The Aw-Rascle traffic model: Enskog-type kinetic derivation and generalisations

We study the derivation of second order macroscopic traffic models from kinetic descriptions. In particular, we recover the celebrated Aw-Rascle model as the hydrodynamic limit of an Enskog-type kinetic equation out of a precise characterisation of the microscopic binary interactions among the vehicles. Unlike other derivations available in the literature, our approach unveils the multiscale physics behind the Aw-Rascle model. This further allows us to generalise it to a new class of second order macroscopic models complying with the Aw-Rascle consistency condition, namely the fact that no wave should travel faster than the mean traffic flow.

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1 Introduction

The kinetic description of vehicular traffic is probably one of the first examples in which methods of the statistical physics were applied to a particle system which was not a standard gas. Such an approach was initiated by the Russian physicist Ilya Prigogine in the sixties 

[17, 32, 33, 34], in an attempt to explain the emergence of collective properties as a result of individual ones in systems composed by human beings instead of molecules. In more recent times, the application of kinetic equations to various systems of interacting agents (see for instance [5, 35, 42] for application to traffic flows) has gained a lot of momentum. These equations, and more in general the mathematical-physical theory on which they are grounded, have proved to be powerful tools to increase the understanding of multi-agent systems, particularly as far as the exploration of the interconnections among their properties at different scales is concerned [24, 29].

Coming back to vehicular traffic, in the literature there exists at least two other modelling approaches based on differential equations. On one hand, there are the so-called microscopic models, which describe the vehicles as point particles moving according to prescribed acceleration/deceleration laws. We recall here, in particular, the well known follow-the-leader and optimal velocity models [2, 16]. On the other hand, there are the macroscopic models, which, inspired by the hyperbolic conservation/balance laws of fluid dynamics, treat the vehicles as a continuum with density [31]. In this case, one distinguishes between first order models, which rely on the mass conservation only, cf. e.g. [7, 22, 36], and second order models, which include also an equation for the conservation or the balance of the mean speed, cf. e.g. [30]. Second order models allow one to overcome the issue of the unbounded acceleration of the vehicles, which first order models may suffer from, see [21] for a very recent contribution on this. However, they may fail to reproduce the correct anisotropy of the interactions among the vehicles, namely the fact that vehicles are mainly influenced by the dynamics ahead than by those behind them. This issue was first pointed out by Daganzo in [8] and later solved by Aw and Rascle [1] and, independently, by Zhang [43]

. They proposed a heuristic second order hyperbolic traffic model, whose characteristic speeds never exceed the speed of the flow. In this way, the small disturbances produced by a vehicle propagate more slowly than the vehicles themselves, thereby guaranteeing that the movement of each vehicle affects only the vehicles behind.

An interesting theoretical problem, left largely unexplored in the original papers [1, 43], is the derivation of the Aw-Rascle macroscopic model from first principles. In [18, 19, 20], the authors were the first to obtain the Aw-Rascle model as a hydrodynamic limit of a kinetic description based on an Enskog-type equation. Their approach is very much inspiring, because it suggests to look at the Enskog-type kinetic description instead of the more classical Boltzmann-type one. On the other hand, in these cited works, the authors did not focus on the explicit characterisation of fundamental microscopic interactions able to generate, at the macroscopic level, the Aw-Rascle model. Moreover, in [19, 20] the hydrodynamic limit is performed by postulating the existence of an equilibrium kinetic distribution function, which is not exhibited explicitly. In addition, partly heuristic closures of other terms appearing in the equations are used. More recently, also the direct link between follow-the-leader microscopic models and the Aw-Rascle macroscopic model has been explored. In particular, in [9, 10] the authors prove that the trajectories of the former converge, in the -Wasserstein metric, to the unique entropy solution of the latter when a suitable large particle limit is considered. Their strategy consists in interpreting the follow-the-leader model as a discrete Lagrangian approximation of the target macroscopic model. We observe that this approach, although successful from the analytical point of view, does not explain the actual multiscale physics behind the derivation of the Aw-Rascle model from a microscopic particle model.

In this paper, we investigate the possibility to obtain the Aw-Rascle model as the hydrodynamic limit of kinetic descriptions of the traffic system. The highlights of our study, which differentiate it from the other contributions recalled above, may be summarised as follows:

  • we derive explicitly a minimal set of fundamental features of the microscopic interactions among the vehicles, necessary to generate the Aw-Rascle macroscopic model. Furthemore, we link some key elements of the Aw-Rascle model, such as e.g., the so-called “traffic pressure”, to precise characteristics of the microscopic interactions. We observe that, in the modelling of multi-agent systems, the microscopic model of the agent behaviour is often largely heuristic and, as such, somewhat arbitrary. In this respect, our result helps to identify a paradigmatic class of interaction rules among the vehicles, which give rise to a physically consistent macroscopic traffic model;

  • we elucidate the multiscale physical structure underlying the Aw-Rascle model. In particular, we show that an Enskog-type kinetic description, as opposed to a Boltzmann-type one, is ultimately necessary to derive it, because the anticipatory nature of the Aw-Rascle dynamics may be understood as the hydrodynamic result of local and non-local microscopic interactions happening on different time scales;

  • taking advantage of the previous analysis, we show how to generalise the Aw-Rascle model to new classes of second order macroscopic traffic models, which take correctly into account the anisotropy of the interactions among the vehicles.

In more detail, the paper is organised as follows: in Section 2, we discuss the microscopic interactions at the basis of the whole theory. In Section 3, we show that a Boltzmann-type kinetic description does not give rise to the Aw-Rascle model in the hydrodynamic limit nor, more in general, to a macroscopic model correctly reproducing the anisotropy of the vehicle interactions Conversely, in Section 4, we prove that the original Aw-Rascle model can be obtained as the hydrodynamic limit of an Enskog-type kinetic description and we stress, in particular, the role played by spatially non-local interactions among the vehicles towards this result. In Section 5, we exploit the Enskog-type hydrodynamics to extend the Aw-Rascle model to a new class of second order macroscopic traffic models, whose characteristic speeds are slower than the mean speed of the flow. We derive these models from a suitable generalisation of the interactions discussed in Section 2 and we establish a direct link between the new terms appearing in the macroscopic equations and the features of the new microscopic interaction rules. In Section 6, we present several numerical experiments, which both validate the theoretical passage from the kinetic to the hydrodynamic descriptions and highlight analogies and differences among the various macroscopic models obtained in the hydrodynamic limit. Finally, in Section 7, we present some concluding remarks and we briefly sketch further research prospects.

2 Microscopic binary interactions

One of the leading ideas in kinetic theory is that the important interactions among the particles of the system are binary, i.e. each of them involves two particles at a time. Interactions involving simultaneously more than two particles are neglected as higher order effects. In our case, taking inspiration from [39], we express a general binary interaction between any two vehicles as

(1)

where and are the pre- and post-interaction speeds, respectively, of the interacting vehicles. Furthermore, is a function modelling the speed variation of the -vehicle due to the leading -vehicle, which, in contrast, does not change speed because of the front-rear anisotropy of the interactions in the traffic stream. We assume that the interaction rule (1) is parametrised by the traffic density :

where is the kinetic distribution function, because the global traffic conditions may influence the reactions of the individual drivers. Finally,

is a centred random variable, i.e. such that

with denoting expectation, taking into account stochastic fluctuations of the driver behaviour with respect to the deterministic law expressed by . We denote by

the variance of

. The function models the speed-dependent intensity of such a stochastic fluctuation. As far as the variables and the coefficients in (1) are concerned, we will assume

In particular, the unitary maximum values of the speed of the vehicles and of the traffic density have to be understood as dimensionless, referred to suitable maximum physical values.

The binary rules (1) do not conserve, either pointwise or on average, the mean speed of the interacting vehicles, indeed

This is clearly reasonable in view of the physics of vehicle interactions as opposed to that of molecule collisions in classical gas dynamics. Nevertheless, as it is well known in the approach to hydrodynamics by local equilibrium closures, see e.g. [3, 4, 14], in order to obtain a second order macroscopic traffic model, namely a model composed of a self-consistent pair of macroscopic equations, it is necessary that the binary interactions (1) conserve locally both the traffic density and the global mean speed defined by

Indeed, in this way the local “Maxwellian”, i.e. the local equilibrium speed distribution generated by (1), is parametrised by the conserved quantities , , which play the role of the unknowns in the hydrodynamic equations.

The local conservation of requires a suitable assumption on the binary interaction rule (1). We recall that if the vehicles are assumed to be homogeneously distributed in space then a statistical description of the superposition of many interactions among them in any point is provided by the homogeneous Boltzmann-type equation, cf. [29]:

(2)

where is any test function, also called an observable quantity, and is the bilinear interaction operator, whose action on a test function is defined as

Here, denotes the average with respect to the distribution of the random variable contained in . Equation (2) is said to be homogeneous because the space variable plays in it the role of a parameter, so that the statistical description of the traffic is actually the same in every point . Choosing , we immediately deduce the local conservation of the traffic density, indeed . With we obtain instead

therefore is locally conserved provided

(3)

A possible class of functions satisfying (3), which we will henceforth consider throughout the paper, is

(4)

for a given .

We conclude this section by observing that, in order to be physically admissible, the binary rules (1) should guarantee for every choice of . This condition is also necessary for the validity of the Boltzmann-type equation in the form (2), namely with a constant (unitary, in this case) collision kernel on the right-hand side, which corresponds to considering vehicles as Maxwellian particles. While is obvious, it may be hard to prove, in general, that the same is a priori true also for . Nevertheless, in the simple prototypical case

(5)

where is a prescribed density-dependent function, it can be proved [39] that a sufficient condition for is that and satisfy

(6)

for an arbitrary constant . This implies that is bounded and vanishes for .

In the particular case , a simpler sufficient condition for is instead .

The choice (4)-(5) leads to the binary interaction

(7)

Apart from the stochastic contribution, by interpreting as the (small) duration of the interaction we see that the acceleration of the -vehicle, i.e. , is proportional to the relative speed with the leading -vehicle, i.e. . This is consistent with the general structure of microscopic follow-the-leader traffic models [16], the function playing the role of the sensitivity of the drivers.

3 Hydrodynamics from a Boltzmann-type description

A local kinetic description of traffic flow is provided by the following inhomogeneous Boltzmann-type equation in weak form:

(8)

which, by means of the second term on the right-hand side, extends (2) taking into account also the transport of the vehicles in space according to the kinematic relationship . See [33, 34].

The usual way to derive macroscopic equations for the hydrodynamic parameters, such as and , is to choose , , in (8

). This procedure is however endless, because the transport term generates systematically a moment of order

in the th equation, thereby never making the latter closed. In order to overcome such a difficulty, a typical strategy consists in introducing the following hyperbolic scaling of space and time:

(9)

with , so that (8) becomes111Also the variables of the distribution function are scaled according to (9). However, in order to avoid introducing additional notations, we still denote by the scaled distribution function.

(10)

In this equation, plays conceptually the role of the Knudsen number of the classical kinetic theory. If is sufficiently small then locally the interactions are much faster than the displacement of the vehicles. As a consequence, a fluid dynamic regime is conceivable, in which the local equilibrium distribution quickly produced by the interactions is simply transported by the traffic stream. This allows one to solve (10) by splitting the contributions of the interactions and of the transport:

(11)
(12)

analogously to what is commonly done in the numerical solution of the Boltzmann equation, see e.g. [11, 12, 28].

The idea is now that if we are able to identify from (11) the local Maxwellian parametrised by the two conserved quantities , , cf. Section 2, then we may plug it into (12) to obtain the hydrodynamic equations satisfied by , .

3.1 The case 

Unfortunately, when in (1) it is in general not possible to compute explicitly the steady distributions of the homogeneous Boltzmann-type equation (11). However, at least in some particular regimes, one may rely on powerful asymptotic procedures, which transform (11

) in partial differential equations more amenable to analytical solutions. One of such procedures is the so-called

quasi-invariant interaction limit, introduced in [6] and reminiscent of the grazing collision limit applied to the classical Boltzmann equation [40, 41].

Let us assume that the system is locally close to equilibrium, so that each binary interaction (1) produces a very small transfer of speed from the leading to the rear vehicle. In particular, we may obtain such an effect by setting

(13)

which, for small, implies that both the deterministic and the stochastic parts of the interaction are small. In this situation, if is sufficiently smooth then the difference in (11) can be expanded in Taylor series about . After some computations, this yields

where is a bilinear reminder, which, under the assumptions that is bounded and has bounded third order moment (i.e., ), is asymptotic to when , see [39] for the details. On the whole, for , so that in such a limit we obtain that satisfies the equation

(14)

Integrating by parts the terms on the right-hand side, along with suitable conditions on at such that the boundary terms vanish (see again [39] for the details), we recognise that this is the weak form of the following Fokker-Planck equation:

(15)

whose solutions approximate the large time behaviour of (11) in the quasi-invariant regime. In particular, the equilibrium solution to (15), i.e. the local Maxwellian , satisfies

which, for the binary interaction (7), cf. also (4)-(5), becomes

whence

being a normalisation constant to be fixed in such a way that .

To proceed further, we have to choose a diffusion coefficient . A closed form of is obtained, for instance, with222We observe that such a function does not comply with (6), because of the vertical tangents at . Nevertheless, it can be obtained as the uniform limit, for , of a sequence of functions , which instead satisfy (6) for every , see [38]. This justifies its use in the Fokker-Planck equation (15), i.e. after performing the quasi-invariant limit. and reads

(16)

where is the beta function. On the whole, we notice that

is the probability density function of a beta random variable, interestingly quite consistent with some recent experimental findings about the speed distribution in traffic flow 

[23, 26]. Entropy arguments can be invoked [15] to prove that (16) is the unique and globally attractive steady solution with mass to (15) with binary rules (7).

Remark 3.1.

Equation (14) may be rewritten as

where the Fokker-Planck operator is defined, in weak form, as

or equivalently, in strong form, as

The quasi-invariant limit performed above implies that can be consistently approximated by in the regime in which , are small and the frequency of the interactions is high.

Plugging (16) into (12) along with the choices , and recalling the known formulas for the moments of a beta random variable, we obtain the following second order macroscopic model:

(17)

Introducing the vector of the conserved quantities

, and assuming for simplicity that is constant, (17) can be rewritten in quasilinear vector form as

(18)

In particular, the eigenvalues of

are

Since obviously , are both real, hence system (17) is hyperbolic. As it is well known, represent the speeds of propagation of the small disturbances in the flow and, in macroscopic traffic models, they are required not to exceed the mean speed of the flow itself. This consistency condition, established in [1] in a successful attempt to cure the drawbacks of second order macroscopic traffic models put in evidence in [8], is meant to preserve, at the macroscopic level, the front-rear anisotropy of the microscopic vehicle interactions. We will henceforth call this the Aw-Rascle condition.

Unfortunately, for model (17) it is immediately evident that

thus the hydrodynamic derivation based on the Boltzmann-type local equilibrium closure fails, in general, to produce macroscopic traffic models consistent with the Aw-Rascle condition.

3.2 The case 

If in (1), i.e. if the stochastic fluctuations ascribable to the driver behaviour are neglected, then it is much easier to compute the Maxwellian directly from (11). In fact, we see straightforwardly that with given by (4) and for any (continuous) the distribution

(19)

where is the Dirac delta distribution, makes the right-hand side of (11) vanish. Moreover, if is given in particular by (5) and then from (11) with we deduce that the energy of the system converges asymptotically in time to regardless of the initial condition. This implies that (19) is the unique and globally attractive steady solution to the interaction step (11). The Maxwellian (19) is also called a monokinetic distribution, because it expresses the fact that all vehicles travel locally at the same speed, which coincides with the mean speed of the flow.

Plugging (19) into (12), we obtain the following second order macroscopic model:

(20)

In particular, taking advantage of the first equation, we can rewrite the second equation in the non-conservative form and, finally, the whole system in quasilinear vector form as

(21)

Notice that, due to the monokinetic Maxwellian (19), system (20) is pressureless. Consequently, the matrix has two coincident eigenvalues , which formally comply with the Aw-Rascle condition. Nevertheless, model (20) is much more trivial than the actual Aw-Rascle model [1].

4 Hydrodynamics from an Enskog-type description

The discussion set forth in Section 3 has shown that, in general, the local equilibrium closure applied to a Boltzmann-type kinetic description of traffic flow fails to yield Aw-Rascle-type hydrodynamic models. On the other hand, in [19] the authors already pointed out some inconsistencies in the fluid dynamic behaviour of second order macroscopic traffic models so derived, such as e.g. the inability to reproduce density waves propagating backwards. In particular, they identified the source of such a problem in the local nature of the interactions, namely the fact that in (8) the two interacting vehicles occupy the same position . Actually, also in the classical Boltzmann equation the colliding gas molecules are supposed to occupy the same space position at the moment of the collision. However, in that case their velocities are not forced to be non-negative, like in the case of the vehicle speeds. This allows one to have, at the macroscopic level, a gas density flowing in principle in any direction.

In order to overcome such difficulties, in [19] the authors suggested to derive macroscopic traffic models from an Enskog-type kinetic description, in which, similarly to the classical Enskog equation for high density gases, the interacting vehicles are not supposed to occupy the same position. Specifically, the Enskog-type equation for vehicular traffic takes the form of a modification of the Boltzmann-type equation (8):

(22)

where is the headway between the -vehicle and the leading -vehicle, which here we assume to be constant for simplicity.

If is small with respect to the characteristic distances along the road, we can write

whence, suppressing the term , we approximate (22) as

From here, performing again the hyperbolic scaling (9) of space and time, we find

(23)

playing again a role analogous to that of the Knudsen number. In particular, if is small we can describe the hydrodynamic regime by means of the following splitting:

(24)
(25)
Notice that (24) is actually the same equation as (11). In particular, if we consider the regime of small , expressed by the scaling (13) then, in view of Remark 3.1, we can consistently replace (viz. approximate) (24) with
(26)

Conversely, unlike (12), the transport step (25) contains a correction on the right-hand side, strictly related to the non-locality of the interactions. With reference to the interaction rules (7), we observe that the correction term is such that

(27)

In practice, the idea behind system (24)-(25) may be paraphrased as follows: one determines a local Maxwellian from (24), as if the interacting vehicles were localised in the same space position. As a matter of fact, this is analytically doable in the quasi-invariant regime, taking advantage of the Fokker-Planck approximation (26) of (24). Next, one transports by means of (25), including a suitable correction to the pure transport (12) due to the actual non-locality of the interactions. In this transport step, the parameter appearing on the right-hand side of (25), cf. (27), will be assumed small, consistently with the quasi-invariant regime invoked to solve (24).

4.1 The case 

In the case , we can repeat the same steps as in Section 3.1 to find the local Maxwellian (16). Plugging it into (25) with , and recalling furthermore the interaction rule (7), we find the following second order macroscopic model:

(28)

which, assuming again constant for simplicity, can be written in quasilinear vector form as

where, as usual, . Notice that, if , both model (28) and the matrix reduce consistently to model (17) and matrix found in Section 3.1.

The eigenvalues of are, in this case,

Considering that

, we estimate:

and, assuming , we continue this computation as

Finally, if we further restrict ourselves to the case , (which is a sub-case of the one previously considered), we have , whence we conclude .

Figure 1: The region of the state space where the greatest eigenvalue of system (28) surely exceeds the flow speed . The dash-dotted line has equation .

As Figure 1 shows, the interpretation is that there exists a non-empty subregion of the state space where certainly violates the Aw-Rascle condition. Notice that, for , such a subregion expands to cover the whole state space, consistently with the fact that, as already observed, model (28) reduces to model (17).

4.2 The case  and the Aw-Rascle model

If then (24) with the interaction rules (7) admits again the Maxwellian (19) as unique and globally attractive local equilibrium. Plugging it into (25) yields

(29)

Using the first equation, the second equation of this system can be rewritten as

(30)

which coincides with the Aw-Rascle equation for the mean speed upon identifying

(31)

where denotes the traffic “pressure”. With this definition, (30) can be formally further recast as , so that finally system (29) can be given the usual form of the Aw-Rascle model:

(32)

In quasilinear vector form this reads

whence we see that the eigenvalues of are , with clearly , because , hence , by assumption (in other words, the traffic pressure is a non-decreasing function of the traffic density ).

Summarising, we have been able to recover the Aw-Rascle model organically from first principles of the kinetic theory out of the following microscopic features of the binary interactions among the vehicles:

  1. [label=()]

  2. interactions change only the speed of the vehicles in such a way that the global mean speed is locally conserved;

  3. the possible randomness in the behaviour of the drivers is neglected, i.e. driver behaviour is modelled as purely deterministic;

  4. interactions are non-local in space, i.e. a headway between the interacting vehicles is taken into account.

Recalling (7) and (31), the first two features are realised by means of the interaction rules

Notice, in particular, that the driver sensitivity turns out to be proportional to the variation of the traffic pressure and inversely proportional to the headway between the interacting vehicles. Thus, the steeper the increase in the traffic pressure, or the closer the leading -vehicle, the prompter the reaction of the -driver, which is a quite meaningful model of driver behaviour. Moreover, the third feature indicates that Enskog-type equations are the natural kinetic setting for the hydrodynamic derivation of the Aw-Rascle model.

5 Generalisations of the Aw-Rascle model

The procedure followed in Section 4.2 to derive the Aw-Rascle model from the microscopic interactions (7) can be fruitfully exploited to obtain classes of second order macroscopic traffic models complying with the Aw-Rascle condition.

Let us consider the interaction rules (1) with given by (4) and , i.e.

The monokinetic Maxwellian (19) is still an equilibrium to (24), indeed for every test function . Moreover, considering that , we compute

whence, plugging into (25) together with , we determine the following macroscopic model:

(33)

Again, using the first equation we can rewrite the second equation in non-conservative form as

which makes it evident that the quasilinear vector form of system (33) is

The eigenvalues of , i.e. and , satisfy the Aw-Rascle condition provided

(34)

for then it results clearly . Under (34), we may therefore call (33) a generalised Aw-Rascle model. We observe that (34) requires essentially that be a non-decreasing function of the speed for all the physically admissible values of the parameter .

Motivated by the introduction of the traffic pressure defined by (31), which allows one to rewrite the Aw-Rascle model in the form (32), we introduce now a generalised traffic pressure defined by the relationships

(35)

which allows us to rewrite the generalised Aw-Rascle model (33) in the form

In practice, generalises the expression in (32). If, for instance, the function is such that does not depend on then from (35) we determine precisely with , thereby recovering the Aw-Rascle model (32) with .

6 Numerical experiments

In this section, we focus on the numerical description of the models introduced so far. We start from an analysis of the microscopic model of Section 2, which describes the interactions among the vehicles through a binary collision approach. Next, we analyse the various mesoscopic approaches detailed in Sections 34, investigating in particular the role of the scaling parameter . Then, we end with some numerical comparisons between the macroscopic traffic models obtained in the hydrodynamic limit and their corresponding kinetic descriptions. In particular, we show that, for so small that the interactions lead quicly to a local equilibrium, the Enskog model is equivalent to the Aw-Rascle macroscopic model, as anticipated by the theoretical results of Section 4. We also show the anticipating nature of the Enskog model compared to the more standard Boltzmann model.

6.1 Test 1: Microscopic model and trend to equilibrium

We consider the binary rule (7), which entails the conservation of both the mass and the global mean speed of the vehicles. In Section 3.1, we have shown that the system reaches a local equilibrium when the number of interactions grows if the effect of each interaction is sufficiently small, i.e. if we are in the so-called quasi-invariant regime. Therefore, we assume, in particular, the quasi-invariant scaling (13) so that, with the choice for the diffusion coefficient, we expect the beta probability density function (16) as the local Maxwellian.

Such a Maxwellian depends, on one hand, on the average speed of the vehicles. Since this parameter does not play an important role in the convergence to equilibrium, in the numerical simulations of this section we consider simply a fixed value, specifically so as to make the resulting distribution asymmetric. On the other hand, the Maxwellian depends also on the sensitivity parameter , whose value strongly affects the shape of the distribution. For the moment, instead of prescribing as a function of , we consider directly several values of , namely , and we compare the corresponding Maxwellians emerging from the microscopic dynamics with the analytical expression (16) found in the quasi-invariant limit. Moreover, we analyse the convergence of the microscopic model to the equilibrium for two different values of the scaling parameter, specifically and . As far as the stochastic fluctuation in (7

) is concerned, we consider a uniformly distributed random variable

. As a matter of fact, we notice that the particular type of distribution of does not affect the final Maxwellian but only the transient regime towards it.

In Figure 2, we show the equilibrium distributions obtained with the microscopic dynamics (7) scaled according to (13). The curves have been obtained by using vehicles and by averaging the steady state solution over realisations. In each plot, we also represent the analytical steady state (16) of the Fokker-Planck equation (15) and the initial distribution of the vehicles, which has been taken uniform in the interval . In all the tested scenarios, an extremely good agreement between the microscopic interaction dynamics and the Fokker-Planck asymptotics is obtained for , which better mimics the quasi-invariant limit .

Finally, we notice that, for , the Maxwellian can be determined directly from the Boltzmann-type equation (11) as explained in Section 3.2, hence, in particular, without resorting to approximate asymptotic procedures. Such a Maxwellian is therefore exact in every regime of the microscopic parameters and, for this reason, there is no need to report here a numerical comparison.

Figure 2: Test 1. Different equilibria of the microscopic interaction model (7) under the quasi-invariant scaling (13). Left column: , right column: . From top to bottom, the interaction parameter increases from to , leading to different shapes of the equilibrium distribution.

6.2 Boltzmann-type model with and without stochasticity

We consider now a space non-homogeneous scenario and we report results for the Boltzmann-type description of traffic flow coming from (10) under the binary interaction model (7) along with the quasi-invariant scaling (13). We start by giving the details of the discretisation technique.

6.2.1 A Monte Carlo method for the Boltzmann model

Equation (10) is discretised using a Monte Carlo approach, in which we define an ensemble of particles (representing the vehicles) , where is the position and the speed of the th car at time . Here, is the space domain. We then approximate the distribution function by means of the empirical distribution

where the mass of a particle (viz. vehicle) is defined as

denoting the initial density of the vehicles. Upon introducing in a space and speed mesh with cell centres and mesh widths , , respectively, an approximation of the particle density can be obtained as an histogram by computing

(36)

Likewise, an empirical position density is obtained as

(37)

We are now ready to describe the details of the Monte Carlo discretisation. This is based on the strong form of (10), which, by choosing formally , can be written as

(38)

where we have denoted by , the pre-interaction speeds (dummy integration variables) and by the post-interaction speed of the first vehicle. The Monte Carlo method corresponding to (38) is obtained by splitting the interaction and the transport steps, exactly in the same spirit as (11), (12), cf. [13, 27].

Transport

Each car advances from time over a time interval of length by changing its position according to

This gives an intermediate empirical distribution: