Multivalued fundamental diagrams of traffic flow in the kinetic Fokker-Planck limit

Multivalued fundamental diagrams of traffic flow in the kinetic Fokker-Planck limit

Giuseppe Visconti
Department of Science and High Technology
University of Insubria
Via Valleggio 11, 22100 Como, Italy
Michael Herty
Department of Mathematics
RWTH Aachen University
Templergraben 55, 52062 Aachen, Germany
Gabriella Puppo
Department of Science and High Technology
University of Insubria
Via Valleggio 11, 22100 Como, Italy
Andrea Tosin
Department of Mathematical Sciences “G. L. Lagrange”
Polytechnic University of Turin
Corso Duca degli Abruzzi 24, 10129 Turin, Italy
Abstract

Starting from interaction rules based on two levels of stochasticity we study the influence of the microscopic dynamics on the macroscopic properties of vehicular flow. In particular, we study the qualitative structure of the resulting flux-density and speed-density diagrams for different choices of the desired speeds. We are able to recover multivalued diagrams as a result of the existence of a one-parameter family of stationary distributions, whose expression is analytically found by means of a Fokker-Planck approximation of the initial Boltzmann-type model.

Keywords

Traffic modeling, kinetic models, Boltzmann, Fokker-Planck limit, multivalued diagrams

Msc

90B20, 65Z05, 35Q20, 35Q70, 35Q84

1 Introduction

In traffic flow theory a basic tool to analyze vehicular traffic problems is given by the so-called fundamental relations among the macroscopic quantities of the flow. Mainly two of them are used: the flux-density diagram (or fundamental diagram) and the speed-density diagram. They allow one to study the macroscopic trends of vehicle dynamics as a result of the interactions taking place at the microscopic scale. Therefore, macroscopic relations can be used for instance to predict the maximum capacity of a road when certain traffic rules are imposed, such as e.g. inflow block of heavy vehicles or speed limits. Since the macroscopic dynamics depend on different properties of the flow, such as the types of vehicles traveling on the road or the external conditions (e.g. weather or environmental conditions), experimental diagrams show a multivalued structure: several values of the flux of vehicles or of the mean speed may correspond to a given value of the vehicle density, see for instance Figure 7 in Section 4 with experimental data published in [37]. The qualitative structure of such diagrams is defined by the properties of different regimes, or phases, of traffic as widely studied, e.g. in [1, 17].

Mathematical models for traffic flow capable of providing fundamental diagrams which reproduce the structure of experimental data play thus an important role. In the current mathematical literature, there are three different approaches to model traffic flow phenomena, which differ from each other in the scale of representation. Microscopic models look at the motion of each single vehicle, predicting the evolution of its position and speed by means of a system of ordinary second-order differential equations, [3, 18, 40]. Clearly, the larger is the number of vehicles, the more computationally expensive is the model. On the opposite end, macroscopic models provide a large-scale aggregate point of view, neglecting the microscopic dynamics. Therefore, they study only the evolution of the macroscopic quantities related to the traffic flow by means of partial differential equations inspired by fluid dynamics laws, [4, 5, 24, 35]. Although, in contrast to microscopic models, macroscopic equations can be easily treated, it is necessary to complete the equations with a closure law. This means that the fundamental diagram is given as an a priori relation derived from heuristic or physical arguments, which therefore usually do not result from microscopic dynamics, see however [20].

In this paper we focus instead on kinetic models (mesoscopic scale), which are based on a statistical mechanics approach. They provide an aggregate description of traffic flow by linking collective dynamics to microscopic interactions among the vehicles. These models are thus characterized by a statistical function giving the distribution of the velocity of the vehicles on the road. Kinetic models provide quite naturally a closure law based on the modeling of microscopic dynamics, thus without assuming previous knowledge on the dependence of the mean velocity on the local density of traffic. In fact, macroscopic quantities are computed as moments, with respect to the microscopic speed, of the distribution function. Several kinetic approaches have been proposed, starting from the pioneering work [30, 31] and later [28]. Recently, kinetic models have been widely used, e.g. to model multilane traffic flow [21, 22], flows on networks [10], homogeneous space problems [13], inhomogeneous space phenomena with non-local interactions [20], heterogeneous compositions of the flow [34]. These models are based on a Boltzmann-type collision term, which models speed changes through a transition probability depending on the local traffic conditions and describing the relaxation in time of the kinetic distribution due to the interactions among vehicles. Although in [33] the analytical expression of the stationary solution is found using a suitable choice of the microscopic interactions, in general the interaction integrals appearing in kinetic Boltzmann-type models for traffic flow typically do not provide explicitly the equilibrium distribution and they are very demanding from a computational point of view, see [19]. For this reason, two main approaches have been taken into account in order to compute the time-asymptotic distribution or to reduce the computational cost: on the one hand, one may consider Vlasov-Fokker-Planck-type models in which the interaction integrals are replaced by differential operators, see [12, 14, 16]; on the other hand, one may consider simplified kinetic models with a small number of velocities, namely the discrete-velocity models, see e.g. [9, 15].

In this paper, starting from a Boltzmann-type model characterized by microscopic interaction rules accounting for two levels of stochasticity in drivers’ subjective decisions, we recover a Fokker-Planck-type kinetic model as a grazing collision limit, see e.g. [7, 8, 25, 27, 39]. In particular we follow the technique introduced in [14] for traffic flow, or in [38] in the context of opinion formation models and in [25] for gas-dynamics. On the one hand, the goal is to find the analytical expression of the steady state distributions starting from the microscopic dynamics. On the other hand, we are interested in reproducing multivalued diagrams without using the multi-population framework [34], in which the scattering behavior of experimental data is explained as a result of the heterogeneous composition of the flow of vehicles on the road. This approach requires the solution of a system of kinetic equations describing the evolution of each kinetic distribution associated to each class of vehicles. Here, instead, assuming that drivers react only to average quantities, multivalued diagrams are obtained as a result of the existence of a one-parameter family of stationary solutions. We observe by numerical evidence that this parameter is strictly linked to the macroscopic characteristics of the flow, as the critical density, the maximum flux, the maximum speed in free-flow regime and the maximum density. This means that the parameter synthesizes the different properties of the flow which can induce different macroscopic dynamics. We also improve the multivalued diagrams obtained in [16], in which the authors introduce a Fokker-Planck-type multilane model and recover scattered data as a consequence of the introduction of a probability of lane changing. As a matter of fact, the diagrams show only a small range of density in which the flux of vehicles is multivalued. In addition to that, they reproduce a capacity drop which is too sharp.

Several Fokker-Planck-type models for traffic flow have been proposed in the literature. For instance in [14] the authors derive a Fokker-Planck-type model as an asymptotic limit of a Boltzmann-type model featuring microscopic interactions in which the drivers react to the mean speed. However, in [14] the authors are mainly interested in establishing links between Fokker-Planck-type models based on a heuristic modeling of braking/acceleration and diffusive operators, see [16], and Fokker-Planck type models obtained as limit of Boltzmann-type equations, thus based on microscopic motivations for the given form of the operators. In this sense, the present work can be considered as a natural sequel to [14]. In fact, on the one hand, here we rewrite the microscopic rules of [14] in order to take into account the influence of the road congestion on the drivers’ decisions. Furthermore, we simplify the grazing collision limit, which now does not depend on free parameters. On the other hand, we focus mainly on the study of macroscopic trends of traffic by means of the fundamental diagrams.

The paper is organized as follows. In Section 2 we introduce the microscopic interaction rules and we discuss the similarities with other models. In particular, we show that the rules can be obtained by averaging the post-interaction speeds prescribed in [33] with the addition of a stochastic perturbation due to the inability of the drivers to adjust precisely their speed. In contrast to [14], another stochastic term is introduced, which models the influence of the road congestion on the drivers’ behavior. In Section 3 we derive the Fokker-Planck model as the grazing collision limit of the Boltzmann-type equation. In Section 4 we recover the time-asymptotic kinetic distributions in general form. Then, we specify the interaction rules by choosing the desired speeds of the drivers in braking and acceleration scenarios and we study the fundamental diagrams resulting from two cases. We show that a particular choice of the desired speeds allows one to reproduce the closures law for macroscopic models, as for instance the Greenshields’ closure [11]. Conversely the diagrams obtained from the second set of desired speeds reproduce very well the qualitative structure of experimental data, in particular their scattering in the congested flow regime. Finally, we end in Section 5 summarizing the main results of this work, and proposing possible applications and further developments.

2 Boltzmann model

Recalling that the core of a kinetic model for traffic flow is the modeling of the microscopic interactions among the vehicles, we first discuss the choice of a set of interaction rules and then we recover a Boltzmann-type equation. As usual, the model is characterized by a collision term which describes the relaxation of the kinetic distribution towards equilibrium.

Here, we assume that drivers react to the mean speed of the surrounding vehicles. Thus, they adjust their velocity by comparing their actual speed with the macroscopic velocity . We recall that, in kinetic theory, the macroscopic variables are recovered as moments of the kinetic distribution function . Here we assume that is the space of the microscopic velocities, being the maximum speed, which usually depends on the mechanical characteristics of the vehicles, on imposed speed limits or on the type of drivers. Then

Remark 1.

Throughout the paper we consider the “spatially homogeneous case”, i.e. we assume that vehicles are uniformly distributed along the road so that the kinetic distribution function depends only on their microscopic speed but not on their position . This assumption allows for a direct focus on the interaction dynamics among the vehicles, neglecting possible space inhomogeneities.

2.1 Microscopic model

Let be the pre-interaction velocity of a vehicle and let be the mean speed of the flow. We distinguish two cases:

Acceleration

Vehicles tend to increase their speed when they are slower than the speed of the flow, namely if ;

Braking

Vehicles tend to decrease their speed when they are faster than the speed of the flow, namely if .

Remark 2.

Notice that we do not model the scenario in which , when it is realistic to assume that vehicles do not change their speed. We will see that this case does not give contribution in the Fokker-Planck approximation, see Section 3.

We point out that the interaction rules in traffic flow are different from the gas dynamics case. In particular, the post-interaction speeds resulting from the two scenarios described above are not necessarily symmetric. Therefore, from a microscopic point of view, we can prescribe the following form of the output velocity resulting from acceleration and braking:

if   (1a)
if   (1b)

where

  • is the jump of velocity in the acceleration interaction. We assume that

    The quantity is the desired speed of a vehicle that increases its velocity . In a realistic framework, cannot be taken as a fixed parameter. Therefore, in contrast to [14], we take dependent on the actual speed , in order to preserve the bounds of the acceleration, see [23]. Potentially, one can consider the case in which the desired speed depends also on the density of vehicles , see Section 4.1.

  • , instead, is the jump of velocity in the braking interaction. As discussed for , we consider

    The quantity is the desired speed of a vehicle which is decreasing its velocity . Notice that, in contrast to the desired speed in acceleration, is not a function of but it depends on the macroscopic flux, in particular on the local density and on the mean speed . In fact, in the braking scenario, the desired speed is mainly influenced by the conditions of the traffic flow. For instance, let us consider the case in which a driver bumps into a jam. Initially the driver travels with a velocity larger than the the speed of the flow in front of him, i.e. . Then, the driver must brake and adjust his speed to the traffic condition ahead independently of the value of his actual velocity . Therefore we expect that the jump of velocity is greater when the difference between the actual speed and the mean speed is larger, as for instance in [14];

  • is a random variable with given distribution , having zero mean and variance . The presence of allows one to include a noise term proportional to the velocity jump. From the modeling point of view, the noise term is introduced in order to consider the fact that drivers are not able to estimate and to reach precisely the desired speed after acceleration and braking, as prescribed in (1);

  • is a deterministic and dimensionless parameter which models the time scaling. We notice that for we have for both and . This means that controls also the strength of the microscopic interactions;

  • is the probability of accelerating, see e.g. [19, 30], taken as a function of the local density . It allows one to define the post-interaction speeds as functions of the density. Thus, the post-interaction speed defined by the rules (1) is not deterministically fixed. Although several laws of the probability function are taken into account in order to better fix experimental data, see e.g. [30, 31, 32], we will consider the simplest choice

    (2)

    where is the maximum density taken as the maximum number of vehicles per unit length in bumper-to-bumper conditions;

  • and are functions of the density which modulate the variation of the velocity due to the noise term. In particular, we assume that the uncertainty in accelerating and braking vanishes in free road and jammed conditions, namely when vehicles tend to travel at the maximum speed or to stop, respectively. From now on, we will always take

    (3)

    These simple choices allow one to simplify the microscopic model (1) and at the same time they reproduce realistic macroscopic behaviors of traffic flow, see Section 4.
    Let us illustrate the model considering a few typical cases. Let , namely let us suppose that the road is almost congested. Then and the second term in (1a) vanishes. Since when , it results if . Thus vehicles tend to keep their velocities even if they are slower than the speed of traffic as the road becomes congested. In contrast, if , the post-interaction speed is less than the pre-interaction speed , see (1b), thus fast vehicles reduce the velocity in order to reach the desired speed . On the contrary, if then , i.e. the probability of accelerating becomes larger as the road becomes free. Therefore, using the rules (1), the post-interaction speed is greater than if , namely slow vehicles increase their velocity in order to reach the desired speed . Conversely, in the opposite case , i.e. fast vehicles tend to keep their high speed.

Using definitions (3), the microscopic interaction rules (1) can be written as

if   (4a)
if   (4b)

It is clear that the above expressions are defined by including two levels of stochasticity. The first one is modeled by the terms at the right-hand side of equations (4) that do not involve the random variable . These terms concern the non-deterministic outcome of the vehicle interactions and which give rise to an average post-interaction speed depending on the probability of accelerating. The second level of stochasticity is instead modeled by the terms in (4) that involve the random variable . These terms perturb the average post-interaction speed as a result of the unpredictability of the individual behavior of the drivers.

Remark 3 (Bounded speeds).

We expect that the post-interaction speed is such that when accelerating (i.e. ) and when braking (i.e. ). We can show that this is indeed guaranteed by making suitable assumptions on the random variable . For the equation (4a) we can write

Thus is a convex combination of the pre-interaction velocity and of the desired speed if and only if

Instead, in the case (4b) we can write

Thus results again in a convex combination of the pre-interaction velocity and of the desired speed if and only if

Finally, the two bounds are simultaneously preserved if

Since in the sequel we will be interested in the behavior of the model for small values of , then in the limit we can allow . However, observe that the post-interaction speeds are again bounded because for .
The above bounds do not preserve in principle that has zero mean. Nevertheless, bounds satisfying this request can be easily obtained. For instance, in the simple case , straightforward computations lead to the following bounds

which ensure that has zero mean.

2.2 Comparison with other models

The interaction rules prescribed in (4) can be obtained as a generalization of the stochastic interaction rules given in [33]. There the post-interaction speed is given by

if   (5)
if   (6)

where is the microscopic velocity of the leading vehicle. In fact, in [33], we consider binary interactions in which a vehicle first compares its velocity with the velocity of the vehicle ahead, and then decides in probability how to change its speed. Since in the present framework we are interested in mean field interactions, is replaced by the macroscopic speed and, apart from the additional noise term and the scaling factor , the rules (4) can be indeed recovered as mean speed of the rules (5)-(6).

Notice that (5)-(6) correspond to the so-called model in [33], which describes the situation of instantaneous change of velocity produced by jumps in velocity, from to (acceleration) or to (braking). Clearly, more refined models are present in the literature, see e.g. [19]. Also in [33] a less simplistic model is considered, in which the output speed is uniformly distributed over a bounded range of velocities, the so-called model. However, looking at the fundamental diagrams of traffic, it is proved that the essential information at equilibrium is already caught by the model.

This fact can be also mathematically investigated by computing the evolution equation for the macroscopic speed. In fact, assuming mean field interactions (in the sense specified above), the model introduced in [33] specializes as

(7)

where is the transition probability modeling the stochastic microscopic interaction rules. It writes as

where and are probability densities prescribing the speed after acceleration and braking, respectively. In the model (see (5)-(6)) it results

(8)

while for the model one has

(9)

Substituting the explicit expression of in (7), multiplying for and integrating over the velocity space , we find the evolution equation for the macroscopic speed. In particular, for the choice (8) we get

while for the choice (9) we obtain

We notice that the two differential equations for differ only in a multiplicative constant. This means that the steady state of the macroscopic speed does not depend on the specific choice, either (8) or (9), of the probability densities and . In fact only the relaxation rate towards the equilibrium is influenced by the presence of the multiplicative factor.

The previous considerations justify the choice of the simpler model as a starting point for the derivation of the interaction rules (4). However, notice that in the present paper we include also the noise term, accounting for output speeds which are not precisely determined.

The microscopic interaction rules given in (4) are very similar to those chosen in [14]. There the post-interaction speed are given by

if   (10a)
(10b)

where is an exponent that calibrates the dependence of the noise on the braking/acceleration dynamics and is a function that vanishes at the extreme values of the velocity.

However, some important differences are introduced in the present work. In fact, in (10) the jumps of velocity modeled by the second terms of the right-hand side are fixed since they do not depend on the level of congestion of the traffic. Moreover, these jumps are modulated by two non-negative constants controlling the strength of the interactions, which, however, do not play the role of the time-scaling parameters. Therefore, when a time scaling is applied, the authors need to make assumptions on the way in which the ratio of these parameters behaves in the grazing collision limit. In this way, they produce some arbitrary constants which are not actually model parameters and they use them in order to recover some Fokker-Planck model already studied in the literature.

2.3 Derivation of the Boltzmann equation with the microscopic interaction rules (4)

As mentioned in the Introduction, in this paper we recover the Fokker-Planck equation as a limit of a Boltzmann model for traffic flow with the aim of making the study of stationary solutions and of fundamental diagrams amenable to analysis. To this end, it is useful to derive preliminarily a weak form of the Boltzmann equation.

The derivation of a general Boltzmann-type equation for the case of binary interactions whose post-interaction states are influenced by random variables is given e.g in [26, Chap. 1, Sect. 1.4] and in [13] for traffic flow models. However, since in the present framework we deal with mean field interactions, we follow a different approach based on a formulation similar to (7).

For the post-interaction rules (4) the transition probability is simply

where is given case-wise by either (4a) or (4b) and includes also the random variable .

The evolution equation for the kinetic distribution becomes then

(11)

To pass to the weak form we multiply equation (11) by a test function and integrate over :

(12)

where the operator denotes the mean with respect to the distribution , namely

From (12) it is clear that is a collision invariant which guarantees the conservation of the total number of vehicles. Moreover, it represents the only conservation property satisfied by the kinetic equation, as usual in a traffic flow model.

Synchronized flow

If we consider in (12) and we divide by the density we can recover the evolution equation for the macroscopic speed:

Using the microscopic rules (4) to express the post-interaction speed , we get

(13)

where we have used the fact that the random variable has zero mean. Suppose that the desired speeds and do not depend on the pre-interaction velocity and take, as assumed for instance in [14], . In this case and using equation (13) a straightforward computation provides

(14)

Observe that if and only if which gives thus a steady solution corresponding to the phenomenon of synchronized traffic flow, in which all vehicles travel at the same speed .
Moreover, notice that if , which means that is increasing, i.e. , until . While, if , , which means that is decreasing, i.e. , until . Therefore the sign of is defined by the density at initial time while the equilibrium value of the speed depends also on the initial condition . We stress that this analysis provides information only on the structure of the steady state of the kinetic distribution but the macroscopic equation (14) for the average speed is not sufficient by itself to compute the equilibrium value of .

Figure 1: Left: evolution towards equilibrium of the macroscopic speed starting from different initial conditions. Right: distribution of the microscopic velocities at equilibrium.

This aspect is also numerically investigated in Figure 1 in which we propose a DSMC (Direct Simulation Monte Carlo) simulation for the kinetic equation (12) with the microscopic rules (4) taking . In particular, as in several kinetic models, we use the Nanbu-like asymptotic method [6] which we reformulate in Algorithm 1 for the model (12)-(4). For further details see e.g. the paper [2] on microscopic models of flocking and swarming or the paper [13] on traffic flow models or the book [26].

1:  Take samples of the microscopic velocities , from the initial density ;
2:  fix being the initial density of vehicles and thus the probability of changing velocity ;
3:  for  to  do
4:     compute the macroscopic speed ;
5:     for  to  do
6:        if  then
7:           sample from a zero mean distribution living on ;
8:        else
9:           sample from a zero mean distribution living on ;
10:        end if
11:        if  then
12:           compute ;
13:        else
14:           compute .
15:        end if
16:     end for
17:  end for
Algorithm 1 Nanbu algorithm for the model (12)-(4) with and .

In Figure 1, Algorithm 1 is applied with samples and time steps. In the left panel we show the evolution towards equilibrium of the macroscopic speed for two values of the density of vehicles. Precisely, the blue lines are referred to , and thus using the definition (2), while the red lines are referred to , and thus . We use different hatchings for the lines in order to mark the evolution obtained with different starting values of the mean speed. We notice that we find different equilibrium speeds depending on the initial condition while the trend is influenced by the value of . In fact, in each case, if (resp. ), the mean speed increases (resp. decreases), see equation (14), until it reaches the equilibrium value which is function of . Thus, as usual, the stationary mean speed can be computed by using the solution of the kinetic model while the macroscopic equation (14) is not sufficient to determine .

For instance, focus on the case . The evolution marked with the solid line is obtained by sampling the particles from a uniform distribution on at initial time. Thus the initial value of the mean speed is . Since , the speed increases during time evolution until the equilibrium value . This value is used as initial condition of the evolution marked with the dashed line. The trend is again increasing, since , and we obtain a different equilibrium value . The same procedure is applied in order to compute the evolution marked with the dotted line and similar considerations hold for the case .

Finally, in the right panel of Figure 1 we show that all the equilibrium values of the mean speed are indeed reached when all vehicles travel at the same velocity, i.e. being function of .

Although we can say nothing on the uniqueness of the synchronized steady solutions, we emphasize that this situation can occur at each density, while in experimental studies it occurs only for certain values of , see e.g. [17]. For this reason, we will not analyze deeply this aspect in the following but we will show that this solution is preserved by the Fokker-Planck approximation.

3 Fokker-Planck approximation

The main drawback of a Boltzmann model is the complexity of the integral collision operator which makes the investigation of steady states difficult. Observe that the explicit knowledge of the asymptotic distribution is crucial both for computing the diagrams of traffic without expensive numerical simulations and for developing closure laws for macroscopic model. In fact, a kinetic model provides quite naturally speed-density relations arising from the microscopic interaction rules among the vehicles rather than from intuitive considerations.

In order to obtain simpler kinetic models, the goal is to replace the integral operator with differential operators. To this end, starting from the Boltzmann model (12) we derive a Fokker-Planck model using a technique similar to the one introduced for instance in [7, 8, 27, 39] and already used in [14] for non-symmetric interaction rules, or in [25] for gas-dynamics and in [38] for opinion formation. However, as discussed at the end of Section 2.2, here, in contrast with [14], the interaction rules (4) allow us to simplify the procedure of the grazing collision limit avoiding the definition of several free parameters and additional assumptions on the test function.

From now on, let us to consider . Starting from (12), we first scale the time using the same parameter appearing in (4), setting

For brevity, in the following we omit the use of the tilde and we indicate by the scaled distribution function. Then, braking up the integrals, equation (12) writes as

where

are the acceleration and braking collision kernels respectively. We study explicitly only the case . Similar computations can be easily repeated for . Since for small values of we have , we can compute the Taylor expansion up to second order of around :

with defined in this case by (4a). Substituting in the above expansion, the acceleration part of the collision kernel becomes

where

is the remaining term that vanishes as . In fact, since is a smooth function, the second derivative is Lipschitz continuous, thus

Recalling that , we have for some . Therefore

where and

Thus, computing the limit for , corresponding to a situation in which the interactions are frequent but they produce only small variations in the output velocities, and using the identities

we obtain

The same considerations apply also to the braking part of the collision kernel. In particular, one verifies that as and

Finally, in order to derive the strong formulation of the Fokker-Planck equation, we write the weak form resulting from the grazing collision limit as

(15)

Integrating by parts each term of the right-hand side of (15) and using the fact that , we recover the following strong form of the Fokker-Planck equation

(16)

as limit of the Boltzmann model (12) based on the microscopic rules (4). Observe that (16) can be rewritten in the usual formulation of a Fokker-Planck-type model for traffic flow:

(17)

see for instance the prototype example introduced in [16]. In the present framework, the acceleration/braking operator is

(18)

while the diffusive operator is

(19)

As we will see in Section 4, the stationary solution of (17) can be obtained avoiding the explicit computation of derivatives of characteristic functions that may result in Dirac functions.

Remark 4 (Boundary conditions).

In contrast to the Boltzmann model, in which is automatically conserved, in the Fokker-Planck model the density remains constant in time and thus it is an invariant of the model (17) if, for any choices of and , the following boundary condition holds

(20)

at each time .

4 Fundamental diagrams

In this section, we investigate the fundamental diagrams of traffic flow resulting from the Fokker-Planck model (17). Since we are interested in reproducing the features of experimental diagrams, as the phase transition and the multivalued behavior, we show that a particular choice of the desired speeds allows us to obtain diagrams which exhibit the qualitative properties of data.

As usual, macroscopic diagrams are recovered by computing the macroscopic quantities as moments of the time-asymptotic kinetic distribution . Notice that, using the Fokker-Planck model, can be computed easily by solving an ordinary differential equation. In fact, at the steady state, the time derivative of the distribution function in (17) must be zero and thus we are led to the following homogeneous ODE:

(21)

whose solution can be written in form of exponentials once the cases and are distinguished.

Using the separation of variables method, the time-asymptotic solution for general jumps of velocity and can be computed explicitly substituting (18) and (19) in (21) and it is given by

if   (22a)
if   (22b)

which indeed depends on three unknown parameters, the mean speed , and . The latter two are the left and right limits, respectively, of at , that here we take as two integration constants.

Equivalently we can think of (22a) as parametrized by , with

(23)

Notice that imposing the constraint on the zeroth moment (i.e. mass conservation) of the distribution function

(24)

is not sufficient to define uniquely both and . In fact, the constraint (24) leads to a single equation

with two unknowns, where

are the partial densities of slow and fast vehicles (with respect to the mean speed). Thus, we obtain a family of steady solutions and without loss of generality we assume that will be parametrized by the ratio , so that

(25)

For we obtain continuous asymptotic distributions at , while for we allow for discontinuous distributions at . This is the mathematical key which allows us to recover multivalued diagrams of traffic as a result of the existence of a one-parameter family of solutions. In fact, for each value of the ratio we obtain a corresponding time-asymptotic solution. Thus different values of give different equilibrium speeds, for a fixed density . Usually, in the literature only continuous solutions were considered, see e.g. [16] in which a “small” multivalued behavior is obtained by means of the probability of lane changing. However, experimental fundamental diagrams typically are discontinuous around a critical density which defines a sort of phase change. Thus it is quite natural to expect discontinuous asymptotic distributions.

The equilibrium value of the macroscopic speed is obtained by solving a non-linear equation coming from the constraint on the first moment of the distribution function and from (24)

(26)

The identity (26) provides the nonlinear equation , with

(27)

Solving it for each value of the density in we find the speed at equilibrium which defines therefore a relation between the density and the mean speed of the flow. Since defines a nonlinear equation, in principle it might have zero or more than one solution.

Finally, observe that without diffusion, namely if , then the two exponentials in (22) tend to . This means that the general steady state (22), and thus the equilibrium speed, is influenced only by the choice of the desired speed and . In fact, in this case the microscopic rules (4) become purely deterministic, provided .

Remark 5 (Synchronized flow).

In Paragraph “Synchronized flow” in Section 2.3 we have proved that the distribution function is a stationary solution of the Boltzmann model (12) based on the interactions (4), when . This solution is preserved in the Fokker-Planck approximation. We use the weak form of the model (17) in order to prove that is, in distributional sense, the trivial steady state of the Fokker-Planck-type equation. In fact, assuming as a test function with compact support in , is a weak stationary solution if

Integrating by parts the left-hand side of the above equation we get

Taking we obtain

In fact, the choice leads to a degeneracy at in the acceleration/braking operator , in the diffusion operator and in its derivative, since:

Now, let us study two particular models obtained with different choices of the desired speeds. In both cases, we formulate the explicit steady state and the expression of the function which defines the equilibrium speed. We show that the two choices provide meaningful diagrams of traffic (flux-density and speed-density relationships).

4.1 Case 1

Here we assume

(28)

This means that, in acceleration, the desired speed is a certain velocity in depending on the value of the probability of accelerating and thus on the density of vehicles. Instead, when braking, the desired speed is a velocity in and depends only on macroscopic quantities, as discussed in Section 2. With this choice and evaluating explicitly the integrals appearing in (22), the asymptotic distribution becomes

if   (29a)
if   (29b)

where and . In the sequel we consider . Recall that the equilibrium speed is found by solving the constraint (26), which leads to the non-linear equation . For the steady solution (29) we have