A hierarchy of heuristic-based models of crowd dynamics

A hierarchy of heuristic-based models of crowd dynamics

P. Degond, C. Appert-Rolland, M. Moussaid,
J. Pettre, G. Theraulaz
Abstract

We derive a hierarchy of kinetic and macroscopic models from a noisy variant of the heuristic behavioral Individual-Based Model of [53] where the pedestrians are supposed to have constant speeds. This IBM supposes that the pedestrians seek the best compromise between navigation towards their target and collisions avoidance. We first propose a kinetic model for the probability distribution function of the pedestrians. Then, we derive fluid models and propose three different closure relations. The first two closures assume that the velocity distribution functions are either a Dirac delta or a von Mises-Fisher distribution respectively. The third closure results from a hydrodynamic limit associated to a Local Thermodynamical Equilibrium. We develop an analogy between this equilibrium and Nash equilibia in a game theoretic framework. In each case, we discuss the features of the models and their suitability for practical use.

1-Université de Toulouse; UPS, INSA, UT1, UTM ;

Institut de Mathématiques de Toulouse ;

F-31062 Toulouse, France.

2-CNRS; Institut de Mathématiques de Toulouse UMR 5219 ;

F-31062 Toulouse, France.

email: pierre.degond@math.univ-toulouse.fr

3- Laboratoire de Physique Théorique, Université Paris Sud,

bâtiment 210, 91405 Orsay cedex, France

4- CNRS, UMR 8627, Laboratoire de physique théorique, 91405 Orsay, France

email: Cecile.Appert-Rolland@th.u-psud.fr

5-1 Center for Adaptive Behavior and Cognition, Max Planck Institute for Human Development, Lentzeallee 94, 14195 Berlin, Germany

email: moussaid@mpib-berlin.mpg.de

6-INRIA Rennes - Bretagne Atlantique, Campus de Beaulieu, 35042 Rennes, France

email: julien.pettre@irisa.fr

4-Centre de Recherches sur la Cognition Animale, UMR-CNRS 5169,

Université Paul Sabatier, Bât 4R3,

118 Route de Narbonne, 31062 Toulouse cedex 9, France.

8- CNRS, Centre de Recherches sur la Cognition Animale, F-31062 Toulouse, France

email: theraula@cict.fr

Acknowledgments: This work has been supported by the French ’Agence Nationale pour la Recherche (ANR)’ in the frame of the contracts ’Pedigree’ (ANR-08-SYSC-015-01) and ’CBDif-Fr’ (ANR-08-BLAN-0333-01)

Key words: Pedestrian dynamics, behavioral heuristics, rational agents, Individual-Based Models, Kinetic Model, Fluid model, Game theory, Closure relation, monokinetic, von Mises-Fisher distribution, Nash equilibrium

AMS Subject classification: 90B20, 35L60, 35L65, 35L67, 35R99, 76L05.

1 Introduction

Understanding and predicting crowd behavior is an extremely important issue in our societies. Public safety concerns have raised considerably after recent major crowd disasters [54]. Public authorities are challenged by increasingly large crowds attending mass events such as sports gatherings. Economic stakes related to crowd management are equally high, as increasing the efficiency of pedestrian infrastructures have important returns in terms of business.

To achieve a better comprehension of crowd behavior and increase the reliability of predictions, numerical modeling and simulation is playing an ever-growing role.

A recent review on crowd modeling can be found in [9]. The most widely used crowd simulation models are Individual-Based Models (or IBM), such as Rule-Based models [61], mechanical models [35, 37, 38], traffic following models [48], optimal control theory models, [40], Cellular-Automata [55] and Vision-Based models [34, 41, 56, 57, 59, 64]. The present paper relies on [53] which is detailed below. Continuum models (CM) are based on a fluid dynamics approach [36, 39]. Other approaches through optimal control theory [42, 43, 44, 46] or exploiting the analogy with car traffic [3, 8, 10, 18, 19, 51, 60] have also been developed. For dense crowds, the handling of the volume exclusion constraint has led to several specific works [26, 27, 50]. Existence theory for some CM of crowds can be found in [30].

Kinetic Models (KM) are intermediate between IBM and CM. As pointed out in the review [9], there are quite few KM of crowds (see an example in [7]). IBM, KM and CM constitute a hierarchy of models in the sense that each level can be deduced from the previous one by a model reduction methodology. Indeed, KM deal with the one-particle probability distributions of IBM. Such a description ignores correlations between the particles (which are described by joint probability distributions of particles for ) and is therefore a reduced description of the IBM. CM are deduced from KM by taking averages over the velocity variable (see section 4.2.1). Therefore, CM involve a reduced description of the velocity statistics of the KM to a small number of its moments.

In general, CM or KM are more efficient than IBM for large crowds because their computational time does not increase with the number of agents. However, they suffer from different drawbacks, such as a reduced validity range due to the necessary recourse to closure relations, as detailed below. Nonetheless, CM are invaluable tools for large-scale analysis and prediction of crowd behavior. Therefore, it is important to firmly base the derivation of CM on their small-scale IBM counterpart. The literature on the derivation of CM from microscopic models (IBM or CA) is scarce (see e.g. [1, 15, 17, 21]). The present paper addresses this question and intends to propose a hierarchy of KM and CM based on the IBM developed in [53].

The psychological literature shows that pedestrians can estimate the positions and velocities of moving obstacles such as other pedestrians with fairly good accuracy [22]. Therefore, the subjects are able to process this information in order to determine the dangerousness level of an encounter [66]. Taking these considerations into account, the heuristic-based model of [53] proposes that pedestrians follow a heuristic rule composed of two phases: a perception phase and a decision-making phase. In the perception phase, the subjects make an assessment of the dangerousness of the possible encounters in all the possible directions of motion. In the decision-making phase, they turn towards the direction which maximizes the distance walked towards their target while avoiding encounters with other pedestrians. In this sense, in the game theoretical sense, the pedestrians choose the Nash equilibrium as the new direction of motion. Game theoretical approaches of traffic have already been considered (see e.g. [62]).

The goal of this paper is to derive a CM from this heuristic-based IBM. With this aim, the time-discrete IBM of [53] is first replaced by a time-continuous IBM and noise is added to account for some uncertainty in the pedestrian velocity. From this time-continuous IBM, a KM is introduced. The KM describes the evolution of the probability distribution function of pedestrians in a phase space composed of position, velocity and target direction. For the sake of simplicity, we assume that the pedestrian speed remains constant and we discard any slowing down induced by close encounters. We do not develop any rigorous theory of the passage from the IBM to the KM [12, 16].

The passage from the KM to the CM is realized by taking the velocity moments of the distribution function. In doing so, some closure relations are needed otherwise the hierarchy of moments is infinite. We propose three distinct closure relations. The first one assumes a monokinetic distribution function. In other words, the velocity distribution is assumed to be a Dirac delta at the mean velocity. Such a monokinetic Ansatz can only be valid in the noiseless case but provides an exact solution of the KM. In the second closure relation, the velocity distribution function is supposed to be a von Mises-Fisher (VMF) distribution [67]. It is adapted to situations where the noise is non-zero. In these first two closures, the resulting macroscopic model is a system consisting of a mass conservation equation and an evolution equation for the mean velocity of each ensemble of pedestrians sharing the same target direction.

The last closure is based on a formal hydrodynamic limit. It can be performed in the restrictive situation where (i) the pedestrian interactions can be approximated by spatially local ones and (ii) the interaction region of the subjects is isotropic (i.e. there is no blind zone behind the subjects). The closure relies on a Local Thermodynamical Equilibrium (LTE) obtained through the solution of a fixed point equation. It expresses that each ensemble of pedestrians has found its optimal mean velocity in the midst of the other ones, i.e. is a Nash equilibrium in the game-theoretical sense. This example fits in the general framework relating kinetic theory and game theory which can be found in [28] and which bears analogies with the theory of Mean-Field Games [47]. In a companion paper [24], the same methodologies are applied to the model of [56] based on a mechanical view of pedestrian encounters.

The outline of the paper is as follows. In section 2 we review the IBM of [53]. In sections 3 and 4, we successively derive the KM and the CM with the three possible closure relations. The resulting models are discussed in 5. Finally, a conclusion is drawn in section 6.

2 The Heuristic-Based model of pedestrian dynamics

2.1 Principles

The heuristic-based model of [53] proposes that pedestrians follow a rule composed of two phases: a perception phase and a decision-making phase.

In the perception phase, the key observables are the distance-to-interaction (DTI), the time-to-interaction (TTI) and the minimal distance (MD). Let us first examine a binary encounter with another pedestrian (see Fig. 1).

Definition 2.1

Consider a pedestrian (the subject) at a given location and time moving with a given velocity. Suppose that this pedestrian interacts with a single other pedestrian (the collision partner) who possibly has a different velocity but whose location at the same time is close. In this encounter, we define the following quantities:

(i) The interaction point is the point on the subject’s future or past trajectory where the distance to the collision partner is minimal, assuming that both agents move in straight line with a constant speed.

(ii) The Minimal Distance (MD) is this minimal distance between the subject and his collision partner.

(iii) The Distance To Interaction (DTI) is the distance which separates the subject’s current position to the interaction point. The DTI is counted positive if the interaction point will be reached in the future of the subject and negative if the interaction point has been crossed in the past.

(iv) The Time-To-Interaction (TTI) is the time needed by the subject to reach this interaction point from his current position (counted positive if this time belongs to the future of the subject and negative if it belongs to the past).

Remark 2.1

(i) If the TTI is negative (i.e. the interaction point has been reached in the past and the subject and his collision partner are now moving away from each other), or if the MD is above a certain threshold (equal to the subjects’ diameter, possibly augmented by some safe-keeping distance), then, no interaction occurs and the DTI is set to infinity.

(ii) Because the subjects have supposedly perfect knowledge of their own and partner’s positions and velocities, we assume that they are able to estimate the DTI, TTI and MD with perfect accuracy.

We now define the DTI and the TTI when several collision partners are present (see Fig. 2). We have:

Definition 2.2

When the subject is interacting with several collision partners at the same time, the subject’s global DTI is the minimum of the DTI of all binary encounters. We denote it by if is the velocity of the subject.

The decision-making phase consists in changing the current cruising direction to a new cruising direction . It is the outcome of an optimization process. From the knowledge of the DTI in each cruising direction, the subject chooses the direction which maximizes the DTI, while keeping his direction of motion close to his target direction. In [53], the decision making phase is performed at discrete times separated by equal time intervals . In this phase, the cruising direction is updated through the following local optimization procedure. Without any obstacle, the subject would choose a target direction ( is a unit vector of ) and cruise with speed . Therefore, after a time interval , in the absence of obstacles, he would find himself at position (assuming that the origin of the coordinate system is placed at his current position). The point is called the target point. Now, in the presence of obstacles, the subject cruising in the direction ( being a unit vector in ) will estimate impossible to move a distance larger that the DTI. Therefore, choosing the cruising direction will place the subject at a position . The point is the estimated point reached in the direction . The decision making consists in choosing for new cruising direction the direction such that the estimated point is the closest to the target point , among a set of test directions belonging to the vision cone about the subject’s current direction of motion . Therefore, is determined by

(2.1)

where arg min denotes the point that realizes the minimum. Such a realization of the minimum may not be unique, but we will discard this possibility as non-generic. This decision-making phase is illustrated on Fig. 3.

This decision-making rule implicitly states how binary interactions are combined. This combination is not a mere superposition, as in the classical social force model [35, 37, 38], but a highly nonlinear operation involving the solution of an optimization problem. In [52], it is shown that this model provides a better account of some of the most striking emergence phenomena in crowds, such as spontaneous lane formation in bidirectional motion.

In the next two sections, we make all these considerations mathematically explicit.

MDDTI
Figure 1: The Minimal Distance MD and the Distance-To-Interaction (DTI). The MD is the smallest distance which separates the two pedestrians and supposing that they cruise on a straight line at constant velocities and . The point on pedestrian ’s trajectory where the minimal distance is attained is the interaction point of pedestrian in his interaction with pedestrian . The MD is the distance between and . The DTI is the distance which separates the current pedestrian position to the interaction point . The Time-To-Interaction (TTI) is the time needed by pedestrian to reach the interaction point from his current position. Clearly, TTI DTI.
DTIDTI
Figure 2: Perception phase: the Distance-To-Interaction (DTI) of a given pedestrian in the case of several simultaneous encounters is the minimum of the DTI of the individual encounters. In this figure, the DTI of pedestrian with pedestrian (denoted by DTI) is smaller than that with pedestrian (denoted by DTI). Therefore, the DTI of pedestrian is DTI.
Target direction New cruising direction The vision cone and the curve Closest distance betweenDirection
Figure 3: Decision-making phase. The new cruising direction is chosen such that the estimated point in this direction is the closest to the target point . The tested cruising directions (with associated points ) are restricted to the vision cone of the pedestrian (where is the current cruising direction).

2.2 Perception phase

We consider a pedestrian located at a position , with a velocity . He interacts with a collision partner located at a position which has a velocity . Figure 4 gives a schematic picture of the geometry of the collision. The goal of this section is to compute , , , respectively the TTI, DTI and MD of walker in his interaction with pedestrian (see Definition 2.1).

Lemma 2.3

We have:

(2.2)
(2.3)
(2.4)

Proof: The distance between the two particles at time is given by

(2.5)

denoting by and the positions of the two particles at time . This quadratic function of time is minimal at the time given by (2.2), which gives the value of the TTI. Then, the DTI of particle is obviously given by the distance traveled by this particle during the TTI, i.e. . This leads to (2.3). Finally, the MD is given by the minimal value of (2.5), i.e. , which leads to (2.4).  

If (2.2) and (2.3) give negative values for the TTI and DTI, it means that there is no threat of collision in the future times, as pedestrians are walking away from each other. Therefore, some interaction occurs in the future if and only if . Furthermore, if the MD is larger than a certain threshold identified as the diameter of the individuals, plus a certain safe-keeping distance, the interaction will no longer be perceived as a collision threat. In both cases, the DTI and TTI are set to infinity. With these additional features, we now define the DTI and TTI as

Definition 2.4

We define: and as:

(2.6)
(2.7)
this distancedivided by TTI = this distanceMD =
Figure 4: Geometry of a collision: The TTI is the projected distance of the two pedestrians on the direction of the relative velocity , divided by the norm of this relative velocity . The DTI is the TTI times the velocity of the pedestrian. The MD is the projected distance of the two pedestrians on the normal direction to the relative velocity.

2.3 Decision-making phase

The collision avoidance model of [53] uses the elements of collision perception reviewed in section 2.2. In this model, the decision of a new cruising direction taken by the pedestrian reflects the balance between two antagonist goals: collision avoidance on the one hand and maintenance of the target direction on the other hand. The goal of the present paper is to investigate the role of the cruising direction. Consequently, we discard the variations of the cruising speed. We assume that all pedestrians move with constant speed equal to . Therefore, and we let

This assumption prevents us to take into account one of the features of the model of [53], namely that pedestrians slow down or stop in case of very close encounters. This restricts the validity of the present paper to low densities. In the present paper, we will also discard fixed obstacles.

The model follows the lines of [53], with some simplifications of the expressions of the collision avoidance response. We assume a time discrete model with time steps . During this time step, the pedestrian moves according to the vector . Then, he updates his velocity and adopts a new velocity. For this purpose, he explores all possible velocity directions and computes the minimum of the DTI with the other pedestrians in the direction . Let us denote by the DTI with pedestrian assuming that moves in the direction . If there are no close encounters, we let this quantity be equal to the distance traveled by the pedestrian during , i.e. . In any case, we limit by this quantity. Then, according to (2.6), (2.7), we have:

(2.8)

For physical consistency, we should have , as the typical diameter of a pedestrian should be much less than the distance traveled by a subject between two velocity updates.

Then, we define the minimum of all the DTI by taking the minimum of over all partner pedestrians. The anisotropy of human vision is taken into account by restricting the set of partner pedestrians to those belonging to the vision cone of pedestrian . Introducing a threshold number , this vision cone is centered at and has half angle about the direction . The minimal DTI of the th-pedestrian is therefore defined by :

(2.9)

where is the total number of pedestrians. Finally, the new direction of motion of the -th pedestrian is the direction that minimizes the distance between the point reached after traveling a distance in this direction and the point reached after traveling a distance in the target direction . Therefore, the new direction of motion is found by solving the following minimization problem:

(2.10)

where again, the test directions are restricted to the vision cone of pedestrian . We denote by the set of vectors of of unit norm.

Remark 2.2

We note that the minimization problem (2.10) is not convex and may have multiple solutions. We will discard this occurrence as non-generic.

2.4 Summary of the Heuristic-Based IBM model

We now consider a collection of point particles with positions , velocity directions at time and target direction , and . The dynamics is as follows:

(2.11)
(2.12)

with

(2.13)

and

(2.14)

We now make some comments on the position update rule (2.11). Since the pedestrian can only walk a distance in the direction before colliding with another pedestrian, it would appear more sensible to use the formula . However, in the present model, the pedestrian speed is supposed equal to one. Therefore, this update can only provide the position at an intermediate time . This leads to position updates at different times for the different pedestrians. In order to derive a time continuous model, it is more convenient to keep position updates at constant time-steps, which justifies the choice made in (2.11). In the limit in (2.11) (but keeping finite in (2.12), (2.14)), the quantity tends to zero and eventually becomes smaller than . Then, the objection formulated at the beginning of this paragraph disappears. In the next section, we first propose a time-continuous model based on this limit, and then deduce a mean-field kinetic model from it.

3 Mean-field kinetic model

3.1 Methodology

The goal of this section is to propose a time and space continuous kinetic model (KM). With this aim, we first convert the previous time-discrete IBM into a time-continuous one. This conversion consists in replacing the sudden change of the velocity every time intervals, by a continuous one.

The difficulty with writing such a time-continuum model comes from the ’roughness’ of the rules of the time-discrete IBM. For this reason, we regularize the time-discrete dynamics in two ways. First, in the perception phase, we replace the distance to the closest encounter by an average distance to the possible encounters in some interaction region. We propose the use of an harmonic average which closely approximates the minimum used in the original model. The use of averages over certain interaction regions is found in many classical swarm models, such as [2, 20, 29, 31, 33, 65], but the introduction of harmonic averages is new. Second, in the decision-making phase, the jump to the direction of motion which maximizes the distance walked towards the target is replaced by a continuous directional change determined by a velocity potential. This supposes that the subjects choose their new cruising direction close to the previous one, which looks realistic.

A final modification of the IBM is to add some noise in the pedestrian velocity updates. This noise accounts for various effects such as the uncertainties in the estimations of the interaction partner velocities, the variability of the subjects’ responses to interactions, the possibility that the subjects react to some unpredicted stimuli, etc. For KM, the introduction of noise in the particle velocity update results in diffusion in velocity space which produces solutions with smooth velocity profiles. This has important consequences for the derivation of CM from KM, as it supports the use of smooth macroscopic closures. Such smooth closures will be at the heart of the VMF closure and of the hydrodynamic limit methodologies which will be described in Sections 4.3 and 4.4 respectively.

3.2 Modified time-continuous IBM

We consider the following time-continuous stochastic model for the pedestrian positions and velocity directions :

(3.1)
(3.2)

where is a force term and is the noise intensity (supposed uniform among pedestrians). The term stands for the standard white noise and the symbol ’’ means that the stochastic differential equation must be understood in the Stratonovich sense. The force term is constructed below in such a way that it remains orthogonal to , i.e. and the noise term is projected onto the orthogonal vector to . These facts, together with the use of the Stratonovich definition of the Stochastic Differential Equation, maintain on the one-dimensional unit sphere i.e. , provided that initially [45].

The force term is defined as follows. First, we replace the ’min’ in (2.13) by an average over neighboring particles located in the -th pedestrian interaction region. We choose an harmonic average, which has the property to give large weights to the small values of the quantity to be averaged. In this way, the harmonic average mimics closely the outcome of the ’min’ operation in (2.13). The -th pedestrian interaction region is defined as the angular sector centered at , with axis , semi-angle and radius . The set of subjects belonging to the -th pedestrian interaction region is:

(3.3)

The angle is the semi-angle of the human vision cone (say typically , i.e. ). The value of is linked to the local inter-particle distance and will be estimated later on. The number of elements of is denoted by .

We then consider the harmonic average of the elementary DTI with all collision partners in the interaction region:

(3.4)

where the ’max’ has been introduced to bound the average for reasons that will be explained below and where the DTI is defined like in (2.14):

(3.5)

The quantity is a lower cut-off for because the elementary DTI can be arbitrarily small. In reality, if the DTI is too small, the pedestrian lowers his velocity or even stops. This feature is not taken into account in the present model, where we only allow directional changes. Therefore, to cope with this situation, our model pedestrian would have to develop very large angular accelerations, which is unrealistic. The parameter is introduced to bound the forces and thus prevent the dynamics to become too singular in this situation. In the situations where the elementary DTI are large (which corresponds to the second alternative of (3.5)), we just set them equal to , so that they are not taken into account in the average (3.4), which computes the global DTI. The bound of the global DTI by is realized by the parameter as described now.

Indeed, the quantity stands for the distance walked by the pedestrian between two velocity updates (i.e. in the discrete model). Of course, if there are no collision partners (i.e. is the vacuum set), or if the elementary DTI with the available collision partners are large, the pedestrian will be able to walk this distance without performing a velocity update. Therefore, we bound by thanks to the ’max’ in (3.4). In practice, we have . Indeed, the lower cut-off for the elementary DTI is of the order of the size of a subject, while the free walking distance between two velocity updates is much larger.

We now define the -th pedestrian potential function for unit vectors by:

(3.6)

The coefficient gives the order of magnitude of the potential and of the force. By (3.2) and the fact that the velocity is dimensionless, the force and consequently have the physical dimension of a reaction rate. Therefore, we can view the quantity as providing the typical magnitude of the pedestrian reaction. The force is defined by

(3.8)

We note that gradients of functions defined on are tangent fields to . Therefore, by formula (3.8), is orthogonal to as it should. This is reflected in (3.8). The first term is proportional to , while the second one is proportional to , and both are orthogonal to .

Definition (3.8) reflects the fact that, under the force , the pedestrian decreases his potential . Therefore, the pedestrian turns towards the direction of the local minimum of the attraction basin of to which he belongs at time . This rule can be viewed as a local version of the global minimum rule set up by (2.12). Of course, this local minimum may not be the global one expressed by (2.12). However, whether in actual life, a pedestrian spontaneously chooses the global minimum or a local one close to his current direction of motion is not clear. Therefore, to our opinion, this local rule is as legitimate as the global one, until experiments can clarify this point. Of course, the two rules coincide if the local minimum is equal to the global one. So the discrepancy between them may be quite small in practice.

3.3 Mean-field kinetic model

We now introduce a statistical description of the system. Instead of using the ’exact’ positions, velocities and preferred directions of pedestrians, we rather describe the system in terms of the probability distribution . Specifically, is the probability of finding pedestrians in a small physical volume about point , within an angular neighborhood of velocity direction , and within an angular neighborhood of preferred direction at time . We recall that , . If there are no interactions between the pedestrians, i.e. if the acceleration term is due to purely external causes, can be rigorously proved to satisfy the following Fokker-Planck equation:

(3.9)

This equation is a consequence of Ito’s formula of stochastic calculus. The left hand-side is a transport operator. It expresses the material derivative of in the phase space spanned by , due to the motion of the particles with velocity and acceleration . The right-hand side is a velocity diffusion term which comes from the velocity noise. Let be the angle between and the first coordinate direction. Then, , , and each term of (3.9) is written as follows:

where the force term is by definition orthogonal to because .

We note that there is no operator acting on the -dependence of . This is because we assume that the target direction is a quantity attached to the agents which is not changed with time. This assumption could easily be modified to take into account a possible change of the target direction with the motion of the pedestrians. However, even with this simplifying hypothesis, the statistics of target directions has a definite influence on the dynamics through the interaction force described below.

Here, the acceleration term is not due to external forces but to interactions between the particles. So the rigorous derivation of (3.9) is more difficult and is left to future work (see e.g. [12]). The acceleration is coupled to through continuous equivalents of (3.8), and is written:

(3.10)

where is the potential of a pedestrian at time located at with velocity and target velocity . The potential is a function of the test direction . It is given by

(3.11)

in terms of the DTI of pedestrians located at position at time with velocity in the test direction .

To compute the DTI, we first define the interaction region of such a pedestrian by:

(3.12)

where will be estimated later on. Then, the continuous equivalent of (3.5) leads to:

(3.13)

We have denoted by the two-dimensional torus . In (3.13), the quantity is the elementary DTI of a pedestrian located at position and velocity in the encounter with a particle located at and having velocity . It is given by:

(3.14)

The significance of formulas (3.13) and (3.14) and the roles of the parameters and are the same as in the time-continuous IBM of section 3.2.

Collecting (3.13) and (3.14) allows us to compute the potential given by (3.11). After computing the gradient, the force (3.10) has the expression:

(3.15)

Now, we can provide an estimate of . As the density increases, the mean inter particle distance decreases like where is the local density:

(3.16)

Therefore, the DTI should decrease in the same proportion. One way to achieve this scaling is by taking . Indeed, since is of the order of (by (3.14)), the average DTI is of the same order. And since , we obtain the expected scaling of like . In practice, we need to take

(3.17)

with sufficiently larger than to ensure that the estimate (3.13) will take into account enough pedestrians.

Finally, the KM consists of the kinetic equation (3.9), with the acceleration computed through (3.15).

3.4 Mean-field kinetic model: discussion

The mean-field model expresses how the statistical distribution of the pedestrians in position, velocity and target direction evolves with time. This evolution combines a transport operator (left-hand side of (3.9)) which describes pedestrian motion towards their target direction and collision avoidance, and a velocity diffusion operator (right-hand side of (3.9)), which models velocity uncertainty. The pedestrian speed is supposed constant because the model focuses on directional changes only. Directional changes are modeled through a force term (3.15), which describes how pedestrians find the best compromise between their target and the necessity of avoiding pedestrians passing by.

The force is tailored to decrease the potential function . This potential describes how well the target point is approached when the pedestrian (initially located at position , velocity and target velocity at time ) moves in direction (formula (3.11)). For a set of test velocities , the pedestrian computes his DTI with a pedestrian located at with velocity (formula (3.14)) and averages it over all pedestrians located in his vision cone (formula (3.12)), giving rise to (formula (3.13)). This averaged DTI provides him with an estimate of the distance he can move in the direction and allows him to compute his potential . Finally, the pedestrian turns to ensure the decay of the potential and to get closer to his goal (formula (3.10)). The interaction term is spatially non-local, through (3.13). In the next section, we derive a spatially local approximation of this non-local term.

3.5 Mean-field kinetic model with local interaction

If we observe the system at a large distance, the various length scales involved in the interaction terms appear to be small. Therefore, under this assumption, it is legitimate to assume that there exists a small dimensionless quantity such that

(3.18)

where all ’hat’ quantities are assumed to be . Simultaneously, we assume that the pedestrian reaction rate remains . We recall that the pedestrian reaction rate is measured by the coefficient (see discussion after Eq. (3.6)). This assumption implies that

(3.19)

We introduce the change of variables , with , in (3.13) and keep only the leading order terms in the expansion in powers of . In this scaling Eqs. (3.9) and (3.10) are unchanged, except that all unknowns , , now depend on . Then, the condition is equivalent to the condition , where

(3.20)

We have

with

(3.21)

Consequently,

with

(3.22)

Finally, we have

Since we look for a local approximation, we assume that . The distribution is assumed to evolve only on the large scale. Therefore, in the Taylor expansion of (3.22) with respect to , we may keep only the leading order term and neglect the higher order ones. As a result of this approximation, disappears from the arguments of the function in both the numerator and denominator. The integration with respect to can thus be performed beforehand, leading to the quantity defined by

(3.23)

and is the two-dimensional area of . This leads to the following expression of , dropping all the hats for simplicity:

(3.24)

Again, is linked to the total density through (3.17). Graphical representations of and can be found in Figs. 5 and 6 respectively. They illustrate that the function only depends on and (i.e. two real variables) while a general function of depends on a vector of and a vector of , i.e. three real variables. This is due to the fact that itself only depends on and . The function can be numerically computed a priori.

We note that the expression of simplifies in the special case . In this case, there is no blind zone: all their collision partners in the disk

(3.25)

are taken into account by the pedestrians. Consequently, the averaging (3.23) is over the whole disk (see Fig. 7). The dependence of upon disappears. The resulting function, denoted by , is given by:

(3.26)

Additionally, it is an elementary matter to remark that when , we have . Another consequence of this simplification is that the potential does not depend on . It can be simply written . This simplification will be exploited in the hydrodynamic limit (see section 4.4).

We summarize this section: due to the assumption that evolves on the large scale only and thus can be taken constant in the interaction region of a given pedestrian, the interaction force only depends on at that location. This local approximation scaling (3.18), (3.19) leads to the kinetic model (3.9) with a local evaluation of the force. The force is still computed from the potential (3.11) through (3.10). However, the evaluation of the DTI is now given by a local velocity average (3.24), where the velocity convolution kernel can be analytically computed. In the simpler case where there is no blind zone, the kernel reduces to a function of only, and the potential does not depend on .

Figure 5: The function . For a fixed value of (in red), the parallel and transverse components of are respectively denoted by and . The domain of definition of (domain where it is finite) is characterized by and (see (3.21)) and is the shaded blue area on the figure. The function is constant along all vertical segments and has value except in the dark blue area where it is constant equal to . The transition happens along the line (the vertical blue line on the figure).
Figure 6: The function is obtained by averaging the function (see Fig. 5) on the cone (represented by the green shaded area). The blue-green shaded area is the intersection of the cone and the support of . It highlights the fact that the function depends on and .
The disk
Figure 7: Case (i.e. there is no blind zone: all collision partners in the disk of radius are taken into account). The function is obtained by averaging the function (see Fig. 5) on the disk (represented by the green shaded area). The blue-green shaded area is the intersection of and the support of . The function depends only on .

4 Macroscopic model

4.1 Necessity of a closure Ansatz

Macroscopic models are obtained by taking averages of functions of the particle velocity over the distribution function . The resulting macroscopic quantities are e.g. the density or the mean velocity of pedestrians at position with target direction at time :

(4.1)
(4.2)

It is necessary to keep the dependence of the macroscopic quantities over the target direction . Indeed, in general, the statistics of the target directions is not known or may change from situation to situation. In situations where the statistics of target directions is known and does not change with time, it is also possible to introduce the total density (already met at (3.16)) and the average velocity of the pedestrians at position and time , irrespective of their target direction. The latter is defined by:

In the present work, we will only consider CM which retain the statistics of target directions.

To pass from the KM (3.9) to a CM for the quantities and , the most direct method is the moment method. It consists in integrating the kinetic equation (3.9) with respect to the particle velocity , after pre-multiplication by polynomial functions of . Unfortunately, in general, this method does not yield a closed model for and because higher order moments (i.e. integrals of higher order polynomials of ) may be involved in the resulting system. These higher moments need to be expressed in terms of and through a suitable closure relation. Closure relations are usually provided through an Ansatz which expresses itself as a function of and . The validity of this Ansatz is subject to caution. When dissipative phenomena are present, such as in gas dynamics, it is possible to justify it through the hydrodynamic limit (see a review on these questions e.g. in [23]). Here, the hydrodynamic limit can be developed solely in the special case where the interaction is local (as in section 3.5) and in the absence of any blind zone behind the subject. We will first propose two other closure methodologies which apply to general cases, but which cannot be justified by a hydrodynamic procedure.

The first closure scheme, referred to as the ’monokinetic closure’, is developed in section 4.2. It is valid when there is rigorously no noise (i.e. no uncertainty in the pedestrian velocities). It postulates a monokinetic distribution function: in the neighborhood of a given location at time , all pedestrians having the same target direction have the same velocity . In other words, in this neighborhood, the statistics of possible velocities is given by a Dirac delta in the velocity variable, located at . The resulting CM belongs to the class of second-order models of traffic: it involves two balance equations for the mass and momentum densities respectively and bears analogies with pressureless gas dynamics models [13, 14]. These models have somehow unpleasant features, such as the possible formation of mass concentrations.

For this reason, a second closure scheme, referred to as the ’VMF closure’, is proposed in section 4.3. The model supposes that some noise is involved in the pedestrian velocities; this is indeed more realistic than the zero-noise assumption of the previous closure. The distribution of velocities is supposed to be a von Mises-Fisher (VMF) distribution. The VMF distribution is a natural extension of the standard Gaussian distribution for random variables defined on the sphere [67]. Like in the monokinetic closure scheme, the resulting CM belongs to the class of second-order models of traffic but the form of the momentum equation has not been previously found anywhere else.

Finally, in section 4.4, we develop the hydrodynamic limit in the special case of a local interaction with no blind zone. The hydrodynamic limit supposes that, for a pedestrian, the process of turning towards the velocity which minimizes the potential is very short. Therefore, the velocity distribution can be approximated by an equilibrium which reflects the instantaneous equilibrium between the turning process and the noise. Such a distribution, which will be our closure Ansatz, is called a ’Local Thermodynamical Equilibrium’ (LTE), by analogy to the standard terminology of statistical mechanics. The LTE is very peaked around the velocity which minimizes the potential, with some spread due to the noise. An important point is that, while the LTE depends on the potential, the potential also depends on the LTE through the definition of the DTI. Therefore, the allowed DTI are determined by a fixed point equation. The resulting LTE can be interpreted as a Nash equilibrium of a game consisting for the pedestrians in finding the best compromise between reaching their target and avoiding collisions with other pedestrians. The framework for a game-theoretic interpretation of LTE can be found in [28]. The resulting model is a first-order model, in the traffic terminology sense. It consists of a conservation equation for the mass density, while the mass flux is determined functionally from the LTE, i.e. from the DTI that have been found by solving the consistency equation.

4.2 Monokinetic closure

4.2.1 Monokinetic closure: derivation

In this section, in order to derive a macroscopic model, we assume a monokinetic distribution function. For the monokinetic assumption to be valid, we need to remove the noise term, and consider the following equation:

(4.3)

coupled to (3.10). The monokinetic closure consists of the Ansatz:

(4.4)

where is the Dirac delta located at (see a graphical representation at Fig. 8 (red arrow)). Note that, by definition, i.e. is a vector of norm . This Ansatz means that there is only one definite velocity at any given point , time , for any preferred direction . It is easily shown [23] that the distribution (4.4) is an exact solution of (4.3) provided that and satisfy the following set of macroscopic equations:

(4.5)
(4.6)

with and given by (3.10). In other words,

(4.7)

and given by (3.11). The potential can be written:

(4.8)

where is given by: