation:(5)
∫
R
d
ϕ
d
μ
t
=
∫
R
d
ϕ
d
μ
0
+
∫
0
t
∫
R
d
∇
ϕ
·
v
μ
s
d
μ
s
d
s
,
∀
ϕ
∈
C
c
∞
R
d
,
C
c
∞
(
R
d
) being the space of infinitely smooth and compactly supported test functions ϕ
:
R
d
→
R. From (5) it can be formally checked that, given an initial measure μ
0 (initial configuration of the crowd), at time t
>
0 the following results:(6)
μ
t
=
γ
t
#
μ
0
that is
μ
t
E
=
μ
0
γ
t
-
1
E
∀
E
∈
B
R
d
,where # is thepush forward operator (see, e.g., [26]) and γ
t is theflow map defined as(7)
γ
t
x
=
x
+
∫
0
t
v
μ
s
γ
s
x
d
s
.It is worth stressing that, under proper regularity conditions on the velocity v, all solutions t
↦
μ
t of the Cauchy problem associated with (3) are continuous functions of time [27, 28] and admit the representations (6)-(7) [26].Such a measure-based framework features an intrinsic generality; indeed it can describe a discrete crowd distribution whenμ
t is an atomic measure:(8)
μ
t
=
ϵ
t
≔
∑
i
=
1
N
δ
X
t
i
,where X
t
i
i
=
1
N
⊂
R
d are the positions of pedestrians at time t, or a continuous crowd distribution when μ
t is an absolutely continuous measure with respect to the d-dimensional Lebesgue measure:(9)
μ
t
=
ρ
twith ρ
t
(
R
d
)
=
∫
R
d
d
ρ
t
=
N for all t
≥
0. In this latter case, by Radon-Nikodym theorem, μ
t admits a density, that is, the crowd density. For ease of notation, we will systematically confuse the measure ρ
t with its density and write d
ρ
t
(
x
) or ρ
t
(
x
)
d
x interchangeably. Moreover, when required by the context, we will write explicitly the number N of pedestrians as superscript of the measures (e.g., μ
t
N, ϵ
t
N, and ρ
t
N).Once plugged into (5), the measures (8), (9) produce models (1), (2), respectively, if the following velocity is used:(10)
v
μ
t
x
=
v
d
x
-
∫
R
d
K
y
-
x
d
μ
t
y
.The interaction kernel K
:
R
d
→
R
d represents pairwise interactions occurring among walking pedestrians, which, in normal conditions (i.e., no panic), tend to be of mutual avoidance and finalized at maintaining a certain comfort distance. As a consequence, they are supposed to be repulsive-like, whence -
K
(
z
)
·
z
≤
0 for all z
∈
R
d. Moreover, they are known to happen within a bounded region in space, the so-calledsensory region [29]. Therefore K has compact support, which, for a pedestrian in position x, we denote by(11)
S
R
x
≔
supp
⁡
K
·
-
x
⊆
B
R
x
,
B
R
(
x
) being the ball centered in x with radius R. Therefore, R is the maximum distance from x at which interactions are effective. If the sensory region is not isotropic (as it is the case for pedestrians, who interact preferentially with people ahead), its orientation is expected to depend on the pedestrian gaze direction [29]. Nonetheless, in the following, we assume for simplicity that S
R
(
x
) is just a rigid translation of a prototypical region S
R
(
0
)
=
supp
⁡
K
⊆
B
R
(
0
). Extending the proposed setting to models featuring fully orientation dependent sensory regions is mainly a technical issue, for which the reader can refer, for example, to [28, 30].According to the arguments set forth, (3) allows for a formal qualitative correspondence between the two modeling scales; nonetheless no quantitative correspondence is established between the actual dynamics. The analysis of quantitative correspondences will be the subject of the next sections.


## 3. The One-Dimensional Stationary Case: Speed Diagrams

In this section we study and compare the stationary behavior of one-dimensional (d
=
1) microscopic and macroscopic homogeneous pedestrian distributions satisfying (3) with velocity (10). Homogeneous conditions, yet to be properly defined at the two considered scales, represent dynamic equilibrium conditions possibly reached asymptotically, after a transient. In homogeneous conditions, the speed of pedestrians is expected to be a constant, depending exclusively on the number of pedestrians and on the length, say L
>
0, of the one-dimensional domain.The evaluation of the pedestrian speed in homogeneous crowding conditions is an established experimental practice which leads to the so-calledspeed diagrams, that is, synthetic quantitative relations between the density of pedestrians and their average speed [31]. Usually, such diagrams feature a decreasing trend for increasing values of the density and are defined up to a characteristic value (stopping density) at which the measured speed is zero. In the following, speed diagrams are studied as a function of the number of pedestrians N (for an analogous experimental case cf. [32]), in the microscopic and macroscopic cases. In this context, N is retained as the common element between the two descriptions, as it remains well-defined independently of the observation scale. It is finally worth pointing out that although speed diagrams have been often used in mathematical models as closure relations (especially for macroscopic models, cf. [33, 34]), in this case they are a genuine output of the considered interaction rules expressed by the integral in (10).


### 3.1. Modeling Hypotheses

For the sake of simplicity, we consider the one-dimensional problem on a periodic domain[
0
,
L
). In order to model homogeneous pedestrian distributions, in the discrete case (1) we consider an equispaced lattice solution translating with a certain constant speed w (to be determined); that is,(12)
ϵ
t
x
=
ϵ
¯
x
-
w
t
,where ϵ
¯ has form (8) with atom locations such that(13)
X
¯
j
-
X
¯
i
=
j
-
i
L
N
,
i
,
j
=
1
,
…
,
N
,for X
¯
1
<
X
¯
2
<
⋯
<
X
¯
N (the ordering is modulo L). Thus the atoms of ϵ
t are X
t
i
=
X
¯
i
+
w
t for i
=
1
,
…
,
N. In the continuous case (2), we consider instead a constant density; that is,(14)
ρ
t
x
≡
ρ
¯
≥
0
,which, owing to (4), is given by ρ
¯
=
N
/
L.We assume that the desired velocity is a positive constantv
d
>
0; therefore the movement is in the positive direction of the real line. Furthermore, we make the following assumptions on the interaction kernel (cf. Figure 2).Figure 2
Prototype of interaction kernelK complying with Assumption 1.Assumption 1 (properties ofK).
We make the following assumptions(i)
Compactness of the support and frontal orientation of the sensory region: the support of K is(15)
S
R
0
=
0
,
R
with0
<
R
<
L.
(ii)
Boundedness and regularity in
(
0
,
R
): pedestrian interactions vary smoothly with the mutual distance of the interacting individuals and have a finite maximum value. Specifically,(16)
K
∈
C
2
0
,
R
,
K
,
K
′
′
∈
L
∞
0
,
R
.
(iii)
Monotonicity in
(
0
,
R
)
and behavior at the endpoints: we assume(17)
K
z
>
0
,
K
′
z
<
0
,
for
z
∈
0
,
R
with, moreover,(18)
K
0
=
K
R
=
0
,
K
0
+
≔
lim
z
→
0
+
⁡
K
z
>
0
.
Thus pedestrian interactions decay in the interior of the sensory region as the mutual distance increases and, moreover, pedestrians do not “self-interact.” This forcesK to be discontinuous in z
=
0.


### 3.2. Stationarity and Stability of Spatially Homogeneous Solutions

Before proceeding with the comparison of asymptotic pedestrian speeds resulting from microscopic and macroscopic dynamics, we ascertain that the spatially homogeneous solutions (12)-(13) and (14) are indeed stable and possibly attractive solutions to either (1) or (2). This ensures that such distributions can indeed be considered as equilibrium distributions and therefore that the evaluation of speed diagrams is well-posed.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 [23, 35–37] and references therein. We consider here much simpler one-dimensional configurations, however useful in this context because they reproduce mathematically the typical experimental setups in which pedestrian speed diagrams are measured; see, for example, [38, 39].Proposition 2 (equilibrium of the microscopic model).
The equispaced lattice solution (12) is a stable solution to (1) for(19)
w
=
v
d
-
∑
h
=
1
N
-
1
K
h
L
N
.It is moreover attractive if R
>
L
/
N.Proposition2 asserts that the equispaced pedestrian distribution is always a stable (quasi)stationary solution to the microscopic model. This is somehow in contrast to what is found in microscopicoptimal-velocity traffic models, where the so-calledPOMs (“Ponies-on-a-Merry-Go-Round”)solutions can generate instabilities (traffic jams) depending on the total number of vehicles [40, 41]. The rationale for this difference is that, unlike the present case, in such models vehicle interactions can be both repulsive and attractive depending on the distance of the interacting pairs.Proposition 3 (equilibrium of the macroscopic model).
The spatially homogeneous solution (14) is a locally stable and attractive solution to (2).For the sake of completeness, we report the proofs of Propositions2 and 3 in Appendix 5.


### 3.3. Discrete and Continuous Speed Diagrams

We now calculate and compare the speed diagrams corresponding to the stable stationary homogeneous solutions studied in the previous sections, that is, the mappingsN
↦
v
[
ϵ
t
N
] and N
↦
v
[
ρ
¯
N
], respectively.Specifically, from Proposition2 we know(20)
v
ϵ
t
N
=
v
d
-
∑
h
=
1
N
-
1
K
h
L
N
.While from (10) with d
=
1, μ
t
=
ρ
¯
N
=
N
/
L and taking Assumption 1 into account we deduce(21)
v
ρ
¯
N
=
v
d
-
N
L
∫
0
L
K
z
d
z
.Notice that bothv
[
ϵ
t
N
] and v
[
ρ
¯
N
] are decreasing functions of N, the trend being definitely linear in the continuous case. This is consistent with typical speed diagrams for pedestrians reported in the experimental literature; see, for example, [42, 43].In order to compare the two speed diagrams we introduce the quantity(22)
Δ
v
N
≔
v
ϵ
t
N
-
v
ρ
¯
N
. Actually, since in view of Proposition 2 the equilibrium speed v
[
ϵ
t
N
] depends only on the headways X
t
j
-
X
t
i, which are constant in time, we can drop the dependence on t by freezing pedestrians in a particular configuration, for instance, the one with X
¯
i
=
(
i
-
1
)
(
L
/
N
). Hence we will write simply ϵ
¯
N
=
∑
i
=
1
N
δ
X
¯
i.From Assumption1(ii) on the regularity of K, we can calculate Δ
v
(
N
) explicitly. To do that, we preliminarily introduce the following pairwise disjoint partition of the interval [
0
,
L
] (cf. Figure 3): (23)
E
1
/
2
1
=
0
,
L
2
N
,
E
i
=
i
-
3
2
L
N
,
i
-
1
2
L
N
,
2
≤
i
≤
N
,
E
1
/
2
N
+
1
=
N
-
1
2
L
N
,
L
,which is such that X
¯
1
∈
E
1
/
2
1, X
¯
i
∈
E
i for i
=
2
,
…
,
N, while E
1
/
2
N
+
1 does not contain any of the atoms of ϵ
¯
N. Then we have (24)
Δ
v
N
=
∫
0
L
K
z
d
ρ
¯
N
-
ϵ
¯
N
z
=
∫
E
1
/
2
1
K
z
d
ρ
¯
N
-
ϵ
¯
N
z
+
∑
i
=
2
N
∫
E
i
K
z
d
ρ
¯
N
-
ϵ
¯
N
z
+
∫
E
1
/
2
N
+
1
K
z
d
ρ
¯
N
z and in particular we compute the following: (i)
For the first integral,(25)
∫
E
1
/
2
1
K
z
d
ρ
¯
N
-
ϵ
¯
N
z
=
N
L
∫
E
1
/
2
1
K
z
d
z
-
K
X
¯
1
=
1
2
K
E
1
/
2
1
-
K
0
=
1
2
K
E
1
/
2
1
becauseX
¯
1
=
0 in the chosen configuration and moreover K
(
0
)
=
0 (cf. Assumption 1(iii)). We have denoted K
E
≔
⨏
E
K
(
z
)
d
z, where ⨏ is the integral mean.
(ii)
For each of the integrals in the sum,(26)
∫
E
i
K
z
d
ρ
¯
N
-
ϵ
¯
N
z
=
N
L
∫
E
i
K
z
d
z
-
K
X
¯
i
=
K
E
i
-
K
X
¯
i
.
(iii)
For the last integral,(27)
∫
E
1
/
2
N
K
z
d
ρ
¯
N
z
=
N
L
∫
E
1
/
2
N
+
1
K
z
d
z
=
1
2
K
E
1
/
2
N
+
1
.Figure 3
The considered partition of the domain0
,
L is shown in the case N
=
5. Solid dots correspond to pedestrian positions. The dashed line portrays an example of interaction kernel K.It follows that(28)
Δ
v
N
=
1
2
K
E
1
/
2
1
+
∑
i
=
2
N
K
E
i
-
K
X
¯
i
+
1
2
K
E
1
/
2
N
+
1
.A numerical evaluation ofΔ
v
(
N
) is reported in Figure 4 (up to a scaling with respect to K
(
0
+
)). We observe that when N grows, the normalized curves approach the constant 1
/
2, thus suggesting that Δ
v
(
N
) does not converge to 0 for N
→
∞. This intuition is confirmed by the following result.Figure 4
Speed diagrams (20) (discrete model, dots) and (21) (continuous model, solid lines) obtained, for different values of L, with v
d
=
1, K
(
z
)
=
(
1
/
5
)
(
1
-
z
2
). The graph in the upper-right box shows the convergence to 0.5 of the quantity Δ
v
(
N
)
/
K
(
0
+
) for N
→
∞.Theorem 4 (nonconvergence of speed diagrams).
One has(29)
lim
N
→
∞
⁡
Δ
v
N
=
1
2
K
0
+
.Proof.
We consider one by one the terms at the right-hand side of (28).
First, the right endpoint ofE
1
/
2
1 approaches the origin when N grows; hence, owing to the mean value theorem and using the continuity of K in (
0
,
R
) (cf. Assumption 1(ii)), we find(30)
lim
N
→
∞
⁡
1
2
K
E
1
/
2
1
=
1
2
K
0
+
.
Second, in view of the smoothness ofK in (
0
,
R
) (cf. Assumption 1(ii)), for each term of the sum we can use a second-order Taylor expansion with Lagrange remainder to discover (31)
K
E
i
-
K
X
¯
i
=
⨏
E
i
K
X
¯
i
+
K
′
X
¯
i
z
-
X
¯
i
+
1
2
K
′
′
ζ
i
z
-
X
¯
i
2
d
z
-
K
X
¯
i(for  some  ζ
i
∈
E
i), (32)
K
E
i
-
K
X
¯
i
=
⨏
E
i
K
′
X
¯
i
z
-
X
¯
i
+
1
2
K
′
′
ζ
i
z
-
X
¯
i
2
d
z
=
1
2
⨏
E
i
K
′
′
ζ
i
z
-
X
¯
i
2
d
z(because z
-
X
¯
i is odd in E
i), and (33)
K
E
i
-
K
X
¯
i
≤
1
2
K
′
′
∞
⨏
E
i
z
-
X
¯
i
2
d
z
=
1
2
K
′
′
∞
L
2
12
N
2
→
N
→
∞
0which, for the arbitrariness of i, implies that the whole sum vanishes for N
→
∞.
Finally,(34)
lim
N
→
∞
⁡
1
2
K
E
1
/
2
N
+
1
=
0 because of Assumption 1(i); indeed for large N the following results: E
1
/
2
N
+
1
∩
[
0
,
R
]
=
∅.From Theorem4 we conclude that the discrete system moves asymptotically at a higher speed than the continuous one because K
(
0
+
)
>
0, as also Figure 4 confirms. Ultimately, the discrete and continuous models (1), (2) predict different walking speeds at equilibrium, which do not match one another even in the limit of a large number of pedestrians. It is worth noticing that this fact does not actually depend on Assumption 1(iii), which states the absence of self-interactions (K
(
0
)
=
0). Indeed, assuming the right continuity of K in 0, that is, K
(
0
)
=
K
(
0
+
), would still lead to a nonzero limit for Δ
v
(
N
),(35)
lim
N
→
∞
⁡
Δ
v
N
=
lim
N
→
∞
⁡
1
2
K
E
1
/
2
1
-
K
0
=
-
1
2
K
0
<
0(cf. (24)). In this case, the discrete system is asymptotically slower than the continuous one, because discrete pedestrians are further slowed down by self-interactions (which, instead, do not affect the continuous system).Additionally, from Theorem4 we infer that speed diagrams approaching one another can be obtained if K
(
0
+
)
=
K
(
0
)
=
0 (cf. Figure 5). This condition however violates the assumption that K is decreasing in (
0
,
R
) (cf. Assumption 1(iii)), which is used to prove the stability and attractiveness of the homogeneous configurations on which speed diagrams are based (cf. Propositions 2 and 3).Figure 5
Speed diagrams (20) (discrete model, dots) and (21) (continuous model, solid lines) obtained, for different values of L, with v
d
=
1, K
(
z
)
=
(
1
/
2
)
z
(
1
-
z
). The graph in the upper-right box shows the convergence to 0 of the quantity Δ
v
(
N
) for N
→
∞.


## 3.1. Modeling Hypotheses

For the sake of simplicity, we consider the one-dimensional problem on a periodic domain[
0
,
L
). In order to model homogeneous pedestrian distributions, in the discrete case (1) we consider an equispaced lattice solution translating with a certain constant speed w (to be determined); that is,(12)
ϵ
t
x
=
ϵ
¯
x
-
w
t
,where ϵ
¯ has form (8) with atom locations such that(13)
X
¯
j
-
X
¯
i
=
j
-
i
L
N
,
i
,
j
=
1
,
…
,
N
,for X
¯
1
<
X
¯
2
<
⋯
<
X
¯
N (the ordering is modulo L). Thus the atoms of ϵ
t are X
t
i
=
X
¯
i
+
w
t for i
=
1
,
…
,
N. In the continuous case (2), we consider instead a constant density; that is,(14)
ρ
t
x
≡
ρ
¯
≥
0
,which, owing to (4), is given by ρ
¯
=
N
/
L.We assume that the desired velocity is a positive constantv
d
>
0; therefore the movement is in the positive direction of the real line. Furthermore, we make the following assumptions on the interaction kernel (cf. Figure 2).Figure 2
Prototype of interaction kernelK complying with Assumption 1.Assumption 1 (properties ofK).
We make the following assumptions(i)
Compactness of the support and frontal orientation of the sensory region: the support of K is(15)
S
R
0
=
0
,
R
with0
<
R
<
L.
(ii)
Boundedness and regularity in
(
0
,
R
): pedestrian interactions vary smoothly with the mutual distance of the interacting individuals and have a finite maximum value. Specifically,(16)
K
∈
C
2
0
,
R
,
K
,
K
′
′
∈
L
∞
0
,
R
.
(iii)
Monotonicity in
(
0
,
R
)
and behavior at the endpoints: we assume(17)
K
z
>
0
,
K
′
z
<
0
,
for
z
∈
0
,
R
with, moreover,(18)
K
0
=
K
R
=
0
,
K
0
+
≔
lim
z
→
0
+
⁡
K
z
>
0
.
Thus pedestrian interactions decay in the interior of the sensory region as the mutual distance increases and, moreover, pedestrians do not “self-interact.” This forcesK to be discontinuous in z
=
0.


## 3.2. Stationarity and Stability of Spatially Homogeneous Solutions

Before proceeding with the comparison of asymptotic pedestrian speeds resulting from microscopic and macroscopic dynamics, we ascertain that the spatially homogeneous solutions (12)-(13) and (14) are indeed stable and possibly attractive solutions to either (1) or (2). This ensures that such distributions can indeed be considered as equilibrium distributions and therefore that the evaluation of speed diagrams is well-posed.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 [23, 35–37] and references therein. We consider here much simpler one-dimensional configurations, however useful in this context because they reproduce mathematically the typical experimental setups in which pedestrian speed diagrams are measured; see, for example, [38, 39].Proposition 2 (equilibrium of the microscopic model).
The equispaced lattice solution (12) is a stable solution to (1) for(19)
w
=
v
d
-
∑
h
=
1
N
-
1
K
h
L
N
.It is moreover attractive if R
>
L
/
N.Proposition2 asserts that the equispaced pedestrian distribution is always a stable (quasi)stationary solution to the microscopic model. This is somehow in contrast to what is found in microscopicoptimal-velocity traffic models, where the so-calledPOMs (“Ponies-on-a-Merry-Go-Round”)solutions can generate instabilities (traffic jams) depending on the total number of vehicles [40, 41]. The rationale for this difference is that, unlike the present case, in such models vehicle interactions can be both repulsive and attractive depending on the distance of the interacting pairs.Proposition 3 (equilibrium of the macroscopic model).
The spatially homogeneous solution (14) is a locally stable and attractive solution to (2).For the sake of completeness, we report the proofs of Propositions2 and 3 in Appendix 5.


## 3.3. Discrete and Continuous Speed Diagrams

We now calculate and compare the speed diagrams corresponding to the stable stationary homogeneous solutions studied in the previous sections, that is, the mappingsN
↦
v
[
ϵ
t
N
] and N
↦
v
[
ρ
¯
N
], respectively.Specifically, from Proposition2 we know(20)
v
ϵ
t
N
=
v
d
-
∑
h
=
1
N
-
1
K
h
L
N
.While from (10) with d
=
1, μ
t
=
ρ
¯
N
=
N
/
L and taking Assumption 1 into account we deduce(21)
v
ρ
¯
N
=
v
d
-
N
L
∫
0
L
K
z
d
z
.Notice that bothv
[
ϵ
t
N
] and v
[
ρ
¯
N
] are decreasing functions of N, the trend being definitely linear in the continuous case. This is consistent with typical speed diagrams for pedestrians reported in the experimental literature; see, for example, [42, 43].In order to compare the two speed diagrams we introduce the quantity(22)
Δ
v
N
≔
v
ϵ
t
N
-
v
ρ
¯
N
. Actually, since in view of Proposition 2 the equilibrium speed v
[
ϵ
t
N
] depends only on the headways X
t
j
-
X
t
i, which are constant in time, we can drop the dependence on t by freezing pedestrians in a particular configuration, for instance, the one with X
¯
i
=
(
i
-
1
)
(
L
/
N
). Hence we will write simply ϵ
¯
N
=
∑
i
=
1
N
δ
X
¯
i.From Assumption1(ii) on the regularity of K, we can calculate Δ
v
(
N
) explicitly. To do that, we preliminarily introduce the following pairwise disjoint partition of the interval [
0
,
L
] (cf. Figure 3): (23)
E
1
/
2
1
=
0
,
L
2
N
,
E
i
=
i
-
3
2
L
N
,
i
-
1
2
L
N
,
2
≤
i
≤
N
,
E
1
/
2
N
+
1
=
N
-
1
2
L
N
,
L
,which is such that X
¯
1
∈
E
1
/
2
1, X
¯
i
∈
E
i for i
=
2
,
…
,
N, while E
1
/
2
N
+
1 does not contain any of the atoms of ϵ
¯
N. Then we have (24)
Δ
v
N
=
∫
0
L
K
z
d
ρ
¯
N
-
ϵ
¯
N
z
=
∫
E
1
/
2
1
K
z
d
ρ
¯
N
-
ϵ
¯
N
z
+
∑
i
=
2
N
∫
E
i
K
z
d
ρ
¯
N
-
ϵ
¯
N
z
+
∫
E
1
/
2
N
+
1
K
z
d
ρ
¯
N
z and in particular we compute the following: (i)
For the first integral,(25)
∫
E
1
/
2
1
K
z
d
ρ
¯
N
-
ϵ
¯
N
z
=
N
L
∫
E
1
/
2
1
K
z
d
z
-
K
X
¯
1
=
1
2
K
E
1
/
2
1
-
K
0
=
1
2
K
E
1
/
2
1
becauseX
¯
1
=
0 in the chosen configuration and moreover K
(
0
)
=
0 (cf. Assumption 1(iii)). We have denoted K
E
≔
⨏
E
K
(
z
)
d
z, where ⨏ is the integral mean.
(ii)
For each of the integrals in the sum,(26)
∫
E
i
K
z
d
ρ
¯
N
-
ϵ
¯
N
z
=
N
L
∫
E
i
K
z
d
z
-
K
X
¯
i
=
K
E
i
-
K
X
¯
i
.
(iii)
For the last integral,(27)
∫
E
1
/
2
N
K
z
d
ρ
¯
N
z
=
N
L
∫
E
1
/
2
N
+
1
K
z
d
z
=
1
2
K
E
1
/
2
N
+
1
.Figure 3
The considered partition of the domain0
,
L is shown in the case N
=
5. Solid dots correspond to pedestrian positions. The dashed line portrays an example of interaction kernel K.It follows that(28)
Δ
v
N
=
1
2
K
E
1
/
2
1
+
∑
i
=
2
N
K
E
i
-
K
X
¯
i
+
1
2
K
E
1
/
2
N
+
1
.A numerical evaluation ofΔ
v
(
N
) is reported in Figure 4 (up to a scaling with respect to K
(
0
+
)). We observe that when N grows, the normalized curves approach the constant 1
/
2, thus suggesting that Δ
v
(
N
) does not converge to 0 for N
→
∞. This intuition is confirmed by the following result.Figure 4
Speed diagrams (20) (discrete model, dots) and (21) (continuous model, solid lines) obtained, for different values of L, with v
d
=
1, K
(
z
)
=
(
1
/
5
)
(
1
-
z
2
). The graph in the upper-right box shows the convergence to 0.5 of the quantity Δ
v
(
N
)
/
K
(
0
+
) for N
→
∞.Theorem 4 (nonconvergence of speed diagrams).
One has(29)
lim
N
→
∞
⁡
Δ
v
N
=
1
2
K
0
+
.Proof.
We consider one by one the terms at the right-hand side of (28).
First, the right endpoint ofE
1
/
2
1 approaches the origin when N grows; hence, owing to the mean value theorem and using the continuity of K in (
0
,
R
) (cf. Assumption 1(ii)), we find(30)
lim
N
→
∞
⁡
1
2
K
E
1
/
2
1
=
1
2
K
0
+
.
Second, in view of the smoothness ofK in (
0
,
R
) (cf. Assumption 1(ii)), for each term of the sum we can use a second-order Taylor expansion with Lagrange remainder to discover (31)
K
E
i
-
K
X
¯
i
=
⨏
E
i
K
X
¯
i
+
K
′
X
¯
i
z
-
X
¯
i
+
1
2
K
′
′
ζ
i
z
-
X
¯
i
2
d
z
-
K
X
¯
i(for  some  ζ
i
∈
E
i), (32)
K
E
i
-
K
X
¯
i
=
⨏
E
i
K
′
X
¯
i
z
-
X
¯
i
+
1
2
K
′
′
ζ
i
z
-
X
¯
i
2
d
z
=
1
2
⨏
E
i
K
′
′
ζ
i
z
-
X
¯
i
2
d
z(because z
-
X
¯
i is odd in E
i), and (33)
K
E
i
-
K
X
¯
i
≤
1
2
K
′
′
∞
⨏
E
i
z
-
X
¯
i
2
d
z
=
1
2
K
′
′
∞
L
2
12
N
2
→
N
→
∞
0which, for the arbitrariness of i, implies that the whole sum vanishes for N
→
∞.
Finally,(34)
lim
N
→
∞
⁡
1
2
K
E
1
/
2
N
+
1
=
0 because of Assumption 1(i); indeed for large N the following results: E
1
/
2
N
+
1
∩
[
0
,
R
]
=
∅.From Theorem4 we conclude that the discrete system moves asymptotically at a higher speed than the continuous one because K
(
0
+
)
>
0, as also Figure 4 confirms. Ultimately, the discrete and continuous models (1), (2) predict different walking speeds at equilibrium, which do not match one another even in the limit of a large number of pedestrians. It is worth noticing that this fact does not actually depend on Assumption 1(iii), which states the absence of self-interactions (K
(
0
)
=
0). Indeed, assuming the right continuity of K in 0, that is, K
(
0
)
=
K
(
0
+
), would still lead to a nonzero limit for Δ
v
(
N
),(35)
lim
N
→
∞
⁡
Δ
v
N
=
lim
N
→
∞
⁡
1
2
K
E
1
/
2
1
-
K
0
=
-
1
2
K
0
<
0(cf. (24)). In this case, the discrete system is asymptotically slower than the continuous one, because discrete pedestrians are further slowed down by self-interactions (which, instead, do not affect the continuous system).Additionally, from Theorem4 we infer that speed diagrams approaching one another can be obtained if K
(
0
+
)
=
K
(
0
)
=
0 (cf. Figure 5). This condition however violates the assumption that K is decreasing in (
0
,
R
) (cf. Assumption 1(iii)), which is used to prove the stability and attractiveness of the homogeneous configurations on which speed diagrams are based (cf. Propositions 2 and 3).Figure 5
Speed diagrams (20) (discrete model, dots) and (21) (continuous model, solid lines) obtained, for different values of L, with v
d
=
1, K
(
z
)
=
(
1
/
2
)
z
(
1
-
z
). The graph in the upper-right box shows the convergence to 0 of the quantity Δ
v
(
N
) for N
→
∞.


## 4. General Nonstationary Dynamics

In this section we consider the Cauchy problem(36)
∂
t
μ
t
+
div
⁡
μ
t
v
μ
t
=
0
,
t
∈
0
,
T
,
x
∈
R
d
μ
0
=
μ
¯
,where v is given by (10), T
>
0 is a certain final time, and μ
¯ is a prescribed measure representing, at the proper scale, the initial distribution of the crowd. Using the results in [1, 28] we can state that, under suitable assumptions encompassing those that we will recall later in Section 4.1, problem (36) admits a unique measure-valued solution μ
·
∈
C
(
[
0
,
T
]
;
M
1
N
(
R
d
)
) in the weak sense (5), M
1
N
(
R
d
) being the space of positive measures on R
d with total mass N and finite first-order moment. Moreover, such a solution preserves the structure of the initial datum: if μ
¯ is discrete, respectively, continuous, then so is μ
t for all t
∈
0
,
T.It makes thus sense to consider sequences of discrete and continuous initial conditionsϵ
¯
N
N
=
1
∞
,
ρ
¯
N
N
=
1
∞, with ϵ
¯
N
,
ρ
¯
N
∈
M
1
N
(
R
d
)
∀
N
≥
1, to which sequences of solutions correspond at the same scales ϵ
·
N
N
=
1
∞
,
{
ρ
·
N
}
N
=
1
∞, with ϵ
·
N
,
ρ
·
N
∈
C
(
[
0
,
T
]
;
M
1
N
(
R
d
)
)
∀
N
≥
1, and to compare them “N-by-N” in order to determine when mutually approaching initial measures, that is,(37)
lim
N
→
∞
⁡
d
ϵ
¯
N
,
ρ
¯
N
=
0for some metric d, generate mutually approaching solutions; that is,(38)
lim
N
→
∞
⁡
d
ϵ
t
N
,
ρ
t
N
=
0
,
∀
t
∈
0
,
T
.Formally speaking, we will operate in the setting of the1-Wasserstein distance W
1, whose definition is as follows:(39)
W
1
μ
,
ν
=
i
n
f
π
∈
Π
μ
,
ν
∫
R
d
×
R
d
x
-
y
d
π
x
,
y
,
μ
,
ν
∈
M
1
N
R
d
,where Π
(
μ
,
ν
) is the set of all transference plans between the measures μ and ν; that is, every π
∈
Π
(
μ
,
ν
) is a measure on the product space R
d
×
R
d with marginal measures μ, ν: π
(
E
×
R
d
)
=
μ
(
E
) and π
(
R
d
×
E
)
=
ν
(
E
) for every E
∈
B
(
R
d
). By Kantorovich duality (cf., e.g., [44]), W
1 admits also the representation(40)
W
1
μ
,
ν
=
sup
ϕ
∈
Lip
1
⁡
R
d
⁡
​
∫
R
d
ϕ
d
ν
-
μ
,where Lip
1
⁡
R
d is the space of Lipschitz continuous functions ϕ
:
R
d
→
R with at most unitary Lipschitz constant. We will use indifferently either expression of W
1 depending on the context.In the following, from Sections4.1–4.3, we recall general results about the solution to (36) independently of the geometric structure of the measure. Such results will allow us to discuss, later in Section 4.4, the limits (37)-(38) previously introduced.


### 4.1. Modeling Hypotheses

Following the theory developed in [1, 27, 28], we assume some smoothness of the transport velocity. Consider, specifically, the following.Assumption 5 (Lipschitz continuity ofv).
There existLip
⁡
(
v
d
)
>
0 and Lip
⁡
(
K
)
>
0 such that(41)
v
d
y
-
v
d
x
≤
Lip
⁡
v
d
y
-
x
,
K
y
-
K
x
≤
Lip
⁡
K
y
-
xfor all x
,
y
∈
R
d.Using the expression (10) of the transport velocity, it is immediate to check that Assumption 5 implies(42)
v
ν
y
-
v
μ
x
≤
Lip
⁡
v
d
+
N
Lip
⁡
K
y
-
x
+
Lip
⁡
K
W
1
μ
,
νfor all x
,
y
∈
R
d and μ
,
ν
∈
M
1
N
(
R
d
), hence letting(43)
ξ
N
≔
2
max
⁡
Lip
⁡
v
d
,
N
Lip
⁡
Kand recalling that we are considering N
≥
1 we finally have(44)
v
ν
y
-
v
μ
x
≤
ξ
N
y
-
x
+
1
N
W
1
μ
,
ν
.


### 4.2. Continuous Dependence on the Initial Datum

The basic tool for the subsequent analysis is the continuous dependence of the solution to (36) on the initial datum, which can be proved using the representation formula (6) based on the flow map (7). The result itself and the analytical technique to obtain it are classical in the theory of problem (36); see, for example, [45–47]. They have also been extensively used to analyze swarming models; see, for example, [48] and the thorough review [49]. However, in the present case it is crucial to obtain the explicit dependence on N of some constants appearing in the final estimates, which is less classical due to the fact that here the total mass of the system is not 1 but N. For this reason, in order to make the paper self-contained, we detail in Appendix 5 the proofs of the following two results.Lemma 6 (regularity of the flow map).
Consider the flow map (7):(i)
For allx
,
y
∈
R
d and t
∈
[
0
,
T
] the following results: (45)
γ
t
y
-
γ
t
x
≤
e
ξ
N
t
y
-
x
.
(ii)
Letμ
·
,
ν
·
∈
C
(
[
0
,
T
]
;
M
1
N
(
R
d
)
) be two solutions to (36) with respective initial conditions μ
¯
,
ν
¯
∈
M
1
N
(
R
d
). Call γ
μ, γ
ν the flow maps associated with either solution. Then for all x
∈
R
d and t
∈
[
0
,
T
] the following results:(46)
γ
t
ν
x
-
γ
t
μ
x
≤
ξ
N
e
ξ
N
t
N
∫
0
t
W
1
μ
s
,
ν
s
d
s
.Proposition 7 (continuous dependence).
Letμ
·
,
ν
·
∈
C
(
[
0
,
T
]
;
M
1
N
(
R
d
)
) be two solutions to (36) corresponding to initial conditions μ
¯
,
ν
¯
∈
M
1
N
(
R
d
). Then(47)
W
1
μ
t
,
ν
t
≤
e
ξ
N
t
1
+
e
ξ
N
T
W
1
μ
¯
,
ν
¯
,
∀
t
∈
0
,
T
.


### 4.3. Sequences of Measures with Growing Mass: Scaling of the Interactions

Let us now consider two sequences of initial conditions of growing mass, sayμ
¯
N
N
=
1
∞
,
ν
¯
N
N
=
1
∞ with μ
¯
N
,
ν
¯
N
∈
M
1
N
(
R
d
)
∀
N
≥
1, and the related sequences of solutions to (36), μ
·
N
N
=
1
∞
,
ν
·
N
N
=
1
∞ with μ
·
N
,
ν
·
N
∈
C
(
[
0
,
T
]
;
M
1
N
(
R
d
)
)
∀
N
≥
1. From Proposition 7 we have(48)
W
1
μ
t
N
,
ν
t
N
≤
e
ξ
N
t
1
+
e
ξ
N
T
W
1
μ
¯
N
,
ν
¯
N
,where the exponential factor estimates the amplification at time t
>
0 of the distance between the initial data. If N is sufficiently large then from (43) we have ξ
N
=
2
N
Lip
⁡
(
K
). In order for the exponential factor to remain bounded for growing N and ensure that W
1
(
μ
¯
N
,
ν
¯
N
) and W
1
(
μ
t
N
,
ν
t
N
) are of the same order of magnitude for all N, it is necessary that(49)
Lip
⁡
K
=
O
N
-
1
for
N
⟶
∞
.In other words, we have to suitably scale the interaction kernel with respect to N. In particular, given a Lipschitz continuous function K
:
R
d
→
R
d compactly supported in B
R
(
0
)
⊂
R
d, to comply with (49) we consider the following two-parameter family of interaction kernels:(50)
K
z
=
K
α
,
β
N
z
≔
1
N
α
K
z
N
β
,
α
,
β
∈
R
,whose Lipschitz constant is(51)
Lip
⁡
K
α
,
β
N
=
Lip
⁡
K
N
α
+
β
.Clearly, they satisfy (49) as long as(52)
α
+
β
≥
1
.Example 8 (role ofα).
An admissible interaction kernel in the family (50) is obtained for α
=
1, β
=
0; that is,(53)
K
1,0
N
z
=
1
N
K
z
.This corresponds to decreasing interpersonal repulsion when the number of pedestrians increases. Notice that this is the same scaling adopted in the mean field limit [23, 24]. In this case, pedestrian velocity (10) reads(54)
v
μ
t
N
x
=
v
d
x
-
1
N
∫
R
d
K
y
-
x
d
μ
t
N
y
.Considering that μ
t
N
(
R
d
)
=
N, the desired velocity and the interactions have commensurable weights for every N.Example 9 (role ofβ).
An interaction kernel somehow opposite toK
1,0
N (cf. Example 8) is obtained for α
=
0, β
=
1; that is,(55)
K
0,1
N
z
=
K
z
N
,whose support is contained in the ball B
N
R
(
0
). This kernel corresponds to pedestrians who interact with an increasing number of individuals as the total number of people increases. The resulting velocity(56)
v
μ
t
N
x
=
v
d
x
-
∫
R
d
K
y
-
x
N
d
μ
t
N
yis such that the component due to interactions tends to predominate over the desired one for growing N.Example 10.
Besides the extreme cases in Examples8 and 9, we may also consider intermediate cases in which both α and β are simultaneously nonzero (and not necessarily positive). For instance, for α
=
2, β
=
-
1 we get(57)
K
2
,
-
1
N
z
=
1
N
2
K
N
z
,which corresponds to interactions that weaken and that are restricted to a contracting sensory region (in fact supp
⁡
K
2
,
-
1
N
⊆
B
R
/
N
(
0
)) as N grows. This models pedestrians who agree to stay closer and closer as their number increases, like, for example, in highly crowded train or metro stations during rush hours. The resulting velocity(58)
v
μ
t
N
x
=
v
d
x
-
1
N
2
∫
R
d
K
N
y
-
x
d
μ
t
N
yis such that v
d dominates for large N, meaning that at high crowding pedestrians tend to be passively transported by the flow without interacting.A scaling of interactions of type (50) is proposed, for example, in [50]. However, it involves only one parameter γ
∈
(
0,1
), corresponding to α
=
-
γ and β
=
-
γ
/
d in (49). As this condition does not satisfy (52), it has to be regarded as a complementary case, not covered by the theory developed here.


#### 4.3.1. Scaling Equivalence

Solutions to (36) with interaction kernel (50) for different values of the parameters α, β account for different interpersonal attitudes of pedestrians in congested crowd regimes. Hence, we may expect significantly different solutions for large N. Nonetheless, in the case v
d
≡
0 a one-to-one correspondence among them exists, up to a transformation of the space variable depending on N. In order to prove it we preliminarily introduce the following.Proposition 11.
LetU
:
R
d
→
R
d be the linear scaling of the space(59)
U
z
=
a
z
,
a
∈
R and let v
^, v
~ be the velocities (10) computed with the following interaction kernels:(60)
K
^
z
=
K
∘
U
-
1
z
=
K
z
a
,
K
~
z
=
U
-
1
∘
K
z
=
1
a
K
z
,where K
:
R
d
→
R
d is Lipschitz continuous. Assume moreover v
d
≡
0.(i)
For allμ
∈
M
1
N
(
R
d
) the following results:(61)
v
~
μ
x
=
1
a
v
^
U
#
μ
a
x
.
Letμ
^
·
,
μ
~
·
∈
C
(
[
0
,
T
]
;
M
1
N
(
R
d
)
) be the solutions to (36) with velocities v
^, v
~, respectively, and initial conditions such that(62)
μ
^
0
=
U
#
μ
~
0
.
(ii)
The flow mapsγ
^
t, γ
~
t correspond to one another as(63)
γ
^
t
=
U
∘
γ
~
t
∘
U
-
1
,
∀
t
∈
0
,
T
.
(iii)
The solutions satisfy(64)
μ
^
t
=
U
#
μ
~
t
,
∀
t
∈
0
,
T
.Proof.
(i) By direct calculation we find(65)
1
a
v
^
U
#
μ
a
x
=
-
1
a
∫
R
d
K
^
y
-
a
x
d
U
#
μ
y
=
-
1
a
∫
R
d
K
y
-
a
x
a
d
U
#
μ
y
=
-
1
a
∫
R
d
K
z
-
x
d
μ
z
=
-
∫
R
d
K
~
z
-
x
d
μ
z
=
v
~
μ
x
.
(ii) We check that(66)
γ
^
t
x
≔
U
∘
γ
~
t
∘
U
-
1
x
=
a
γ
~
t
x
acomplies with definition (7). Using (7) for γ
~
t, we write(67)
γ
^
t
x
=
x
+
a
∫
0
t
v
~
γ
~
s
#
μ
~
0
γ
~
s
x
a
d
s
.
Next we observe that(68)
γ
~
s
#
μ
~
0
=
γ
~
s
∘
U
-
1
∘
U
#
μ
~
0
=
γ
~
s
∘
U
-
1
#
μ
^
0
;hence from the previous point (i) we deduce (69)
γ
^
t
x
=
x
+
a
∫
0
t
1
a
v
^
U
#
γ
~
s
∘
U
-
1
#
μ
^
0
a
γ
~
s
x
a
d
s
=
x
+
∫
0
t
v
^
γ
^
s
#
μ
^
0
γ
^
s
x
d
s
,as desired.
(iii) Due to the result in (ii) we have(70)
μ
^
t
=
γ
^
t
#
μ
^
0
=
U
∘
γ
~
t
∘
U
-
1
#
U
#
μ
~
0
=
U
∘
γ
~
t
#
μ
~
0
=
U
#
μ
~
t
.As a consequence of Proposition11, we can prove a correspondence among the dynamics governed by different interaction kernels of family (50).Theorem 12 (scaling equivalence).
Letμ
·
N
,
ν
·
N
∈
C
(
[
0
,
T
]
;
M
1
N
(
R
d
)
) be the solutions to (36) corresponding to interaction kernels K
α
,
β
N, K
α
′
,
β
′
N, with(71)
α
+
β
=
α
′
+
β
′
,and to initial conditions μ
¯
N
,
ν
¯
N
=
U
N
#
μ
¯
N
∈
M
1
N
(
R
d
), respectively, where(72)
U
N
z
=
N
β
′
-
β
z
.Then(73)
ν
t
N
=
U
N
#
μ
t
N
,
∀
t
∈
0
,
T
.Proof.
By (50) we have, on the one hand,(74)
K
α
,
β
N
z
=
1
N
α
K
z
N
β
=
1
N
β
′
-
β
·
1
N
α
′
K
z
N
β
,where we have used the fact that α
=
α
′
+
β
′
-
β, and, on the other hand,(75)
K
α
′
,
β
′
N
z
=
1
N
α
′
K
z
N
β
′
=
1
N
α
′
K
1
N
β
′
-
β
·
z
N
β
.Thus, (76)
K
α
,
β
N
z
=
1
a
·
1
N
α
′
K
z
N
β
,
K
α
′
,
β
′
N
z
=
1
N
α
′
K
1
a
·
z
N
βfor a
=
N
β
′
-
β and the thesis follows from Proposition 11.


### 4.4. Back to Discrete and Continuous Models

Conditions (50) and (52) imply(77)
ξ
N
≤
ξ
∗
≔
2
max
⁡
Lip
⁡
v
d
,
Lip
⁡
K
,
∀
N
≥
1
.When using (48) for sequences of discrete and continuous measures we can incorporate this fact and write(78)
W
1
ϵ
t
N
,
ρ
t
N
≤
e
ξ
∗
t
1
+
e
ξ
∗
T
W
1
ϵ
¯
N
,
ρ
¯
N
,
∀
t
∈
0
,
T
,
∀
N
≥
1
.Hence mutually approaching sequences of discrete and continuous solutions to (36) (cf. (38)) are possible, provided one is able to construct mutually approaching sequences of initial conditions at the corresponding scales (cf. (37)). In the following we discuss a possible procedure leading to the desired result.LetX
¯
1
,
…
,
X
¯
N
∈
R
d be the initial positions of N distinct microscopic pedestrians. The associated discrete distribution ϵ
¯
N is constructed from (8), while, given r
>
0, we define(79)
ρ
¯
N
=
∑
i
=
1
N
ρ
i
,
ρ
i
x
=
1
r
d
f
x
-
X
¯
i
r
,where f
:
R
d
→
R is a nonnegative function such that(80)
supp
⁡
f
⊆
B
1
0
¯
,
∫
B
1
0
f
x
d
x
=
1
.Consequently, supp
⁡
ρ
i
⊆
B
r
(
X
¯
i
)
¯ and moreover ρ
i
(
B
r
(
X
¯
i
)
)
=
1 each i; thus ρ
¯
N is the superposition of N piecewise constant density bumps, each of which carries a unit mass representative of one pedestrian. We assume(81)
r
<
1
2
m
i
n
i
≠
j
X
¯
j
-
X
¯
iwhich ensures no overlapping of the supports of the ρ
i’s.In the remaining part of this section we compute the Wasserstein distance betweenϵ
¯
N and ρ
¯
N and we study its trend with respect to N.Proposition 13 (distance between the initial conditions).
Let(82)
m
f
≔
∫
B
1
0
x
f
x
d
x
.The distance between ϵ
¯
N and ρ
¯
N is(83)
W
1
ϵ
¯
N
,
ρ
¯
N
=
m
f
N
r
.Proof.
Here we use expression (40) of W
1. Let us consider the case N
=
1 first. Since ϵ
¯
1 is a single Dirac mass, there is no ambiguity in the construction of the optimal transference plan π between ϵ
¯
1 and ρ
¯
1 (i.e., the transference plan which realizes the infimum in (39)), which is just the tensor product of the measures:(84)
π
x
,
y
=
ρ
¯
1
x
⊗
ϵ
¯
1
y
. By substitution in (39) we find (85)
W
1
ϵ
¯
1
,
ρ
¯
1
=
∫
R
d
×
R
d
x
-
y
d
ρ
¯
1
x
⊗
ϵ
¯
1
y
=
∫
B
r
X
¯
1
x
-
X
¯
1
ρ
1
x
d
x
=
1
r
d
∫
B
r
X
¯
1
x
-
X
¯
1
f
x
-
X
¯
1
r
d
x
=
r
∫
B
1
0
y
f
y
d
y
s
e
t
y
≔
x
-
X
¯
1
r
=
m
f
r
.
We pass now to characterize transference plans betweenϵ
¯
N and ρ
¯
N in the case N
>
1. Every element of the continuous mass ρ
¯
N is transported onto a delta, a condition that, in the spirit of the so-calledsemidiscrete Monge-Kantorovich problem [51], can be expressed by a measure π on R
d
×
R
d of the form(86)
π
x
,
y
=
ρ
¯
N
x
⊗
∑
j
=
1
N
α
j
x
δ
X
¯
j
y
,where, in order to ensure conservation of the mass, the α
j’s are such that(87)
∑
j
=
1
N
α
j
x
=
1
,
α
j
x
≥
0
,
∀
j
=
1
,
…
,
N
,
∀
x
∈
supp
⁡
ρ
¯
N
.Representation (86) means that the infinitesimal element of continuous mass d
ρ
¯
N
(
x
) located in x
∈
supp
⁡
ρ
¯
N is split in the points X
¯
j
j
=
1
N following the convex combination given by the coefficients α
j
(
x
) (cf. Figure 6). The measure π is generally not a transference plan between ϵ
¯
N and ρ
¯
N, because it is not guaranteed to have marginal ϵ
¯
N with respect to y. In particular, a nonunit mass might be allocated in every X
¯
j. In order to have a transference plan, the further condition,(88)
∫
R
d
α
j
x
d
ρ
¯
N
x
=
1
,
∀
j
=
1
,
…
,
N
,needs to be enforced.
Let us consider, for a transference plan of the form (86) with ρ
¯
N as in (79), the global transportation cost: (89)
∫
R
d
×
R
d
x
-
y
d
π
x
,
y
=
∫
R
d
×
R
d
x
-
y
d
∑
i
=
1
N
ρ
i
x
⊗
∑
j
=
1
N
α
j
x
δ
X
¯
j
y
=
∑
i
,
j
=
1
N
∫
R
d
×
R
d
x
-
y
d
ρ
i
x
⊗
α
j
x
δ
X
¯
j
y
=
∑
i
=
1
N
∫
B
r
X
¯
i
∑
j
=
1
N
α
j
x
x
-
X
¯
j
d
ρ
i
x
. Owing to (87) we have (90)
∑
j
=
1
N
α
j
x
x
-
X
¯
j
=
∑
j
≠
i
α
j
x
x
-
X
¯
j
+
1
-
∑
j
≠
i
α
j
x
x
-
X
¯
iwhence, for x
∈
B
r
(
X
¯
i
) and taking (81) into account,(91)
∑
j
=
1
N
α
j
x
x
-
X
¯
j
≥
r
∑
j
≠
i
α
j
x
+
x
-
X
¯
i
-
r
∑
j
≠
i
α
j
x
=
x
-
X
¯
i
;thus ultimately (92)
∫
R
d
×
R
d
x
-
y
d
π
x
,
y
≥
∑
i
=
1
N
∫
B
r
X
¯
i
x
-
X
¯
i
d
ρ
i
x
=
∑
i
=
1
N
∫
R
d
×
R
d
x
-
y
d
ρ
i
x
⊗
δ
X
¯
i
y
.
Notice that the transference plan(93)
π
=
∑
i
=
1
N
ρ
i
⊗
δ
X
¯
iis of the form (86) for, for example, the coefficients α
j
(
x
)
=
χ
B
r
(
X
¯
j
)
(
x
) which fulfill both (87) and (88). The previous calculation says that it is actually an optimal transference plan between ϵ
¯
N and ρ
¯
N, that is, one which ensures the minimum transportation cost. Thus(94)
W
1
ϵ
¯
N
,
ρ
¯
N
=
∑
i
=
1
N
∫
R
d
×
R
d
x
-
y
d
ρ
i
x
⊗
δ
X
¯
i
y
=
N
W
1
ϵ
¯
1
,
ρ
¯
1
,whence the thesis follows.Figure 6
Transference plans (86) in R
d
×
R
d for N
=
3. Each continuous mass element is within a distance r from its corresponding Dirac mass. Therefore, owing to (81), it is farther than r from all other Dirac masses.Thanks to Proposition13 we finally obtain the main result of the paper.Theorem 14 (discrete-continuous convergence).
Letϵ
¯
N
∈
M
1
N
(
R
d
) be given and let ρ
¯
N
∈
M
1
N
(
R
d
) be constructed as in (79)-(80). Set r
=
c
N
N
-
γ, γ
>
1, where 0
<
c
N
≤
1 is possibly used to enforce condition (81). Let moreover the interaction kernel satisfy (50) and (52). Then(95)
lim
N
→
∞
⁡
W
1
ϵ
t
N
,
ρ
t
N
=
0
,
∀
t
∈
0
,
T
.Proof.
From Proposition13 with the given r we obtain (96)
W
1
ϵ
¯
N
,
ρ
¯
N
=
m
f
c
N
N
γ
-
1
→
N
→
∞
0because γ
>
1. Hence the thesis follows from (78).It is worth remarking that, because of (81), in dimension d a bound on the radius r of the form r
<
C
N
-
1
/
d, where C
>
0 is a constant, holds true, for example, when considering homogeneous distributions of pedestrians in bounded domains (cf. the next example of regular lattices). This is not sufficient by itself to comply with the hypotheses of Theorem 14, but choosing r as(97)
r
=
C
N
-
1
+
h
/
d
,
N
>
1
,where h
>
0, is instead sufficient if(98)
h
>
d
-
1
.Then, according to (78) and to Proposition 13, the distance between the discrete and continuous solutions to (36) scales with N as(99)
W
1
ϵ
t
N
,
ρ
t
N
≤
O
N
d
-
1
-
h
/
d
f
o
r
N
⟶
∞
.Example 15 (regular lattices).
Homogeneous pedestrian distributions can be obtained, for instance, by considering regular lattices.
LetΩ
=
[
0,1
]
d. We partition it in N
k
=
2
k
d equal hypercubes of edge size 2
-
k; then we position N
k discrete pedestrians X
¯
i
i
=
1
N
k in their centroids (cf. Figure 7), so that(100)
min
i
≠
j
⁡
X
¯
j
-
X
¯
i
=
2
-
k
.Owing to (81) we need then to take(101)
r
<
2
-
k
+
1
=
1
2
N
k
-
1
/
d
,which, following (97), we satisfy by setting r
=
2
-
(
1
+
(
h
+
1
)
k
) for h
>
0. For this value of r, let us set(102)
ρ
i
x
=
d
ω
d
r
d
χ
B
r
X
¯
i
x
,
i
=
1
,
…
,
N
,where ω
d is the surface area of the unit ball in R
d and χ the characteristic function. These ρ
i’s are of the form (79) for f
(
x
)
=
(
d
/
ω
d
)
χ
B
1
(
0
)
(
x
), which complies with (80) and moreover is such that m
f
=
d
/
(
1
+
d
). This entails the following:(i)
In dimensiond
=
1,(103)
W
1
ϵ
¯
N
k
,
ρ
¯
N
k
=
2
-
2
-
h
k
,
which converges to zero fork
→
∞ if h
>
0.
(ii)
In dimensiond
=
2,(104)
W
1
ϵ
¯
N
k
,
ρ
¯
N
k
=
2
1
-
h
k
3
,
which converges to zero fork
→
∞ if h
>
1.
(iii)
In dimensiond
=
3,(105)
W
1
ϵ
¯
N
k
,
ρ
¯
N
k
=
3
·
2
2
-
h
k
-
3
,
which converges to zero fork
→
∞ if h
>
2.Figure 7
Discrete pedestrians (dots) and their continuous counterparts (circular regions delimited by dashed lines) inΩ
=
[
0,1
]
2 (d
=
2) and for k
=
0,1
,
2.


## 4.1. Modeling Hypotheses

Following the theory developed in [1, 27, 28], we assume some smoothness of the transport velocity. Consider, specifically, the following.Assumption 5 (Lipschitz continuity ofv).
There existLip
⁡
(
v
d
)
>
0 and Lip
⁡
(
K
)
>
0 such that(41)
v
d
y
-
v
d
x
≤
Lip
⁡
v
d
y
-
x
,
K
y
-
K
x
≤
Lip
⁡
K
y
-
xfor all x
,
y
∈
R
d.Using the expression (10) of the transport velocity, it is immediate to check that Assumption 5 implies(42)
v
ν
y
-
v
μ
x
≤
Lip
⁡
v
d
+
N
Lip
⁡
K
y
-
x
+
Lip
⁡
K
W
1
μ
,
νfor all x
,
y
∈
R
d and μ
,
ν
∈
M
1
N
(
R
d
), hence letting(43)
ξ
N
≔
2
max
⁡
Lip
⁡
v
d
,
N
Lip
⁡
Kand recalling that we are considering N
≥
1 we finally have(44)
v
ν
y
-
v
μ
x
≤
ξ
N
y
-
x
+
1
N
W
1
μ
,
ν
.


## 4.2. Continuous Dependence on the Initial Datum

The basic tool for the subsequent analysis is the continuous dependence of the solution to (36) on the initial datum, which can be proved using the representation formula (6) based on the flow map (7). The result itself and the analytical technique to obtain it are classical in the theory of problem (36); see, for example, [45–47]. They have also been extensively used to analyze swarming models; see, for example, [48] and the thorough review [49]. However, in the present case it is crucial to obtain the explicit dependence on N of some constants appearing in the final estimates, which is less classical due to the fact that here the total mass of the system is not 1 but N. For this reason, in order to make the paper self-contained, we detail in Appendix 5 the proofs of the following two results.Lemma 6 (regularity of the flow map).
Consider the flow map (7):(i)
For allx
,
y
∈
R
d and t
∈
[
0
,
T
] the following results: (45)
γ
t
y
-
γ
t
x
≤
e
ξ
N
t
y
-
x
.
(ii)
Letμ
·
,
ν
·
∈
C
(
[
0
,
T
]
;
M
1
N
(
R
d
)
) be two solutions to (36) with respective initial conditions μ
¯
,
ν
¯
∈
M
1
N
(
R
d
). Call γ
μ, γ
ν the flow maps associated with either solution. Then for all x
∈
R
d and t
∈
[
0
,
T
] the following results:(46)
γ
t
ν
x
-
γ
t
μ
x
≤
ξ
N
e
ξ
N
t
N
∫
0
t
W
1
μ
s
,
ν
s
d
s
.Proposition 7 (continuous dependence).
Letμ
·
,
ν
·
∈
C
(
[
0
,
T
]
;
M
1
N
(
R
d
)
) be two solutions to (36) corresponding to initial conditions μ
¯
,
ν
¯
∈
M
1
N
(
R
d
). Then(47)
W
1
μ
t
,
ν
t
≤
e
ξ
N
t
1
+
e
ξ
N
T
W
1
μ
¯
,
ν
¯
,
∀
t
∈
0
,
T
.


## 4.3. Sequences of Measures with Growing Mass: Scaling of the Interactions

Let us now consider two sequences of initial conditions of growing mass, sayμ
¯
N
N
=
1
∞
,
ν
¯
N
N
=
1
∞ with μ
¯
N
,
ν
¯
N
∈
M
1
N
(
R
d
)
∀
N
≥
1, and the related sequences of solutions to (36), μ
·
N
N
=
1
∞
,
ν
·
N
N
=
1
∞ with μ
·
N
,
ν
·
N
∈
C
(
[
0
,
T
]
;
M
1
N
(
R
d
)
)
∀
N
≥
1. From Proposition 7 we have(48)
W
1
μ
t
N
,
ν
t
N
≤
e
ξ
N
t
1
+
e
ξ
N
T
W
1
μ
¯
N
,
ν
¯
N
,where the exponential factor estimates the amplification at time t
>
0 of the distance between the initial data. If N is sufficiently large then from (43) we have ξ
N
=
2
N
Lip
⁡
(
K
). In order for the exponential factor to remain bounded for growing N and ensure that W
1
(
μ
¯
N
,
ν
¯
N
) and W
1
(
μ
t
N
,
ν
t
N
) are of the same order of magnitude for all N, it is necessary that(49)
Lip
⁡
K
=
O
N
-
1
for
N
⟶
∞
.In other words, we have to suitably scale the interaction kernel with respect to N. In particular, given a Lipschitz continuous function K
:
R
d
→
R
d compactly supported in B
R
(
0
)
⊂
R
d, to comply with (49) we consider the following two-parameter family of interaction kernels:(50)
K
z
=
K
α
,
β
N
z
≔
1
N
α
K
z
N
β
,
α
,
β
∈
R
,whose Lipschitz constant is(51)
Lip
⁡
K
α
,
β
N
=
Lip
⁡
K
N
α
+
β
.Clearly, they satisfy (49) as long as(52)
α
+
β
≥
1
.Example 8 (role ofα).
An admissible interaction kernel in the family (50) is obtained for α
=
1, β
=
0; that is,(53)
K
1,0
N
z
=
1
N
K
z
.This corresponds to decreasing interpersonal repulsion when the number of pedestrians increases. Notice that this is the same scaling adopted in the mean field limit [23, 24]. In this case, pedestrian velocity (10) reads(54)
v
μ
t
N
x
=
v
d
x
-
1
N
∫
R
d
K
y
-
x
d
μ
t
N
y
.Considering that μ
t
N
(
R
d
)
=
N, the desired velocity and the interactions have commensurable weights for every N.Example 9 (role ofβ).
An interaction kernel somehow opposite toK
1,0
N (cf. Example 8) is obtained for α
=
0, β
=
1; that is,(55)
K
0,1
N
z
=
K
z
N
,whose support is contained in the ball B
N
R
(
0
). This kernel corresponds to pedestrians who interact with an increasing number of individuals as the total number of people increases. The resulting velocity(56)
v
μ
t
N
x
=
v
d
x
-
∫
R
d
K
y
-
x
N
d
μ
t
N
yis such that the component due to interactions tends to predominate over the desired one for growing N.Example 10.
Besides the extreme cases in Examples8 and 9, we may also consider intermediate cases in which both α and β are simultaneously nonzero (and not necessarily positive). For instance, for α
=
2, β
=
-
1 we get(57)
K
2
,
-
1
N
z
=
1
N
2
K
N
z
,which corresponds to interactions that weaken and that are restricted to a contracting sensory region (in fact supp
⁡
K
2
,
-
1
N
⊆
B
R
/
N
(
0
)) as N grows. This models pedestrians who agree to stay closer and closer as their number increases, like, for example, in highly crowded train or metro stations during rush hours. The resulting velocity(58)
v
μ
t
N
x
=
v
d
x
-
1
N
2
∫
R
d
K
N
y
-
x
d
μ
t
N
yis such that v
d dominates for large N, meaning that at high crowding pedestrians tend to be passively transported by the flow without interacting.A scaling of interactions of type (50) is proposed, for example, in [50]. However, it involves only one parameter γ
∈
(
0,1
), corresponding to α
=
-
γ and β
=
-
γ
/
d in (49). As this condition does not satisfy (52), it has to be regarded as a complementary case, not covered by the theory developed here.


### 4.3.1. Scaling Equivalence

Solutions to (36) with interaction kernel (50) for different values of the parameters α, β account for different interpersonal attitudes of pedestrians in congested crowd regimes. Hence, we may expect significantly different solutions for large N. Nonetheless, in the case v
d
≡
0 a one-to-one correspondence among them exists, up to a transformation of the space variable depending on N. In order to prove it we preliminarily introduce the following.Proposition 11.
LetU
:
R
d
→
R
d be the linear scaling of the space(59)
U
z
=
a
z
,
a
∈
R and let v
^, v
~ be the velocities (10) computed with the following interaction kernels:(60)
K
^
z
=
K
∘
U
-
1
z
=
K
z
a
,
K
~
z
=
U
-
1
∘
K
z
=
1
a
K
z
,where K
:
R
d
→
R
d is Lipschitz continuous. Assume moreover v
d
≡
0.(i)
For allμ
∈
M
1
N
(
R
d
) the following results:(61)
v
~
μ
x
=
1
a
v
^
U
#
μ
a
x
.
Letμ
^
·
,
μ
~
·
∈
C
(
[
0
,
T
]
;
M
1
N
(
R
d
)
) be the solutions to (36) with velocities v
^, v
~, respectively, and initial conditions such that(62)
μ
^
0
=
U
#
μ
~
0
.
(ii)
The flow mapsγ
^
t, γ
~
t correspond to one another as(63)
γ
^
t
=
U
∘
γ
~
t
∘
U
-
1
,
∀
t
∈
0
,
T
.
(iii)
The solutions satisfy(64)
μ
^
t
=
U
#
μ
~
t
,
∀
t
∈
0
,
T
.Proof.
(i) By direct calculation we find(65)
1
a
v
^
U
#
μ
a
x
=
-
1
a
∫
R
d
K
^
y
-
a
x
d
U
#
μ
y
=
-
1
a
∫
R
d
K
y
-
a
x
a
d
U
#
μ
y
=
-
1
a
∫
R
d
K
z
-
x
d
μ
z
=
-
∫
R
d
K
~
z
-
x
d
μ
z
=
v
~
μ
x
.
(ii) We check that(66)
γ
^
t
x
≔
U
∘
γ
~
t
∘
U
-
1
x
=
a
γ
~
t
x
acomplies with definition (7). Using (7) for γ
~
t, we write(67)
γ
^
t
x
=
x
+
a
∫
0
t
v
~
γ
~
s
#
μ
~
0
γ
~
s
x
a
d
s
.
Next we observe that(68)
γ
~
s
#
μ
~
0
=
γ
~
s
∘
U
-
1
∘
U
#
μ
~
0
=
γ
~
s
∘
U
-
1
#
μ
^
0
;hence from the previous point (i) we deduce (69)
γ
^
t
x
=
x
+
a
∫
0
t
1
a
v
^
U
#
γ
~
s
∘
U
-
1
#
μ
^
0
a
γ
~
s
x
a
d
s
=
x
+
∫
0
t
v
^
γ
^
s
#
μ
^
0
γ
^
s
x
d
s
,as desired.
(iii) Due to the result in (ii) we have(70)
μ
^
t
=
γ
^
t
#
μ
^
0
=
U
∘
γ
~
t
∘
U
-
1
#
U
#
μ
~
0
=
U
∘
γ
~
t
#
μ
~
0
=
U
#
μ
~
t
.As a consequence of Proposition11, we can prove a correspondence among the dynamics governed by different interaction kernels of family (50).Theorem 12 (scaling equivalence).
Letμ
·
N
,
ν
·
N
∈
C
(
[
0
,
T
]
;
M
1
N
(
R
d
)
) be the solutions to (36) corresponding to interaction kernels K
α
,
β
N, K
α
′
,
β
′
N, with(71)
α
+
β
=
α
′
+
β
′
,and to initial conditions μ
¯
N
,
ν
¯
N
=
U
N
#
μ
¯
N
∈
M
1
N
(
R
d
), respectively, where(72)
U
N
z
=
N
β
′
-
β
z
.Then(73)
ν
t
N
=
U
N
#
μ
t
N
,
∀
t
∈
0
,
T
.Proof.
By (50) we have, on the one hand,(74)
K
α
,
β
N
z
=
1
N
α
K
z
N
β
=
1
N
β
′
-
β
·
1
N
α
′
K
z
N
β
,where we have used the fact that α
=
α
′
+
β
′
-
β, and, on the other hand,(75)
K
α
′
,
β
′
N
z
=
1
N
α
′
K
z
N
β
′
=
1
N
α
′
K
1
N
β
′
-
β
·
z
N
β
.Thus, (76)
K
α
,
β
N
z
=
1
a
·
1
N
α
′
K
z
N
β
,
K
α
′
,
β
′
N
z
=
1
N
α
′
K
1
a
·
z
N
βfor a
=
N
β
′
-
β and the thesis follows from Proposition 11.


## 4.3.1. Scaling Equivalence

Solutions to (36) with interaction kernel (50) for different values of the parameters α, β account for different interpersonal attitudes of pedestrians in congested crowd regimes. Hence, we may expect significantly different solutions for large N. Nonetheless, in the case v
d
≡
0 a one-to-one correspondence among them exists, up to a transformation of the space variable depending on N. In order to prove it we preliminarily introduce the following.Proposition 11.
LetU
:
R
d
→
R
d be the linear scaling of the space(59)
U
z
=
a
z
,
a
∈
R and let v
^, v
~ be the velocities (10) computed with the following interaction kernels:(60)
K
^
z
=
K
∘
U
-
1
z
=
K
z
a
,
K
~
z
=
U
-
1
∘
K
z
=
1
a
K
z
,where K
:
R
d
→
R
d is Lipschitz continuous. Assume moreover v
d
≡
0.(i)
For allμ
∈
M
1
N
(
R
d
) the following results:(61)
v
~
μ
x
=
1
a
v
^
U
#
μ
a
x
.
Letμ
^
·
,
μ
~
·
∈
C
(
[
0
,
T
]
;
M
1
N
(
R
d
)
) be the solutions to (36) with velocities v
^, v
~, respectively, and initial conditions such that(62)
μ
^
0
=
U
#
μ
~
0
.
(ii)
The flow mapsγ
^
t, γ
~
t correspond to one another as(63)
γ
^
t
=
U
∘
γ
~
t
∘
U
-
1
,
∀
t
∈
0
,
T
.
(iii)
The solutions satisfy(64)
μ
^
t
=
U
#
μ
~
t
,
∀
t
∈
0
,
T
.Proof.
(i) By direct calculation we find(65)
1
a
v
^
U
#
μ
a
x
=
-
1
a
∫
R
d
K
^
y
-
a
x
d
U
#
μ
y
=
-
1
a
∫
R
d
K
y
-
a
x
a
d
U
#
μ
y
=
-
1
a
∫
R
d
K
z
-
x
d
μ
z
=
-
∫
R
d
K
~
z
-
x
d
μ
z
=
v
~
μ
x
.
(ii) We check that(66)
γ
^
t
x
≔
U
∘
γ
~
t
∘
U
-
1
x
=
a
γ
~
t
x
acomplies with definition (7). Using (7) for γ
~
t, we write(67)
γ
^
t
x
=
x
+
a
∫
0
t
v
~
γ
~
s
#
μ
~
0
γ
~
s
x
a
d
s
.
Next we observe that(68)
γ
~
s
#
μ
~
0
=
γ
~
s
∘
U
-
1
∘
U
#
μ
~
0
=
γ
~
s
∘
U
-
1
#
μ
^
0
;hence from the previous point (i) we deduce (69)
γ
^
t
x
=
x
+
a
∫
0
t
1
a
v
^
U
#
γ
~
s
∘
U
-
1
#
μ
^
0
a
γ
~
s
x
a
d
s
=
x
+
∫
0
t
v
^
γ
^
s
#
μ
^
0
γ
^
s
x
d
s
,as desired.
(iii) Due to the result in (ii) we have(70)
μ
^
t
=
γ
^
t
#
μ
^
0
=
U
∘
γ
~
t
∘
U
-
1
#
U
#
μ
~
0
=
U
∘
γ
~
t
#
μ
~
0
=
U
#
μ
~
t
.As a consequence of Proposition11, we can prove a correspondence among the dynamics governed by different interaction kernels of family (50).Theorem 12 (scaling equivalence).
Letμ
·
N
,
ν
·
N
∈
C
(
[
0
,
T
]
;
M
1
N
(
R
d
)
) be the solutions to (36) corresponding to interaction kernels K
α
,
β
N, K
α
′
,
β
′
N, with(71)
α
+
β
=
α
′
+
β
′
,and to initial conditions μ
¯
N
,
ν
¯
N
=
U
N
#
μ
¯
N
∈
M
1
N
(
R
d
), respectively, where(72)
U
N
z
=
N
β
′
-
β
z
.Then(73)
ν
t
N
=
U
N
#
μ
t
N
,
∀
t
∈
0
,
T
.Proof.
By (50) we have, on the one hand,(74)
K
α
,
β
N
z
=
1
N
α
K
z
N
β
=
1
N
β
′
-
β
·
1
N
α
′
K
z
N
β
,where we have used the fact that α
=
α
′
+
β
′
-
β, and, on the other hand,(75)
K
α
′
,
β
′
N
z
=
1
N
α
′
K
z
N
β
′
=
1
N
α
′
K
1
N
β
′
-
β
·
z
N
β
.Thus, (76)
K
α
,
β
N
z
=
1
a
·
1
N
α
′
K
z
N
β
,
K
α
′
,
β
′
N
z
=
1
N
α
′
K
1
a
·
z
N
βfor a
=
N
β
′
-
β and the thesis follows from Proposition 11.


## 4.4. Back to Discrete and Continuous Models

Conditions (50) and (52) imply(77)
ξ
N
≤
ξ
∗
≔
2
max
⁡
Lip
⁡
v
d
,
Lip
⁡
K
,
∀
N
≥
1
.When using (48) for sequences of discrete and continuous measures we can incorporate this fact and write(78)
W
1
ϵ
t
N
,
ρ
t
N
≤
e
ξ
∗
t
1
+
e
ξ
∗
T
W
1
ϵ
¯
N
,
ρ
¯
N
,
∀
t
∈
0
,
T
,
∀
N
≥
1
.Hence mutually approaching sequences of discrete and continuous solutions to (36) (cf. (38)) are possible, provided one is able to construct mutually approaching sequences of initial conditions at the corresponding scales (cf. (37)). In the following we discuss a possible procedure leading to the desired result.LetX
¯
1
,
…
,
X
¯
N
∈
R
d be the initial positions of N distinct microscopic pedestrians. The associated discrete distribution ϵ
¯
N is constructed from (8), while, given r
>
0, we define(79)
ρ
¯
N
=
∑
i
=
1
N
ρ
i
,
ρ
i
x
=
1
r
d
f
x
-
X
¯
i
r
,where f
:
R
d
→
R is a nonnegative function such that(80)
supp
⁡
f
⊆
B
1
0
¯
,
∫
B
1
0
f
x
d
x
=
1
.Consequently, supp
⁡
ρ
i
⊆
B
r
(
X
¯
i
)
¯ and moreover ρ
i
(
B
r
(
X
¯
i
)
)
=
1 each i; thus ρ
¯
N is the superposition of N piecewise constant density bumps, each of which carries a unit mass representative of one pedestrian. We assume(81)
r
<
1
2
m
i
n
i
≠
j
X
¯
j
-
X
¯
iwhich ensures no overlapping of the supports of the ρ
i’s.In the remaining part of this section we compute the Wasserstein distance betweenϵ
¯
N and ρ
¯
N and we study its trend with respect to N.Proposition 13 (distance between the initial conditions).
Let(82)
m
f
≔
∫
B
1
0
x
f
x
d
x
.The distance between ϵ
¯
N and ρ
¯
N is(83)
W
1
ϵ
¯
N
,
ρ
¯
N
=
m
f
N
r
.Proof.
Here we use expression (40) of W
1. Let us consider the case N
=
1 first. Since ϵ
¯
1 is a single Dirac mass, there is no ambiguity in the construction of the optimal transference plan π between ϵ
¯
1 and ρ
¯
1 (i.e., the transference plan which realizes the infimum in (39)), which is just the tensor product of the measures:(84)
π
x
,
y
=
ρ
¯
1
x
⊗
ϵ
¯
1
y
. By substitution in (39) we find (85)
W
1
ϵ
¯
1
,
ρ
¯
1
=
∫
R
d
×
R
d
x
-
y
d
ρ
¯
1
x
⊗
ϵ
¯
1
y
=
∫
B
r
X
¯
1
x
-
X
¯
1
ρ
1
x
d
x
=
1
r
d
∫
B
r
X
¯
1
x
-
X
¯
1
f
x
-
X
¯
1
r
d
x
=
r
∫
B
1
0
y
f
y
d
y
s
e
t
y
≔
x
-
X
¯
1
r
=
m
f
r
.
We pass now to characterize transference plans betweenϵ
¯
N and ρ
¯
N in the case N
>
1. Every element of the continuous mass ρ
¯
N is transported onto a delta, a condition that, in the spirit of the so-calledsemidiscrete Monge-Kantorovich problem [51], can be expressed by a measure π on R
d
×
R
d of the form(86)
π
x
,
y
=
ρ
¯
N
x
⊗
∑
j
=
1
N
α
j
x
δ
X
¯
j
y
,where, in order to ensure conservation of the mass, the α
j’s are such that(87)
∑
j
=
1
N
α
j
x
=
1
,
α
j
x
≥
0
,
∀
j
=
1
,
…
,
N
,
∀
x
∈
supp
⁡
ρ
¯
N
.Representation (86) means that the infinitesimal element of continuous mass d
ρ
¯
N
(
x
) located in x
∈
supp
⁡
ρ
¯
N is split in the points X
¯
j
j
=
1
N following the convex combination given by the coefficients α
j
(
x
) (cf. Figure 6). The measure π is generally not a transference plan between ϵ
¯
N and ρ
¯
N, because it is not guaranteed to have marginal ϵ
¯
N with respect to y. In particular, a nonunit mass might be allocated in every X
¯
j. In order to have a transference plan, the further condition,(88)
∫
R
d
α
j
x
d
ρ
¯
N
x
=
1
,
∀
j
=
1
,
…
,
N
,needs to be enforced.
Let us consider, for a transference plan of the form (86) with ρ
¯
N as in (79), the global transportation cost: (89)
∫
R
d
×
R
d
x
-
y
d
π
x
,
y
=
∫
R
d
×
R
d
x
-
y
d
∑
i
=
1
N
ρ
i
x
⊗
∑
j
=
1
N
α
j
x
δ
X
¯
j
y
=
∑
i
,
j
=
1
N
∫
R
d
×
R
d
x
-
y
d
ρ
i
x
⊗
α
j
x
δ
X
¯
j
y
=
∑
i
=
1
N
∫
B
r
X
¯
i
∑
j
=
1
N
α
j
x
x
-
X
¯
j
d
ρ
i
x
. Owing to (87) we have (90)
∑
j
=
1
N
α
j
x
x
-
X
¯
j
=
∑
j
≠
i
α
j
x
x
-
X
¯
j
+
1
-
∑
j
≠
i
α
j
x
x
-
X
¯
iwhence, for x
∈
B
r
(
X
¯
i
) and taking (81) into account,(91)
∑
j
=
1
N
α
j
x
x
-
X
¯
j
≥
r
∑
j
≠
i
α
j
x
+
x
-
X
¯
i
-
r
∑
j
≠
i
α
j
x
=
x
-
X
¯
i
;thus ultimately (92)
∫
R
d
×
R
d
x
-
y
d
π
x
,
y
≥
∑
i
=
1
N
∫
B
r
X
¯
i
x
-
X
¯
i
d
ρ
i
x
=
∑
i
=
1
N
∫
R
d
×
R
d
x
-
y
d
ρ
i
x
⊗
δ
X
¯
i
y
.
Notice that the transference plan(93)
π
=
∑
i
=
1
N
ρ
i
⊗
δ
X
¯
iis of the form (86) for, for example, the coefficients α
j
(
x
)
=
χ
B
r
(
X
¯
j
)
(
x
) which fulfill both (87) and (88). The previous calculation says that it is actually an optimal transference plan between ϵ
¯
N and ρ
¯
N, that is, one which ensures the minimum transportation cost. Thus(94)
W
1
ϵ
¯
N
,
ρ
¯
N
=
∑
i
=
1
N
∫
R
d
×
R
d
x
-
y
d
ρ
i
x
⊗
δ
X
¯
i
y
=
N
W
1
ϵ
¯
1
,
ρ
¯
1
,whence the thesis follows.Figure 6
Transference plans (86) in R
d
×
R
d for N
=
3. Each continuous mass element is within a distance r from its corresponding Dirac mass. Therefore, owing to (81), it is farther than r from all other Dirac masses.Thanks to Proposition13 we finally obtain the main result of the paper.Theorem 14 (discrete-continuous convergence).
Letϵ
¯
N
∈
M
1
N
(
R
d
) be given and let ρ
¯
N
∈
M
1
N
(
R
d
) be constructed as in (79)-(80). Set r
=
c
N
N
-
γ, γ
>
1, where 0
<
c
N
≤
1 is possibly used to enforce condition (81). Let moreover the interaction kernel satisfy (50) and (52). Then(95)
lim
N
→
∞
⁡
W
1
ϵ
t
N
,
ρ
t
N
=
0
,
∀
t
∈
0
,
T
.Proof.
From Proposition13 with the given r we obtain (96)
W
1
ϵ
¯
N
,
ρ
¯
N
=
m
f
c
N
N
γ
-
1
→
N
→
∞
0because γ
>
1. Hence the thesis follows from (78).It is worth remarking that, because of (81), in dimension d a bound on the radius r of the form r
<
C
N
-
1
/
d, where C
>
0 is a constant, holds true, for example, when considering homogeneous distributions of pedestrians in bounded domains (cf. the next example of regular lattices). This is not sufficient by itself to comply with the hypotheses of Theorem 14, but choosing r as(97)
r
=
C
N
-
1
+
h
/
d
,
N
>
1
,where h
>
0, is instead sufficient if(98)
h
>
d
-
1
.Then, according to (78) and to Proposition 13, the distance between the discrete and continuous solutions to (36) scales with N as(99)
W
1
ϵ
t
N
,
ρ
t
N
≤
O
N
d
-
1
-
h
/
d
f
o
r
N
⟶
∞
.Example 15 (regular lattices).
Homogeneous pedestrian distributions can be obtained, for instance, by considering regular lattices.
LetΩ
=
[
0,1
]
d. We partition it in N
k
=
2
k
d equal hypercubes of edge size 2
-
k; then we position N
k discrete pedestrians X
¯
i
i
=
1
N
k in their centroids (cf. Figure 7), so that(100)
min
i
≠
j
⁡
X
¯
j
-
X
¯
i
=
2
-
k
.Owing to (81) we need then to take(101)
r
<
2
-
k
+
1
=
1
2
N
k
-
1
/
d
,which, following (97), we satisfy by setting r
=
2
-
(
1
+
(
h
+
1
)
k
) for h
>
0. For this value of r, let us set(102)
ρ
i
x
=
d
ω
d
r
d
χ
B
r
X
¯
i
x
,
i
=
1
,
…
,
N
,where ω
d is the surface area of the unit ball in R
d and χ the characteristic function. These ρ
i’s are of the form (79) for f
(
x
)
=
(
d
/
ω
d
)
χ
B
1
(
0
)
(
x
), which complies with (80) and moreover is such that m
f
=
d
/
(
1
+
d
). This entails the following:(i)
In dimensiond
=
1,(103)
W
1
ϵ
¯
N
k
,
ρ
¯
N
k
=
2
-
2
-
h
k
,
which converges to zero fork
→
∞ if h
>
0.
(ii)
In dimensiond
=
2,(104)
W
1
ϵ
¯
N
k
,
ρ
¯
N
k
=
2
1
-
h
k
3
,
which converges to zero fork
→
∞ if h
>
1.
(iii)
In dimensiond
=
3,(105)
W
1
ϵ
¯
N
k
,
ρ
¯
N
k
=
3
·
2
2
-
h
k
-
3
,
which converges to zero fork
→
∞ if h
>
2.Figure 7
Discrete pedestrians (dots) and their continuous counterparts (circular regions delimited by dashed lines) inΩ
=
[
0,1
]
2 (d
=
2) and for k
=
0,1
,
2.


## 5. Discussion

In this paper we have investigated microscopic and macroscopic differential models of systems of interacting particles, chiefly inspired by human crowds, for an increasing numberN of total agents. The main novelty was the consideration ofmassive particles, that is, particles whose mass does not scale with the number N. This implies that the continuous model is not obtained in the limit N
→
∞ from the discrete model; rather it is postulatedper se for every value of N. The following question then arises: under which conditions are the discrete and continuous models counterparts of one another at smaller and larger scales?In particular, we have proved the solutions to the following two models:(106)
X
˙
t
i
=
v
d
X
t
i
+
1
N
α
∑
j
=
1
N
K
X
t
j
-
X
t
i
N
β
,
i
=
1
,
…
,
N
,
∂
t
ρ
t
+
div
⁡
ρ
t
v
d
+
1
N
α
∫
R
d
K
y
-
·
N
β
ρ
t
y
d
y
=
0
,
ρ
t
R
d
=
N
,where the velocity field v
d and the interaction kernel K are assumed to be Lipschitz continuous, converging to one another in the sense of the 1-Wasserstein distance when N
→
∞ if the parameters α, β are such that α
+
β
≥
1 and if, in addition, the respective initial conditions approximate each other. This fact, which is schematically illustrated in Figure 8, has implications from the modeling point of view, especially as far as the role of single scales and their possible coupling are concerned.Figure 8
A schematic illustration of the result of the paper. The ODE and PDE solutions approach each other forN
→
∞ under certain conditions on the scaling of the interactions and of the initial conditions.First of all, we point out that interactions are modeled in anN-dependent way by means of the kernel (107)
K
α
,
β
N
z
=
1
N
α
K
z
N
β
.More specifically, the function K expresses the basic interaction trend (e.g., repulsion) while the factors N
-
α, N
-
β modulate it depending on the total number of particles. This is consistent with the idea that particles like, for example, pedestrians do not behave the same regardless of their number. The strength of their mutual repulsion or the acceptable interpersonal distances may vary considerably from free to congested situations. We model this aspect by acting on the values of α
,
β. Hence, the various scalings contained in the two-parameter family of kernels K
α
,
β
N correspond to differentinterpersonal attitudes of the particles for growing N. As our analysis demonstrates, the latter need to be taken into account for ensuring consistency of a given interaction model at different scales. In this respect, we have shown that if such a scaling is neglected the discrete and continuous models predict quantitatively different, albeit qualitatively analogous, emerging equilibria. In particular, they yield different asymptotic speeds of the particles, which do not approach each other as N grows.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 theinterchangeability of the two models at sufficiently high numbers of particles, with (i) the possibility to switch from a microscopic to a macroscopic description, which may be more convenient for thea posteriori calculation of observable quantities and statistics of interest for applications; (ii) the possibility to infer qualitative properties at one scale from their rigorous knowledge at the other scale (e.g., the microscopic model can be expected to exhibit qualitatively the same nonlinear diffusive behavior for large N which is typically proved quantitatively for macroscopic conservation laws with nonlocal repulsive flux). On the other hand, it allows one to motivate and supportmultiscale couplings of microscopic and macroscopic models [1, 22], which are supposed to provide adual representation of the same particle system at different scales. In this case, the interest does not lie as much in the congested regime (large N), where the two models have been proved to be equivalent, but rather in the moderately crowded one, where the discrete and continuous solutions can complement each other with effects which would not be recovered at a single scale.



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*Source: 6902086-2016-03-20.xml*