Gas-kinetic derivation of Navier-Stokes-like traffic equations

Abstract

Macroscopic traffic models have recently been severely criticized to base on lax analogies only and to have a number of deficiencies. Therefore, this paper shows how to construct a logically consistent fluid-dynamic traffic model from basic laws for the acceleration and interaction of vehicles. These considerations lead to the gas-kinetic traffic equation of Paveri-Fontana. Its stationary and spatially homogeneous solution implies equilibrium relations for the `fundamental diagram', the variance-density relation, and other quantities which are partly difficult to determine empirically. Paveri-Fontana's traffic equation allows the derivation of macroscopic moment equations which build a system of non-closed equations. This system can be closed by the well proved method of Chapman and Enskog which leads to Euler-like traffic equations in zeroth-order approximation and to Navier-Stokes-like traffic equations in first-order approximation. The latter are finally corrected for the finite space requirements of vehicles. It is shown that the resulting model is able to withstand the above mentioned criticism. Comment: For related work see http://www.theo2.physik.uni-stuttgart.de/helbing.html

Full text

arXiv:cond-mat/9806026v1 [cond-mat.stat-mech] 2 Jun 1998
Gas-kinetic derivation of Navier-Stokes-like traffic equations
Dirk Helbing
II. Institute of Theoretical Physics, University of Stuttgart, 70550 Stuttgart, Germany
Abstract
Macroscopic traffic mo dels have recently been severely criticized to base on
lax analogies only and to have a number of deficiencies. Therefore, this paper
shows how to construct a logically consistent fluid-dynamic traffic model from
basic laws for the acceleration and interaction of vehicles. These considera-
tions lead to the gas-kinetic traffic equation of Paveri-Fontana. Its stationary
and spatially homogeneous solution implies equilibrium relations for the ‘fun-
damental diagram’, the variance-density relation, and other quantities which
are partly difficult to determine empirically.
Paveri-Fontana’s traffic equation allows th e derivation of macroscopic mo-
ment equ ations which build a system of non-closed equations. This system
can be closed by the well proved method of C hapman and Enskog which leads
to Euler-like traffic equations in zeroth-order app roximation and to Navier-
Stokes-like traffic equations in first-order approximation. The latter are finally
corrected for the finite space requirements of vehicles. I t is shown that the
resulting model is able to withstand the above mentioned criticism.
Typeset using REVT
E
X
1

I. INTRODUCTION
Because of analogies with gas theory [1–4] a nd fluid dynamics [5–9,3,10] modeling a nd
simulating traffic flow increasingly att racts the attention of physicists [1,5,8,9,11–14]. How-
ever, due to the great importance of efficient traffic for modern industrialized countries,
the investigation of traffic flow has a lready a long tradition. In the 1950s Lighthill and
Whitham [10] as well as Richards [15] proposed a first fluid-dynamic (macroscopic) traffic
model. During the 1960s traffic research focused on microscopic follow-the-leader models
[16–23]. Mesoscopic models of a gas-kinetic (Boltzm ann-like) type came up in the 1 970s
[24,25,4,3,2,26]. Since the 1980s simulation models [27,28] play the most important role due
to the availability of cheap, fast, and powerful computers. We can distinguish macroscopic
traffi c simulation models [29–32], mi c roscopic simulation models [33–36] which include cel-
lular a utomaton models [37–39,11–14], and mixtures of both [40].
In high-fidelity microscopic tra ffic models each car is describ ed by its own equation(s)
of motion. Consequently, computer time and memory requirements of corresponding traffic
simulations grow proportional to the number N of simulated cars. Therefore, this kind
of models is mainly suitable for off -line traffic simulations, detail studies (for example of
on-ramps or lane mergings), or the numerical evaluation of collective quantities [33] like
the density-dependent velocity distribution, the distribution of headway distances etc., and
other quantities that are difficult to determine empirically.
For this reason, fast low-fidelity microsimulation models that allow bit-handling have
been developed for the simulation of large freeways or freeway networks [37,38]. However,
although they reproduce the main effects of traffic flow, they are not very suitable for detailed
predictions because of their coarse-grained description.
Therefore, some authors prefer ma cro scopic traffic models [10,41–43,30,44,5–9]. These
base on equations for collective quantities like the average spatial density ρ(r, t) per lane (at
place r and time t), t he average v elocity V (r, t), and maybe also the velocity variance Θ(r, t).
Here, simulation time a nd memory requirements mainly depend on the discretization ∆r
2

and ∆t of space r and time t, but not on the number N of cars. Therefore, macroscopic
traffic models are suitable for real-time traffic simulations. The quality and reliability of the
simulation results mainly depend on the correctness of the applied macroscopic equations
and the choice of a suitable numerical integration method. The rather old and still continuing
controversy on these problems [41,45,46,42,43,47,30,44,48–50,8,9,5,1,51] shows tha t they are
not at all trivial.
Some of the most important points of this controversy will be outlined in Section II. It
will be shown that even the most a dvanced models still have some serious shortcomings.
The main reason for this is that the proposed macroscopic traffic equations were founded on
heuristic arguments or based on analogies with the equations for ordinary fluids. In contrast
to these approaches, this paper will present a m athematical derivation of macroscopic traffic
equations starting from the gas-kinetic traffic equation of Paveri-Fo ntana [2] which is very
reasonable and seems to be superior to the one of Prigogine and co-workers [24,25,4]. The
applied method is analogous to the derivation of the Navier-Stokes equations for ordinary
fluids from the Boltzmann equation [52–55]. It bases on a Chapman-Enskog expansion [56,57]
which is known from kinetic gas theory and leads to idealized, Euler-like equations in zeroth-
order approximation and to Navier-Stokes-like equations in first-order approximation [58,55].
In this respect, the paper puts into effect the method suggested by Nelson [1]. A similar
method was already applied to the derivation of fluid-dynamic equations for the motion of
pedestrian crowds [59], but it assumed some dissatisfactory approximations.
The further procedure of this paper is as follows: Section II presents a short history of
macroscopic traffic models and discusses the abilities and weaknesses of the different ap-
proaches. Section III introduces the Boltzmann-like model of Prigogine [4] and compares
it with the one of Paveri-Fontana [2]. From their gas-kinetic equations macroscopic (‘fluid-
dynamic’) traffic equations will be derived in Section IV. Unfortunately, they turn out to
build a hierarchy of non-closed equations, i.e. the density equation depends on average veloc-
ity V , t he velocity equation on velocity variance Θ, etc. Therefore, a suitable approximation
must be found to obtain a set of closed equations. It will be shown that some of the traffic
3

models introduced in section II correspond to zeroth-order approximations of different kinds.
These, however, are not very well justified. A similar thing holds for the Euler-like traffic
equations which, apart from a complementary covariance equation, contain additional terms
compared with the Euler equations of ordinary fluids [58]. These are, on the one hand, due
to a relaxation term which describes the drivers’ acceleration towards their desired velocities.
On the other hand, they are due to interactions which are connected with deceleration pro-
cesses since these do not satisfy momentum and energy conservatio n in contrast to atomic
collisions.
A very realistic, first-order approximation which is, in a certain sense, self-consistent can
be found by solving the reduced Paveri-Fontana equation which is obta ined from the original
one by integration with respect to desired velocity. We will utilize the fa ct that, according
to empirical traffic data [60,61,3,62,33], the equilibrium velocity distribution has a Gaussian
form. This allows the derivation of mathematical expressions for the equilibrium velocity-
density relation, the ‘fundamental diagram’ of traffic flow, and the equilibrium variance-
density relation (cf. Sec. IV C). Afterwards an approximate t ime-dependent solution of
Paveri-Fontana’s equation will be calculated by use of the Euler-like equations. D ue to
the additional terms in Paveri-Fontana’s equation compared with the Boltzmann equation
the corresponding mathematical procedure is more complicated than the Chapman-Enskog
expansion for or dinary gases (cf. Sec. V).
Nevertheless, it is still possible to derive correction terms of the Euler-like macroscopic
traffic equations (cf. Sec. VI). These have the meaning of transport terms (like e.g. the
flux density of velocity variance) and are related with the finite skewness γ of the velocity
distribution in non- equilibrium situations. The resulting equations are Navier-Stokes-like
traffic equations which, in comparison with the ordinary Navier-Stokes equations [58 ], con-
tain additional terms arising from the acceleration and interaction of vehicles. Additionally,
they are complemented by a covariance equation which takes into account the tendency of
drivers to adapt t o their desired velocities.
Because of the one-dimensionality of the Navier-Stokes-like traffic equations no shear
4

viscosity term occurs. However, in Section VII it is indicated how transitions between
different driving modes can cause a bulk viscosity term. Furthermore, corrections due to
finite space requirements of each vehicle (vehicle length plus safe distance) are introduced.
The resulting model overcomes the shortcomings of the former macroscopic traffic models
(that are mentioned in Sec. II). Section VIII summarizes the r esults of the paper and gives
a short outlook.
II. SHORT HISTORY OF MACROSCO PIC TRAFFIC MODELS
In 1955 Lighthill a nd Whitham [10] propo sed the first macroscopic (fluid-dynamic) traffic
model. This bases on the continuity equation
∂ρ
∂t
+
∂(ρV )
∂r
= 0 (1)
which reflects a conservation of the number of vehicles. For the average velocity V , Lighthill
and Whitham assumed a static velocity-density relation:
V (r, t) := V
e
[ρ(r, t)] . (2)
Inserting (2) into (1) we obtain
∂ρ
∂t
+
"
V
e
+ ρ
∂V
e
∂ρ
#
∂ρ
∂r
= 0 . (3)
Equation (3) describes the propagation of non-linear ‘kinematic wave s’ with velocity c(ρ) =
V
e
(ρ) + ρ ∂V
e
/∂ρ [10,63]. In the course of time the waves develop a sh ock structure, i.e. their
back becomes steeper and steeper until it becomes perpendicular, leading to discontinuous
wave profiles [10,15,63].
In reality, density changes are not so extreme. Therefore, it was suggested to add a
diffusion term D∂
2
ρ/∂r
2
which smoothes out the shock structures somewhat [63,64]. The
resulting equation r eads
∂ρ
∂t
+ V
e
∂ρ
∂r
= −ρ
∂V
e
∂ρ
∂ρ
∂r
+ D
∂
2
ρ
∂r
2
. (4)
5

For the case of a linear velocity-density relation [65]
V
e
(ρ) := V
max
1 −
ρ
ρ
max
!
(5)
it can be transformed into the Burgers equation [66]
∂g
∂t
+ g
∂g
∂r
= D
∂
2
g
∂r
2
(6)
which is analytically solvable [63]. Here, we have intro duced the function
g[ρ(r, t)] := V
max
1 −
2ρ(r, t)
ρ
max
!
. (7)
The most important restriction of models (1), (2) and (4), (2) is relation (2) which
assumes that average speed V (r, t) is always in equilibrium with density ρ(r, t). Therefore,
these models are not suitable for the description of non-equilibrium situations occuring at
on-ramps, changes of the number of lanes, or stop-and-go traffic.
Consequently, it was suggested to replace relation (2) by a dynamic equation for the
average velocity V . In 1971 , Payne [41] introduced the velocity equation
∂V
∂t
+ V
∂V
∂r
= −
C(ρ)
ρ
∂ρ
∂r
+
1
τ
[V
e
(ρ) − V ] (8a)
with
C(ρ) := −
1
2τ
∂V
e
∂ρ
=
1
2τ





∂V
e
∂ρ





(8b)
which he motivated by a heuristic derivation from a microscopic follow-the-leader model
[67]. Here, V ∂V/∂r is called the ‘convection term’ and describes velocity changes at place
r that are caused by average vehicle motion. The ‘anticipation term’ −(C/ρ)∂ρ/∂r was
intended to account f or the drivers’ awareness of the traffic conditions ahead. Finally, t he
‘relaxation term’ [V
e
(ρ) −V ]/τ delineates an (exponential) adaptation of average velocity V
to the equilibrium velocity V
e
(ρ) with a relaxation time τ.
Unfortunately, fo r bottlenecks the corresponding computer simulation program ’FRE-
FLO’ suggested by Payne [29] produces output that “does not seem to reflect what really
6

happens even in a qualitative manner” [46]. As a consequence, several authors have sug-
gested a considerable number of modifications of Payne’s numerical integration method or
of his equations [68,42,43,47,30,4 4,48,49,69]. A more principal weakness of Payne’s equa-
tions is that their stationary and homogeneous solution is stable with respect t o fluctuations
over the whole density range which can be shown by a linear stability analysis [68,45,41].
Therefore, Payne’s model (1), (8) does not describe the well-known self-organization of stop-
and-go waves above a critical density [43,70]. This problem is removed [45] by substituting
relation (8b) by
C(ρ) :=
∂P
e
∂ρ
(9)
with the equilibrium ‘traffic pre ssure’
P
e
(ρ) := ρΘ
e
(ρ) . (10)
The modified velocity equation reads
∂V
∂t
+ V
∂V
∂r
= −
1
ρ
∂P
e
∂r
+
1
τ
[V
e
(ρ) − V ] (11a)
and can be derived f rom the gas-kinetic (Boltzmann-like) traffic models [4,3,2] (cf. Section
IV). For Θ
e
(ρ), Phillips [3 ,7 1] suggested a relation of the form
Θ
e
(ρ) := Θ
0
1 −
ρ
ρ
max
!
. (11b)
In contrast, K¨uhne [72] as well as Kerner and Konh¨auser [8,9] assumed, as a first approach,
Θ
e
to be a positive constant:
Θ
e
(ρ) := Θ
0
. (12)
Unfortunately, equations (1), (11a ) predict the formation of shock waves like Lighthill and
Whitham’s equation does [43,5 ]. For this r eason, K¨uhne [43,70] suggested to add a small
viscosity term ν∂
2
V/∂r
2
which smoothes out sudden density and velocity changes somewhat.
Then, the velocity equation
7

∂V
∂t
+ V
∂V
∂r
= −
Θ
0
ρ
∂ρ
∂r
+ ν
∂
2
V
∂r
2
+
1
τ
[V
e
(ρ) −V ] (13)
results. A linear stability analysis of K¨uhne’s equations (1), (13) shows that these predict
the self-organization of stop-and-go waves or of so-called ‘phantom traffic jams’ (i.e. unstable
traffic) on the condition
ρ
e





∂V
e
∂ρ





>
q
Θ
0
(1 + τνk
2
) (14)
where k denotes the wave number of the perturbation [73,5]. This condition is fulfilled if the
equilibrium density ρ
e
corresponding to the stationary and spatially homogeneous solution
exceeds a critical density ρ
cr
that depends on the concrete form of V
e
(ρ).
For reasons of compatibility with the Navier-Stokes equations for ordinary fluids Kerner
and Konh¨auser replaced K¨uhne’s constant ν by the density-dependent relation
ν(ρ) =
ν
0
ρ
(15)
with the constant viscosity coefficien t ν
0
. Computer simulations of their equations (1) and
(13), (15) show the development of density clusters [8,9] if the critical density ρ
cr
given by
(14) and (15) is exceeded. On the basis of a very comprehensive study of cluster-formation
phenomena, Kerner and Konh¨auser [9] presented a detailed interpretation of stop-and-go
traffic.
Despite the considerable variety of proposed macroscopic traffic models, even the most
adva nced of them have still some shortcomings. For example, for a certain set of par ameters
the mentioned mo dels predict traffic densities that exceed the maximum admissible density
ρ
bb
= 1/l
0
which is the bumper-to-bumper de nsity (l
0
= average vehicle length) [5]. Fur-
thermore, in certain situations even negative velocities may occur [51]. To illustrate this,
imagine a queue of vehicles of constant density ρ
0
. Assume that, e.g. due to an accident
that blocks the road, this queue has come to rest (i.e. V = 0) and that it ends at r = r
0
which shall imply ρ(r, t) = 0 for r < r
0
. Then, ∂ρ/∂r diverges at place r
0
(or is at least very
large) and equations (8), (11 ), (13) all predict ∂V (r
0
, t)/∂t < 0 if Θ 6= 0.
8

Of course, we wish to have a model that is not only valid in standard situations, but also
in extreme ones. Moreover, the model should provide reasonable results not only for certain
parameter values. This is particularly important for the reason that technical measures like
automatic distance control may change some pa rameter values considerably. Nobody knows
if the existing phenomenological models ar e still applicable, then. Therefore, we will derive
the specific structure of the traffic model from basic principles regarding the behavior of the
single driver-vehicle units and their interactions.
III. GAS-KINETIC (BOLTZMANN- LIKE) TRAFFIC MO DE LS
Let us assume that the motion of an individual vehicle α can be described by several
va riables like its place r
α
(t), its velocity v
α
(t), and maybe other quantities which characterize
the vehicle type or driving style (the driver’s p ersonality). We can combine these quantities
in a vector
~x
α
(t) :=

r
α
(t), v
α
(t), . . .

(16)
that denotes the state of vehicle α at a given time t. The time-dependent phase-space density
ˆρ(~x, t) ≡ ˆρ(r, v, . . . , t) (17)
is then determined by the mean number ∆n(r, v, . . . , t
′
) of vehicles that are at a place
between r − ∆r/ 2 and r + ∆r/2, driving with a velocity between v − ∆v/2 and v + ∆v/2,
. . . at a time t
′
∈ [t −∆t/2, t + ∆t/2]:
ˆρ(r, v, . . . , t) ∆r ∆v . . . :=
1
∆t
t+∆t/2
Z
t−∆t/2
dt
′
∆n(r, v, . . . , t
′
) . (18)
For vehicles, the phase-space densitiy ˆρ is a very small quantity. Therefore, in the limit
∆r → 0, ∆v → 0, . . ., ∆t → 0 it is only meaningful in the sense of the e xpected value of
an ensemb l e of macroscopically identical systems [1]. The interpretation of ˆρ as a quantity
which can describe single traffic situations is only possible for “coarse-grained averaging”
9

where ∆r, ∆v, . . ., and ∆t must be chosen “microscopically large but macroscopically small”
[1,59] or, more exactly,
1. smaller than the scale o n which variations of the corresponding macroscopic quantities
occur,
2. so large that ∆n(r, v, . . . , t) ≫ 1 which is not always compatible with the first condi-
tion.
However, in any case a suitable gas-kinetic equation for the phase-space density ˆρ allows the
derivation of meaningful equations for collective (‘macroscopic’) quantities like the spatial
density ρ(r, t) p er lane, the average velocity V (r, t), and the velocity variance Θ(r, t). To
obtain an equation of this kind, we will bring in the well-known fact that the tempora l
evolution of phase-space density ˆρ is given by the continuity equation [74]
∂ ˆρ
∂t
+ ∇
~x
ˆρ
d~x
dt
!
=
∂ ˆρ
∂t
!
tr
(19)
which again describes a conservation of the number of vehicles, but this time in phase-space
Ω = {all admissible states ~x}. Whereas ∇
~x
(ˆρd~x/dt) reflects changes of phase-space density
ˆρ due to a motion in phase space Ω with velocity d~x/dt, the term (∂ ˆρ/∂t)
tr
delineates changes
of ˆρ due to discontinuous transitions between states.
A. Prigogine’s model
In Prigogine’s model the state ~x is given by the place r and velocity v = dr/dt of a vehicle.
The transition term (∂ ˆρ/∂t)
tr
consists of a relaxation term (∂ ˆρ/∂t)
rel
and an interaction term
(∂ ˆρ/∂t)
int
[24,25,4]. Therefore, equation (19) assumes the explicit form
∂ ˆρ
∂t
+
∂(ˆρv)
∂r
+
∂
∂v
ˆρ
dv
dt
!
=
∂ ˆρ
∂t
!
rel
+
∂ ˆρ
∂t
!
int
. (20)
The interaction term (∂ρ/∂t)
int
is intended to describe t he deceleration of vehicles to the
velocity of the next car ahead in situations when this moves slower and cannot be overtaken.
Prigogine [24,4] suggests to describ e processes of this kind by the Boltzmann equation
10

∂ ˆρ
∂t
!
int
:=
∞
Z
v
dw (1 − p)|v − w|ˆρ(r, v, t)ˆρ(r, w, t) (21a)
−
v
Z
0
dw (1 − p)|w − v|ˆρ(r, w, t)ˆρ(r, v, t) (21b)
= (1 −p)ˆρ(r, v, t)
∞
Z
0
dw (w − v)ˆρ(r, w, t) .
where p denotes the probability that a slower car can be overtaken. Functional relations for
p ≡ p(ρ, V, Θ) (22)
are propo sed in Refs. [4,3,75]. The term (21a) corresponds to situations where a vehicle
with speed w > v must decelerate to speed v, causing an increase of phase-space density
ˆρ(r, v, t). The rate of these situations is proportional
1. to the probability (1 −p) that passing is not possible (which corresponds to the ‘scat-
tering cross section’ in kinetic gas theory),
2. to the relative velocity |v − w | of the interacting vehicles,
3. to the phase-space density ˆρ(r, v, t) of vehicles which may hinder a vehicle with velocity
w > v, and
4. to the phase-space density ˆρ(r, w, t) of vehicles with velocity w > v that may be
affected by an interaction.
Term (21b) describes a decrease of phase-space density ˆρ(r, v, t) due to situations in which
vehicles with velocity v must decelerate to a velocity w < v. A more detailed discussion of
interaction term (21) can be found in Refs. [4,2].
Note that approach (21) a ssumes an instantaneous adaptation of velocity which does not
take any bra king time. Moreover, the deceleration process of the faster vehicle is assumed
to happen at the location r of the slower vehicle, i.e. vehicles are implicitly modelled as
point-like objects without any space requirements. The first assumption is only justified
for braking times that are short compared to temporal changes of phase-space density ˆρ,
11

but modifications for finite braking times are possible [7 5]. The second assumption is only
acceptable for very small densities at which the average headway distance is much larger
than average vehicle length plus safe distance. It will, therefore, be corrected in Section
VII. The corresponding modifications also implicitly take into account the pair correlations
of succeeding vehicles [76]. These are neglected by approach (2 1) due to its assumption of
‘vehicular chaos’, according to which the velocities of vehicles are not correlated until they
interact with each other [2,1].
Now, we come to the description of acceleration processes by vehicles that do not move
with their desired speeds. In this connection, Prigogine suggests a collective relaxation of
the actual velocity distribution
P (v; r, t) :=
ˆρ(r, v, t)
ρ(r, t)
(23)
towards an equilibrium velocity distribution P
0
(v) instead of a n individual speed adjustment
so that
dv
dt
:= 0 . (24)
In detail, Prigogine starts from the observation that free traffic is characterized by a certain
velocity distribution P
0
(v) which corresponds to the distribution P
0
(v
0
) of desired velocities
v
0
. Moreover, he assumes that the drivers’ intention to get ahead with their desired speeds
causes the phase-space density ˆρ(r, v, t) to approach the equilibrium phase-space densi ty
ˆρ
0
(r, v, t) := ρ(r, t)P
0
(v) (25)
(exponentially) with a certain relaxation time τ which is given by the average duration of
acceleration processes. Therefore, Prigogine’s relaxation term has the for m [24 ,2 5,4]
∂ ˆρ
∂t
!
rel
:=
ρ(r, t)P
0
(v) − ˆρ(r, v, t)
τ
. (26)
Despite the merits of Prigogine’s stimulating model, this approach has been severely crit-
icized [2,51]. In a clear a nd detailed paper [2] Paveri-Fontana showed that Prigogine’s model
12

has a number of peculiar properties which are not compatible with empirical findings. For
example, he demonstrates that the relaxation term (26) correspo nds to discontinuous veloc-
ity changes which take place with a certain, time-dependent rate. Furthermore, Daganzo
criticized that, a ccording to (26), “the desired speed distribution is a property of the road
and not the drivers” [51] which was already noted by Paveri-Fontana [2]. In reality, however,
one can distinguish different ‘personalities’ of drivers: ‘aggressive’ ones desire to drive faster,
‘timid’ ones slower. Therefore, Paveri-Fontana [2] developed a n improved gas-kinetic tra ffic
model which corrects the deficiencies of Prigogine’s approach.
B. Paveri-Fontana’s model
Paveri-Fontana assumes that each driver has an individual, characteristic desired velocity
v
0
. Consequently, the associated states ~x are given by place r, velocity v, and desired
velocity v
0
so that Prigogine’s phase-space density ˆρ(r, v, t) is replaced by ˆρ(r, v, v
0
, t). The
corresponding gas-kinetic equation (19) explicitly reads [77]
∂ ˆρ
∂t
+
∂(ˆρv)
∂r
+
∂
∂v
ˆρ
dv
dt
!
+
∂
∂v
0
ˆρ
dv
0
dt
!
=
∂ ˆρ
∂t
!
tr
. (27a)
The term ∂(ˆρdv
0
/dt)/∂v
0
can be neglected since the desired velocity of each driver is nor-
mally time-independent during a trip which implies
dv
0
dt
:= 0 . (27b)
In contrast to Prigogine, Paveri-Fontana describes the acceleration towards the desired ve-
locity v
0
by
dv
dt
:=
1
τ
(v
0
− v) (27c)
which means an individual instead o f a collective relaxation. Relation (27c) can b e easily re-
placed by other acceleration laws dv/dt or density-dependent driving programs as suggested
by Alberti a nd Belli [26]. Alternatively, for acceleration processes a n interaction a pproa ch
can be formulated which was recently proposed by Nelson [1]. However, the assumption
13

(27c) of exponential relaxation is a relatively good approximation since drivers gradually
reduce the acceleration as they approach their desired velocity v
0
.
Paveri-Fontana needs the transition term (∂ˆρ/∂t)
tr
only for the description of deceleration
processes due to vehicular interactions. For these he assumes the Boltzmann equation [2]
∂ ˆρ
∂t
!
tr
:= (1 −p)
∞
Z
v
dw
Z
dw
0
|v − w|ˆρ(r, v, w
0
, t)ˆρ(r, w, v
0
, t)
− (1 − p)
v
Z
0
dw
Z
dw
0
|w − v|ˆρ(r, w, w
0
, t)ˆρ(r, v, v
0
, t) (27d)
which has an analogous interpretation as (21). (For details cf. Ref. [2].) Note that, according
to (27d), “the velocity of the slow car is unaffected by the interaction or by the fact of being
passed” [2] and t hat “no driver changes his desired speed” [2] during interactions. Therefore,
the interaction term (27d) fulfils the requirements called for by Daganzo [51]:
1. that “a car is an anisotropic particle that mostly responds to frontal stimuli” [51]
and that “a slow car should be virtually unaffected by its interaction with faster cars
passing it (o r queueing behind it)” [51].
2. that “interactions do not change the ‘personality’ ( aggressive/timid) of any car” [51].
Finally, note that the proportion of vehicles jamming behind slower cars cannot accelerate.
This circumstance can be taken into account by a density- and maybe velocity- or variance-
dependence of t he relaxation time [4,3,75]:
τ ≡ τ(ρ, V, Θ) . (28)
In order to compare Paveri-Fontana’s t r affic equation with Prigogine’s one we integrate
equation (27) with respect to v
0
and obtain the reduced Paveri-Fontana equation
∂ ˜ρ
∂t
+
∂(v ˜ρ)
∂r
+
∂
∂v
"
˜ρ(r, v, t)
˜
V
0
(v; r, t) − v
τ
#
= (1 − p)˜ρ(r, v, t)
∞
Z
0
dw (w − v)˜ρ(r, w, t) . (29)
Here, we have introduced the reduced phase-space densi ty
14

˜ρ(r, v, t) :=
Z
dv
0
ˆρ(r, v, v
0
, t) (30)
and the quantity
˜
V
0
(v; r, t) :=
Z
dv
0
v
0
ˆρ(r, v, v
0
, t)
˜ρ(r, v, t)
. (31)
The only difference with respect to Prigogine’s formulation (20) to (26) is obviously the
other relaxation term.
IV. DERIVATION OF MACROSCOPIC TRAFFIC EQUATIONS
Since we are mainly interested in the temporal evolution of collective (‘macroscopic’)
quantities like the spatial de nsity
ρ(r, t) :=
Z
dv ˜ρ(r, v, t) (32)
per lane, the average velocity
V (r, t) ≡ hvi :=
Z
dv v
˜ρ(r, v, t)
ρ(r, t)
, (33)
and the velocity va riance
Θ(r, t) ≡ h[v − V (r, t)]
2
i :=
Z
dv [v − V (r, t)]
2
˜ρ(r, v, t)
ρ(r, t)
= hv
2
i −[V (r, t)]
2
(34)
we will now derive equations for the moments m
k,0
with
m
k,l
(r, t) ≡ ρ(r, t)hv
k
(v
0
)
l
i :=
Z
dv
Z
dv
0
v
k
(v
0
)
l
ˆρ(r, v, v
0
, t) . (35)
By multiplying Paveri-Fontana’s equation (29) with v
k
and integrating with respect to v we
obtain [2], via partial integration,
∂
∂t
m
k,0
+
∂
∂r
m
k+1,0
+
Z
dv v
k
∂
∂v
˜ρ
˜
V
0
(v) −v
τ
!
=
∂
∂t
m
k,0
+
∂
∂r
m
k+1,0
−
Z
dv kv
k−1
˜ρ
˜
V
0
(v) −v
τ
!
15

=
∂
∂t
m
k,0
+
∂
∂r
m
k+1,0
−
k
τ
(m
k−1,1
− m
k,0
) (36a)
= (1 − p)
Z
dv ˜ρ(r, v, t)
Z
dw (wv
k
− v
k+1
)˜ρ(r, w, t)
= (1 − p)(m
1,0
m
k,0
− m
k+1,0
m
0,0
) . (36b)
Applying the analogous procedure to Prigogine’s model (20) to (26), for the moments
m
k,0
(r, t) ≡ ρ(r, t)hv
k
i :=
Z
dv v
k
ˆρ(r, v, t) (37)
one can derive t he equations
∂
∂t
m
k,0
+
∂
∂r
m
k+1,0
=
1
τ
(m
0,k
− m
k,0
)
+ (1 − p)(m
1,0
m
k,0
− m
k+1,0
m
0,0
) (38)
(cf. [2]) where
m
0,k
(r, t) :=
Z
dv
0
(v
0
)
k
ˆρ
0
(r, v
0
, t)
= ρ(r, t)
Z
dv
0
(v
0
)
k
P
0
(v
0
) . (39)
A comparison of moment equations (36) with (3 8) shows that Prigogine’s and Paveri-
Fontana’s model lead to identical equations for spatial density ρ(r, t) = m
0,0
(r, t) and average
velocity V (r, t) = m
1,0
(r, t)/ρ(r, t), despite the different approaches for the relaxation term.
However, the equations for higher order moments m
k,0
(r, t) with k ≥ 2 differ.
Obviously, equations (36) as well as (38) represent a hierarchy of non-closed equations
since the equation for the kth moment m
k,0
depends on the (k + 1)st moment m
k+1,0
. As a
consequence, the density equation
∂ρ
∂t
+
∂(ρV )
∂r
= 0 (40)
depends on average velocity V , the velocity equation
∂V
∂t
+ V
∂V
∂r
= −
1
ρ
∂(ρΘ)
∂r
+
1
τ
(V
0
− V ) − (1 − p)ρΘ
= −
1
ρ
∂P
∂r
+
1
τ
[V
e
(ρ, V, Θ) −V ] (41)
16

on variance Θ, etc. Here, we have introduced the average desired velocity
V
0
(r, t) :=
Z
dv
Z
dv
0
v
0
ˆρ(r, v, v
0
, t)
ρ(r, t)
, (42)
the so-called ‘traffic pre ssure’ [25,3,71]
P(r, t) :=
1
ρ(r, t)
Z
dv (v − V )˜ρ(r, v, t)
Z
dw (v − w)˜ρ(r, w, t)
=
Z
dv (v − V )
2
˜ρ(r, v, t) = ρ(r, t)Θ(r, t) , (43)
and the equilibrium velocity
V
e
(ρ, V, Θ) := V
0
− τ(ρ, V, Θ)[1 − p(ρ, V, Θ)]P (44)
which is related with stationary and spatially homogeneous tra ffic flow.
Equations (40) and (41) are easily derivable from the moment equations (36) and (38)
respectively by use of m
0,0
= ρ and
∂m
1,0
∂t
=
∂(ρV )
∂t
= ρ
∂V
∂t
+ V
∂ρ
∂t
. (45)
The variance equation is obtained analogously. For the traffic equation of Paveri-Fontana it
reads
∂Θ
∂t
+ V
∂Θ
∂r
= −2Θ
∂V
∂r
−
1
ρ
∂(ρΓ)
∂r
+
2
τ
(C − Θ) − (1 − p)ρΓ
= −
2P
ρ
∂V
∂r
−
1
ρ
∂J
∂r
+
2
τ
[Θ
e
(ρ, V, Θ, C, J) −Θ] (46)
and dep ends on the covariance
C(r, t) ≡ h(v − V )(v
0
− V
0
)i
:=
Z
dv
0
Z
dv (v − V )(v
0
− V
0
)
ˆρ(r, v, v
0
, t)
ρ(r, t)
=
Z
dv (v − V )[
˜
V
0
(v) − V
0
]
˜ρ(r, v, t)
ρ(r, t)
(47)
17

as well as the third centra l moment
Γ(r, t) ≡ h(v − V )
3
i :=
Z
dv (v − V )
3
˜ρ(r, v, t)
ρ(r, t)
. (48)
In addition, we have introduced the flux density of velocity variance
J(r, t) :=
1
ρ(r, t)
Z
dv (v − V )
2
˜ρ(r, v, t)
Z
dw (v − w)˜ρ(r, w, t)
=
Z
dv (v − V )
3
˜ρ(r, v, t) = ρ(r, t)Γ(r, t) (49)
(which corresponds to the ‘heat flow’ in conventional fluid-dynamics) and the equilbrium
variance
Θ
e
(ρ, V, Θ, C, J) := C −
τ(ρ, V, Θ)
2
[1 −p(ρ, V, Θ)]J . (50)
A. Approximate closed macroscopic traffic equations
We will now fa ce the problem of closing the hierarchy of moment equations by a suitable
approximation. The simplest approximations replace a macroscopic quantity Q(r, t) (which
would be determined by a dynamic equation) by its equilibrium value Q
e
which belongs to
the stationary and spatially homogeneous solution. Approximations of this kind are zeroth-
order approximations. The simplest one is obtained by a substitution of V (r, t) (which
actually obeys Eq. (41)) by the equilibrium velocity
V
e
(ρ) := V
0
− τ(ρ)[1 − p(ρ)]ρΘ
e
(ρ) (51)
(cf. (44 ) ) . Equations (40), (51) obviously correspond to the model (1), (2) of Lighthill and
Whitham. Relation (51) specifies the equilibrium velocity-density relation (2) in accordance
with Paveri-Fontana’s traffic equation. It could be interpreted as a theoretical result con-
cerning the dependence of V
e
(ρ) on the microscopic processes of traffic flow: According to
(51), the equilibrium velocity V
e
(ρ) is given by the average desired velocity V
0
diminished
by a term arising from necessary deceleration maneuvers due to interactions of vehicles.
18

However, according to equation (41), the approximation V (r, t) ≈ V
e
[ρ(r, t)] is only
justified for τ → 0 which is not compatible with empirical data. Consequently, the latter
does not adequately describe non-equilibrium situations like on-ramp t raffic or stop-and- go
traffic where the velocity is not uniquely given by the spatial density ρ(r, t).
Another zeroth-order approximation is found by leaving Eq. (41) unchanged but replac-
ing the dynamic variance Θ(r, t) by the equilibrium variance
Θ
e
(ρ, V ) := C
e
(ρ, V ) −
τ(ρ, V )
2
[1 −p(ρ, V )]ρΓ
e
(ρ, V ) (52)
(cf. (50)). (Here, t he subscript e shall again indicate the equilibrium-value or -relation of a
function.) The resulting model (40), (41), (52) obviously corresponds to the model (1), (11)
of Phillips, this time specifying the equilibrium variance-density relation in accordance with
Paveri-Fontana’s traffic model. A complete agreement between (52) and (11b) results f or
C
e
(ρ, V ) ≡ C
e
(ρ), Γ
e
(ρ, V ) ≡ Γ
e
(ρ), and a special choice of the functional relation τ(ρ, V )[1−
p(ρ, V )] ≡ τ(ρ)[1 −p(ρ)].
However, it is not fully justified to assume that the variance Θ(r, t) is always in equilib-
rium Θ
e
(ρ, V ), since the corresponding relaxation time 2/τ is of the order of the relaxation
time 1/τ for the velocity V (r, t). Moreover, the approximation Θ(r, t) ≈ Θ
e
[ρ(r, t), V (r, t)]
does not describe the empirically observed increase of variance Θ directly before a traffic jam
develops [43,5]. Therefore, we also need the dynamic variance equation (46). The remaining
problem is how to obtain suitable relations for Γ(r, t) and C(r, t).
B. Euler-like traffic equations
Before looking for dynamic relations for Γ(r, t) a nd C(r, t), it is plausible first to look
for equilibrium relations which apply to stationary and spatially homogeneous tra ffic. For
this purpose we require the equilibrium solution ˆρ
e
(v, v
0
) of Paveri-Fontana’s traffic equation
(27).
Unfortunately, it seems impossible to find an analytical expression for ˆρ
e
(v, v
0
), but in
order to derive equations for the velocity moments hv
k
i we are mainly interested in, it is
19

sufficient to find the stationary and spatially homogeneous solution ˜ρ
e
(v) of the reduced
Paveri-Fontana equation (29). For this we need to know the relation
˜
V
0
(v) = a
0
+ a
1
δv + a
2
(δv)
2
+ . . . + a
n
(δv)
n
(53)
with
δv := v − V (54)
and arbitrary n. However, the equation that determines
˜
V
0
(v) depends on the unknown
quantity
˜
Θ
0
(v) :=
Z
dv
0
(v
0
− V
0
)
2
ˆρ
e
(v, v
0
)
˜ρ
e
(v)
(55)
etc. so that we are again confronted with a non- closed hierarchy of equations.
Luckily, f r om empirical data and microsimulatio ns we know that the equilibrium velocity-
distribution
P
e
(v) :=
˜ρ
e
(v)
ρ
e
(56)
(at least in the range of stable traffic without stop-and-go waves) is approximately a Gaussian
distribution [60,61,3,62,33]:
P
e
(v) =
1
√
2πΘ
e
e
−(v−V
e
)
2
/(2Θ
e
)
. (57)
Inserting (53) and (57) into the equation
∂
∂v
˜ρ
e
˜
V
0
(v) −v
τ
!
= −(1 − p)˜ρ
e
ρ
e
δv (58)
which corresponds to equation (29) in the stationary and spatially homogeneous case, we
find the condition
∂
∂v
˜ρ
e
˜
V
0
(v) − v
τ
!
=
˜
V
0
(v) −v
τ
∂ ˜ρ
e
∂v
+
˜ρ
e
τ
∂
˜
V
0
(v)
∂v
− 1
!
=
˜ρ
e
τ

(a
1
− 1) +

2a
2
−
a
0
− V
e
Θ
e

δv +

3a
3
−
a
1
− 1
Θ
e

(δv)
2
. . . +

ka
k
−
a
k−2
Θ
e

(δv)
k−1
. . . −
a
n−1
Θ
e
(δv)
n
−
a
n
Θ
e
(δv)
n+1

(59a)
!
= −(1 − p)˜ρ
e
ρ
e
δv . (59b)
20

A comparison o f the coefficients of (δv)
k
in (59 a) and (59b) leads to
a
n
= 0 , a
n−1
= 0 , . . . a
2
= 0 , a
1
= 1 , (60)
and
a
0
= V
e
+ τ(1 − p)ρ
e
Θ
e
= V
0
, (61)
where we have utilized relation (44) with ( 43). Consequently, for equilibrium situations
velocity distribution (57) implies
˜
V
0
(v) = V
0
+ δv . (62)
With (57) and (62) we can now derive equilibrium relations for C and Γ. One obtains
Γ
e
= 0 (63)
and
C
e
= Θ
e
. (64)
Next, we are looking for r elations for non-equilibrium cases. Assuming that the velocity
distribution
P (v; r, t) :=
˜ρ(r, v, t)
ρ(r, t)
(65)
locally approaches the equilibrium distribution P
e
[V ( r, t), Θ(r, t)] very rapidly, we can apply
the zeroth-order approximation of local equilibrium:
P (v; r, t) ≈ P
(0)
(v; r, t) := P
e
[V ( r, t), Θ(r, t)]
=
1
√
2πΘ(r,t)
e
−[v−V (r,t)]
2
/[2Θ(r,t)]
. (66)
Furthermore, in order to fulfil the compatibility condition
C(r, t) =
Z
dv[v − V (r, t)][v
0
−
˜
V
0
(v; r, t)]P (v; r, t) (67)
21

(cf. (47)), we must generalize relation (62) to
˜
V
0
(v; r, t) = V
0
+
C(r, t)
Θ(r, t)
δv (68)
which is fully consistent with (64). Relations (66) and (68) yield zeroth-order relations for
the spatio-temporal variation of C(r, t) and J(r, t): For the flux density of velocity variance
we find
J(r, t) ≈ J
(0)
(ρ, V, Θ) = ρΓ
(0)
(ρ, V, Θ) = 0 , (69)
whereas for the covariance the dynamic equation
∂C
∂t
+ V
∂C
∂r
= −C
∂V
∂r
−
P
ρ
∂V
0
∂r
+
1
τ
(Θ
0
− C) −2(1 − p)ρC
s
Θ
π
(70)
can be derived from the reduced Paveri-Fontana equation (29) due to
Z
dv
Z
dv
0
(δv)
2
δv
0
ˆρ(r, v, v
0
, t)
=
Z
dv (δv)
2
[
˜
V
0
(v) −V
0
]˜ρ(r, v, t)
=
Z
dv (δv)
3
C
Θ
˜ρ(r, v, t) = J
C
Θ
(71)
(δv
0
:= v
0
− V
0
). (The somewhat lengthy but straightforward calculation is presented in
Ref. [79].)
In the zeroth-order covariance equation (70) the quantity
Θ
0
(r, t) :=
Z
dv
Z
dv
0
[v
0
− V
0
(r, t)]
2
ˆρ(r, v, v
0
, t)
ρ(r, t)
(72)
denotes the variance of desired velocities. The term −Θ∂V
0
/∂r no rmally vanishes since the
average desired velocity V
0
is approximately constant almost everywhere (cf. [77]). Due to
(64), the equilibrium variance related to stationary and homogeneous traffic is obviously
determined by the implicit relation
Θ
e
(ρ
e
, V
e
, Θ
e
) = C
e
(ρ
e
, V
e
, Θ
e
) = Θ
0
− 2τ(1 − p)ρ
e
Θ
e
s
Θ
e
π
. (73)
Inserting the above results into equations (40) , (41 ) , and (46), we obtain the f ollowing
zeroth-order approximations of the density-, velocity-, and variance-equation respectively:
22

∂ρ
∂t
+ V
∂ρ
∂r
= −ρ
∂V
∂r
, (74)
∂V
∂t
+ V
∂V
∂r
= −
1
ρ
∂(ρΘ)
∂r
+
1
τ
(V
0
− V ) − (1 − p)ρΘ
= −
1
ρ
∂P
∂r
+
1
τ
[V
e
(ρ, V, Θ) −V ] , (75)
∂Θ
∂t
+ V
∂Θ
∂r
= −2Θ
∂V
∂r
+
2
τ
(C − Θ)
= −
2P
ρ
∂V
∂r
+
2
τ
(C −Θ) . (76)
Equations (74), (75), and (76) are the ‘Euler-like equations’ of vehicular traffic [58].
In comparison with the Euler equations for ordinary fluids [52–55] they contain additional
terms:
1. The terms (V
0
−V )/τ and 2(C−Θ)/τ arise from the acceleration of vehicles towards the
drivers’ desired velocities v
0
, i.e. they are a consequence of the fa ct that driver-vehicles
units are active systems.
2. The term −(1 − p)ρΘ results from the vehicles’ interactions. It would vanish if mo-
mentum would be a collisiona l i nvariant during vehicular interactions like this is the
case for atomic collisions [74]. However, without this term the ‘vehicular fluid’ would
speed up at bottlenecks which is, of course, unrealistic.
Moreover, the covariance equation (70) is a complementary equation which arises from the
drivers’ tendency to move with their desired velocities v
0
.
C. Equilibrium relations and fundamental diagram
For vehicular traffic, the only dynamic quantity that remains unchanged in a closed sys-
tem (i.e. a circular road) is the average spatial density ¯ρ (due to the conservation of the
number of vehicles). As a consequence, the equilibrium traffic situation is uniquely deter-
mined by ¯ρ which obviously agrees with the equilibrium density ρ
e
. Equilibrium relations
23

for the average velocity V
e
(ρ
e
) and the velocity variance Θ
e
(ρ
e
) in dependence of ρ
e
= ¯ρ
can be obtained from equations (4 4) and (73) if the relations p(ρ, V, Θ) and τ(ρ, V, Θ) are
given (cf. [4,3]). A simple procedure for finding a solution of these implicit equations is to
numerically integrate the equations
dV
dy
= V
e
[ρ
e
, V (y), Θ(y)] − V (y)
= V
0
− τ(ρ
e
, V, Θ)[1 − p(ρ
e
, V, Θ)]̺
e
Θ −V , (77)
dΘ
dy
= Θ
e
[ρ
e
, V (y), Θ(y)] − Θ(y)
= Θ
0
− 2τ(ρ
e
, V, Θ)[1 − p(ρ
e
, V, Θ)]̺
e
Θ
s
Θ
π
− Θ (78)
until dV/dy = 0 and dΘ/dy = 0. Here, we have replaced ρ
e
by ̺
e
= ̺
e
(ρ
e
, V ) in accor-
dance with section VII B in order to take into account the space requirements of vehicles.
The theoretical results for the equilibrium v elocity-density relation V
e
(ρ
e
) = lim
y →∞
V (y), the
equilibrium variance-density relation Θ
e
(ρ
e
) = lim
y →∞
Θ(y), and the fundamental diagram
q
e
(ρ
e
) := ρ
e
V
e
(ρ
e
) (79)
can be directly compared with empirical data.
If, however, p(ρ, V, Θ) or τ(ρ, V, Θ) are unknown r elations, it is still possible to derive
from the fundamental diagram q
e
(ρ
e
) the equilibrium variance-density relation Θ
e
(ρ
e
) for
which an empirical relation seems to be missing: From (77) and (79) we g et
τ(1 − p)̺
e
Θ
e
(ρ
e
) = V
0
− V
e
(ρ
e
) = V
0
−
q
e
(ρ
e
)
ρ
e
. (80)
Inserting this into (73) we find
Θ
e
(ρ
e
) = Θ
0
− 2τ(1 − p)̺
e
Θ
e
(ρ
e
)
s
Θ
e
(ρ
e
)
π
= Θ
0
− 2[V
0
− V
e
(ρ
e
)]
s
Θ
e
(ρ
e
)
π
. (81)
This results in a quadratic equation for the standard devi ation
q
Θ
e
(ρ
e
) of vehicle velocities
which is solved by
q
Θ
e
(ρ
e
) = −
V
0
− V
e
(ρ
e
)
√
π
+
s
[V
0
− V
e
(ρ
e
)]
2
π
+ Θ
0
. (82)
24

V. APPROXIMATE SOLUTION OF PAVERI-FONTANA’S TRAFFIC EQUATION
The traffic equation of Paveri-Fontana was mathematically investigated in several papers
dealing with the existence, uniqueness, and numerical determination of a solution which
satisfies t he non-linear initial-value boundary problem [80–82]. However, the approximate
dynamic solution of the reduced Paveri-Fo ntana equation (29) which will be presented in
this section has not been proposed before.
As one would expect, in non-equilibrium situations the zeroth-order approximation (66)
does not solve the reduced Paveri-Fontana equation (29) exactly. Therefore, we write
˜ρ(r, v, t) =: ˜ρ
(0)
(r, v, t) + ˜ρ
(1)
(r, v, t) (83)
with
˜ρ
(0)
(r, v, t) := ρ(r, t)P
(0)
(v; r, t) =
ρ(r,t)
√
2πΘ(r,t)
e
−[v−V (r,t)]
2
/[2Θ(r,t)]
(84)
and try to derive a relation for the deviation ˜ρ
(1)
(r, v, t). Utilizing that the correction term
˜ρ
(1)
(r, v, t) will usually be small compared to ˜ρ
(0)
(r, v, t) we have
˜ρ
(1)
(r, v, t) ≪ ˜ρ
(0)
(r, v, t) (85)
and get
∂ ˜ρ
∂t
+ v
∂ ˜ρ
∂r
+
∂
∂v
˜ρ
˜
V
0
(v) − v
τ
!
≈
∂ ˜ρ
(0)
∂t
+ v
∂ ˜ρ
(0)
∂r
+
∂
∂v
˜ρ
(0)
˜
V
0
(v) − v
τ
!
=
∂ ˜ρ
(0)
∂t
+ v
∂ ˜ρ
(0)
∂r
+
˜
V
0
(v) −v
τ
∂ ˜ρ
(0)
∂v
+
˜ρ
(0)
τ
∂
˜
V
0
(v)
∂v
− 1
!
. (86)
(For a detailled discussion of this approximation cf. [52,53,55].) Now, introducing the ab-
breviation
d
dt
:=
∂
∂t
+ v
∂
∂r
(87)
we can write
25

∂ ˜ρ
(0)
∂t
+ v
∂ ˜ρ
(0)
∂r
=
d˜ρ
(0)
dt
=
∂ ˜ρ
(0)
∂ρ
dρ
dt
+
∂ ˜ρ
(0)
∂V
dV
dt
+
∂ ˜ρ
(0)
∂Θ
dΘ
dt
=
˜ρ
(0)
ρ
dρ
dt
+
˜ρ
(0)
Θ
δv
dV
dt
+
˜ρ
(0)
2Θ
(δv)
2
Θ
− 1
!
dΘ
dt
. (88)
Relations for dρ/dt, dV/dt, and dΘ/dt can be obtained from the Euler-like equations (74),
(75), and (76) via
d
dt
=
∂
∂t
+ V
∂
∂r
+ δv
∂
∂r
. (89)
We find
dρ
dt
= δv
∂ρ
∂r
− ρ
∂V
∂r
, (90a)
dV
dt
= δv
∂V
∂r
−
1
ρ
∂(ρΘ)
∂r
+
1
τ
[V
e
(ρ, V, Θ) −V ] , (90b)
and
dΘ
dt
= δv
∂Θ
∂r
− 2Θ
∂V
∂r
+
2
τ
(C −Θ) . (90c)
For the interaction term we apply a linear approximation in ˜ρ
(1)
(r, v, t) which is justified by
relation (85). The result is
(1 −p)˜ρ(r, v, t)
Z
dw (w − v)˜ρ(r, w, t)
≈ (1 −p)˜ρ
(0)
(r, v, t)ρ(V − v) −
Z
dw L(v, w; r, t)˜ρ
(1)
(r, w, t) (91a)
where we have introduced a linear operator L
with the components
L(v, w; r, t) := (1 − p)ρ(r, t){[v − V (r, t)]δ(v − w) + P
(0)
(v; r, t)(v − w )}. (91b)
Here, δ(v − w) denotes Dirac’s delta function. The linear operator L
possesses an infinite
number of eigenvalues 1/ τ
µ
(cf. [55,83–86]). The r elevant eigenvalue is the smallest one since
it characterizes temporal changes that take place on t he time scale we are interested in. It
is of the order of the average interaction rate per vehicle [53,52,54]
26

1
τ
0
:=
1 −p
ρ(r, t)
Z
dv
Z
w<v
dw |w − v|˜ρ(r, w, t)˜ρ(r, v, t)
≈ (1 −p)ρ(r, t)
Z
dv
Z
w<v
dw |w − v|P
(0)
(w; r, t)P
(0)
(v; r, t)
= (1 −p)ρ(r, t)
s
Θ
π
. (91c)
The other eigenvalues are somewhat larger [55,83–86] (i.e. τ
µ
< τ
0
for µ 6= 0) and they
describe fast fluctuations which can be adiabatically eliminated [78]. As a consequence, we
can make the so-called ‘relaxation time approximation’ [87]
Z
dw L(v, w; r, t)˜ρ
(1)
(r, w, t) ≈
˜ρ
(1)
(r, v, t)
τ
0
. (91d)
Now, we calculate
˜
V
0
(v) −v
τ
∂ ˜ρ
(0)
∂v
+
˜ρ
(0)
τ
∂
˜
V
0
(v)
∂v
− 1
!
− (1 − p)˜ρ
(0)
ρ(V − v)
=
1
τ

V
0
+
C
Θ
δv − v

−
˜ρ
(0)
Θ
δv

+
˜ρ
(0)
τ

C
Θ
− 1

+ (1 − p)˜ρ
(0)
ρ δv
=
˜ρ
(0)
τΘ
"
(C − Θ)
1 −
(δv)
2
Θ
!
− (V
e
− V )δv
#
. (92)
Inserting (86), (88 ) , and ( 90) to (92) into the reduced Paveri-Fontana equation (29) we
finally obtain
˜ρ
(1)
(r, v, t) ≈ −τ
0
(
˜ρ
(0)
ρ
δv
∂ρ
∂r
− ρ
∂V
∂r
!
+
˜ρ
(0)
Θ
δv
δv
∂V
∂r
−
Θ
ρ
∂ρ
∂r
−
∂Θ
∂r
+
1
τ
(V
e
− V )
!
+
˜ρ
(0)
2Θ
(δv)
2
Θ
− 1
!
δv
∂Θ
∂r
− 2Θ
∂V
∂r
+
2
τ
(C −Θ)
!
−
˜ρ
(0)
Θ
"
C −Θ
τ
(δv)
2
Θ
− 1
!
+
V
e
− V
τ
δv
#)
= −˜ρ
(0)
τ
0
(δv)
3
2Θ
2
−
3 δv
2Θ
!
∂Θ
∂r
. (93)
Obviously, the correction term ˜ρ
(1)
(r, v, t) is a consequence of the finite interaction free
time τ
0
which causes a delayed adjustment of ˜ρ(r, v, t) to the local equilibrium ˜ρ
(0)
(r, v, t).
However, in order to take into account the effects of finite reaction time and braking time we
27

must add a time period τ
′
> 0 t o t he interaction free time τ
0
. Hence, τ
0
must be replaced
by the a daptation time
τ
∗
= τ
0
+ τ
′
. (94)
VI. NAVIER-STOKES-LIKE TRAFFI C EQUATIONS
With the corrected phase-space density
˜ρ(r, v, t) ≈ ˜ρ
(0)
(r, v, t) + ˜ρ
(1)
(r, v, t)
≈ ˜ρ
(0)
(r, v, t)
"
1 −τ
∗
(δv)
3
2Θ
2
−
3 δv
2Θ
!
∂Θ
∂r
#
(95)
we can calculate corrected relations for the collective (‘macroscopic’) quantities
F (r, t) ≡ hf (v)i :=
Z
dv f(v)
˜ρ(r, v, t)
ρ(r, t)
≈ F
(0)
(r, t) + F
(1)
(r, t) (96)
where
F
(i)
(r, t) ≡ hf(v)i
(i)
:=
Z
dv f(v)
˜ρ
(i)
(r, v, t)
ρ(r, t)
. (97)
We find
ρ(r, t) ≈ ρ
(0)
(r, t) , V (r, t) ≈ V
(0)
(r, t) ,
Θ(r, t) ≈ Θ
(0)
(r, t) , P(r, t) ≈ P
(0)
(r, t) , (98)
and
C(r, t) ≈ C
(0)
(r, t) . (99)
However, for the flux density of velocity variance we get
J(r, t) ≈ J
(1)
(ρ, V, Θ) ≡ ρΓ
(1)
(ρ, V, Θ) = −κ
∂Θ
∂r
, (100)
where
28

κ := 3ρτ
∗
Θ (101)
is called a ‘kine tic coeffici ent’. Therefore, the macroscopic traffic equations (40), (41), and
(46) assume f orm
∂ρ
∂t
+
∂(ρV )
∂r
= 0 , (102)
∂V
∂t
+ V
∂V
∂r
= −
1
ρ
∂P
∂r
+
1
τ
[V
e
(ρ, V, Θ) − V ] , (103)
and
∂Θ
∂t
+ V
∂Θ
∂r
= −
2P
ρ
∂V
∂r
+
1
ρ
∂
∂r
κ
∂Θ
∂r
!
+
2
τ
(C − Θ) + (1 − p)κ
∂Θ
∂r
. (104)
Additionally, the corrected covariance equation becomes
∂C
∂t
+ V
∂C
∂r
= −C
∂V
∂r
−
P
ρ
∂V
0
∂r
+
1
ρ
∂
∂r
ζ
∂Θ
∂r
!
+
1
τ
[C
e
(ρ, V, Θ, C) − C] +
(1 − p)
2
ζ
∂Θ
∂r
(105)
with the kinetic coefficient
ζ := κ
C
Θ
= 3ρτ
∗
C (106)
and the equilibrium covariance
C
e
(ρ, V, Θ, C) := Θ
0
− 2τ(1 − p)ρC
s
Θ
π
. (107)
(For a detailed derivation of (105) to (107) cf. Ref. [79].)
Equations (102), (103), and (104) are the Navier-Stokes-like traffic equations [58]. Com-
pared with the Navier-Stokes equations for ordinary fluids they possess the additional terms
(V
e
− V )/τ and 2(Θ
e
− Θ)/τ with Θ
e
= C + (τ/2)(1 − p)κ∂Θ/∂r which are due to accel-
eration and interaction processes. Because of the spatial one-dimensionality of the consid-
ered traffic equations, the velocity equation (103) does not include a shear viscosity term
29

(1/ρ)∂/∂r(ν
0
∂V /∂r). The variance equation (104) is related to the equation of heat cond uc-
tion. However, Θ does not have the interpretation of ‘heat’ but only of velocity variance,
here. Finally, the Navier-Stokes-like traffic equations are complemented by the a dditional
cova r ia nce equation (105) arising from t he tendency of drivers to get ahead with a certain
desired velocity v
0
.
We recognize that the first-order macroscopic traffic equations (102), (10 3), (104), and
(105) build a closed system of equations. Moreover, according to (98), the relations for the
spatial density, average velocity, velocity variance, and traffic pressure did not change. In
this sense, the chosen Chapman-Enskog method for closing the hierarchy of macroscopic
equations is consistent with its assumption, according to which only the expressions for the
flux density of velocity variance J ≡ ρΓ and the covariance C were to be improved by the
non-equilibrium correction ˜ρ
(1)
(r, v, t). However, note that another relation for
˜
V
0
(v) than
(68) would have led to modifications of ρ, V , and/or Θ.
We also recognize that the finite adaptation time τ
∗
for approaching the equilibrium
distribution (66) causes a finite skewness
γ :=
Γ
Θ
3/2
=
J
ρΘ
3/2
= −
κ
ρΘ
3/2
∂Θ
∂r
= −
3τ
∗
√
Θ
∂Θ
∂r
(108)
of the non- equilibrium velocity distribution
P (v; r, t) ≈
˜ρ
(0)
(r, v, t) + ˜ρ
(1)
(r, v, t)
ρ(r, t)
. (109)
This leads to the so-called transport terms
− κ
∂Θ
∂r
and − ζ
∂Θ
∂r
. (110)
The effect o f these terms in equations (104) and (105) is to smooth out sudden changes of
va riance and covariance via second spatial derivatives of Θ(r, t), namely
∂
∂r
κ
∂Θ
∂r
!
and
∂
∂r
ζ
∂Θ
∂r
!
. (111)
30

VII. CORRECT IONS OF THE MODEL
A. Driver behavior and bulk viscosity
We remember that the term −(1/ρ)∂P/∂r describes an anticipation effect. It reflects
that drivers accelerate when the ‘traffic pressure’ P = ρΘ lessens, i.e. when the density ρ or
the variance Θ decreases. However, drivers additionally r eact to a spatial change of averag e
velocity. This effect can be modelled by the modified pressure relation
P(ρ, V, Θ) := ρΘ − η
∂V
∂r
(112)
which g ives velocity equation (1 03) a similar form like variance equation (104) a nd covariance
equation (105).
In order to present reasons for relation (112) let us a ssume that drivers switch between
two d riv ing modes m ∈ {1, 2} depending on the traffic situation. Let m = 1 characterize
a brisk, m = 2 describe a careful driving mode. Then, we can split the density ρ(r, t) into
partial densities ρ
m
(r, t) that delineate drivers who are in state m:
ρ
1
(r, t) + ρ
2
(r, t) = ρ(r, t) . (113)
Both densities are governed by a continuity equation, but this time we have transitions
between the two driving modes with a rate R(ρ
1
, V ) so that
∂ρ
1
∂t
= −
∂
∂r
(ρ
1
V ) − R(ρ
1
, V ) , (114a)
∂ρ
2
∂t
= −
∂
∂r
(ρ
2
V ) + R(ρ − ρ
2
, V ) . (114b)
Adding both equations we see that the original continuity equation (102) is still valid. Now,
defining the substantial time derivative
D
Dt
:=
∂
∂t
+ V
∂
∂r
(115)
we can rewrite (114a) and obtain
31

Dρ
1
Dt
= −ρ
1
∂V
∂r
− R(ρ
1
, V ) . (116)
D/Dt describes temporal changes in a coordinate system that moves with velocity V . As-
suming that ρ
1
relaxes rapidly we can apply the adiabatic approximation [78]
Dρ
1
Dt
≈ 0 (117)
which is va lid on the slow time-scale of the macroscopic changes of traffic flow. This leads
to
R(ρ
1
, V ) ≈ −ρ
1
∂V
∂r
. (118)
Relation (117) implies that t he density ρ
1
of briskly behaving drivers is approximately con-
stant in the moving coordinate system whereas the density ρ
2
= ρ −ρ
1
of carefully b ehaving
drivers varies with the traffic situation:
Dρ
2
Dt
≈ −ρ
∂V
∂r
. (1 19)
ρ
2
increases when the average velocity spatially decreases (∂V/∂r < 0) since this may
indicate a critical traffic situation.
According to relations (114), (118) incessant transitions between the two driving modes
take place as long as traffic flow is spatially non-homogeneous (i.e. ∂V/ ∂r 6= 0). This leads
to corrections of the pressure relation. Expanding P with respect to the variable R which
characterizes the disequilibrium between the two driving modes we find [74]
P(ρ, Θ, R) = P(ρ, Θ, 0) −
∂P
∂R





R=0
ρ
1
∂V
∂r
+ . . . . (120)
With the equilibrium relation P(ρ, Θ, 0) = ρΘ and
η := ρ
1
∂P
∂R





R=0
(121)
we finally obtain the desired result
P(ρ, Θ, R) ≡ P(ρ, V, Θ) = ρΘ − η
∂V
∂r
. (122)
A more detailed discussion can be found in Ref. [74].
32

B. Modifications due to finite space requirements
We will now introduce some corrections that are due to the fact that vehicles are no
point-like objects but need, on average, a space of
s(V ) = l + V T (123)
each. Here, l ≥ l
0
is about the average ve hicle l ength whereas V T corresponds to the s afe
distance each driver should keep to the next vehicle ahead. T is ab out the reaction time.
Consequently, if ∆N(r, t) := ρ(r, t) ∆r means the number of vehicles that are at a place
between r − ∆r/2 and r + ∆r/2, the effective density is
̺(r, t) =
∆N(r, t)
∆r − ∆N(r, t)s[V (r, t)]
=
ρ(r, t)
1 − ρ(r, t)s[V (r, t)]
. (124)
Since ∆N(r, t)s(V ) is the space which is occupied by ∆N(r, t) vehicles, the effective density
is the number ∆N(r, t) of vehicles per effective free space ∆r − ∆N(r, t)s(V ).
The reduction o f available space by the vehicles leads to an increase of their interaction
rate. Therefore, we have
∂ ˆρ
∂t
!
tr
:= (1 − p)
∞
Z
v
dw
Z
dw
0
|v − w|ˆ̺(r, v, w
0
, t)ˆρ(r, w, v
0
, t)
− (1 − p)
v
Z
0
dw
Z
dw
0
|w − v|ˆ̺(r, w, w
0
, t)ˆρ(r, v, v
0
, t) (125)
with
ˆ̺(r, v, v
0
, t) :=
ˆρ(r, v, v
0
, t)
1 − ρ(r, t)s[V (r, t)]
. (126)
Consequently, we obtain the corrected relation
1
τ
0
:= (1 − p)̺
s
Θ
π
. (127)
In addition, we must replace P and J by
P
′
:=
P
1 − ρs(V )
and J
′
:=
J
1 − ρs(V )
(128)
33

respectively [76]. For the kinetic coefficients η, κ, and ζ we obtain the corrected relations
η
′
:=
η
1 − ρs(V )
, κ
′
:=
κ
1 − ρs(V )
= 3̺τ
∗
Θ ,
and ζ
′
:=
ζ
1 − ρs(V )
= 3̺τ
∗
C . (129)
The corrected fo r mula
̺Θ =
ρΘ
1 − ρs(V )
(130)
for the equilibrium pressure corresponds to the pressure relation of van der Waals for a ‘real
gas’. According to (130), the traffic pressure diverges for ρ → ρ
max
:= 1/l which causes a
deceleration of vehicles.
The corrected kinetic coefficients η
′
(ρ, V, Θ), κ
′
(ρ, V, Θ), and ζ
′
(ρ, V, Θ, C) also diverge
for ρ → ρ
max
[76]. We find for example
κ
′
ρ≈ρ
max
−→ 3̺τ
′
Θ =
3ρτ
′
Θ
1 − ρs(V )
(131)
so that the divergence of κ
′
is a consequence of the finite reaction- and braking-time τ
′
.
This divergence causes a homogenization of traffic flow since the second spatial derivatives
∂/∂r(η∂V/∂r), ∂/∂r(κ∂Θ/∂r), and ∂/∂r(ζ∂Θ/∂r) produce a spatial smoothing of average
velocity V , variance Θ, and covariance C respectively.
It is the divergence of ‘traffic pressure’ and kinetic coefficients for ρ → ρ
max
that prevents
the spatial density ρ from exceeding the maximum density ρ
max
[5].
VIII. SUMMARY AND O UTLOOK
This paper started with a discussion of the most widespread macroscopic traffic mo dels.
Each of them is suitable for the description of certain traffic situations on freeways but fails
for others. Therefore, an improved fluid-dynamic model was derived fro m the gas-kinetic
traffic equation of Paveri-Fontana [2] which is very well justified and does not show the
peculiar properties of Prigo gine’s Boltzmann-like approach [4].
34

For the derivation of the improved traffic model, moment equations for collective (’macro-
scopic’) quantities like the spatial density, averag e velocity, and velo city variance had to be
calculated. The system of macroscopic equations turned out to be non-closed so that a
suitable approximation was necessary. Here, the well proved Chapman-Enskog method was
applied. In zeroth-order approximation the velocity distribution is assumed to be in ‘local
equilibrium’. According to empirical data, the latter is characterized by a Gaussian velocity
distribution. D epending on the respective kind of zeroth-order approximation one arrives
at the Lighthill-Whitham model [10], the model of Phillips [3,71], or the Euler-like traffic
equations.
For the derivation of a first-order approximation, the reduced Paveri-Fontana equation
was linearized ar ound the local equilibrium solution and solved by a pplication of the Euler-
like traffic equations. The resulting correction term for the non-equilibrium velocity distri-
bution allowed the calculation of additional transpor t terms which describe a flux density of
velocity variance and covariance in spatially non-homogeneous situations. They are related
with a finite skewness of the velocity distribution. The shear-viscosity term vanishes because
of the one-dimensionality of the considered traffic equations. Nevertheless, a bulk-viscosity
term results from transitions between two different driving modes: a brisk and a careful one.
The resulting Navier-Stokes-like traffic equations were finally corrected in order to take
into account the finite space requirements of vehicles. They overcome the deficiencies of the
former macroscopic traffic models so that the criticism by Daganzo [51] and others could be
invalidated:
1. The anticipation term which, in other models, is responsible for the prediction of
negative velocities vanishes in problematic situations like the one described at the end
of Sec. II since the variance becomes zero, then.
2. The density ρ(r, t) does not exceed the maximum a dmissible density ρ
bb
(= bumper-
to-bumper density) [5] since the diverging viscosity term causes a homogenization
of t raffic flow and the diverging traffic pressure suppresses an unrealistic growth of
35

velocity which stops a further increase of traffic density.
3. The model takes into a ccount different driving styles by a distribution o f desired ve-
locities v
0
which are directly associated with the individual drivers. An extension of
the Navier-Stokes-like traffic equations to different vehicle types (cars and trucks) is
possible [88].
4. The interaction between drivers is modelled a nisotropically since the slower vehicle is
assumed not to be affected by a faster vehicle behind it or overtaking it.
5. According to the Navier-Stokes-like equations, disturbances may propagate with a
velocity c > V since a certain proportion of vehicles moves faster than the average
velocity V due to the finite velocity variance Θ. Therefore, in contrast to what was
claimed by Dag anzo [51], it is admissible that macroscopic traffic models “exhibit one
characteristic speed greater than the macroscopic fluid velocity” [51,89].
Present investigations focus on the computer simulation of the Navier-Stokes-like traffic
equations. This work has already been successfully started f or a circular road [5,90] and is
now extended to complex freeway networks.
Moreover, the gas-kinetic and Navier-Stokes-like traffic models can be generalized to
models for multi-lane traffic where overtaking and lane-changing is explicitly taken into
account [88]. By this, formulas for the relations τ(ρ, V, Θ) and p(ρ, V, Θ) can be derived [91].
ACKNOWLEDGEMENTS
The author wants to thank M. Hilliges, R. K¨uhne, and P. Konh¨auser for valuable and
inspiring discussions.
36

REFERENCES
[1] P. Nelson, Transport Theory and Statistical Physics 24, 383 (1995).
[2] S. L. Paveri-Fontana, Transportation Research 9, 225 (1975).
[3] W. F. Phillips, Kinetic Model for Traffic Flow (Report No. DOT/RSPD/DPB/50–
77/17, Nat io nal Technical Information Service, Springfield, Virginia 22161, 1977 ).
[4] I. Prigogine and R. Herman, Kin e tic Theory of Vehicular Traffic (American Elsevier,
New York, 1971).
[5] D. Helbing, Phys. Rev. E 51, 3164 (1995).
[6] M. Hilliges, “ A phenomenological model for dynamic traffic flow in networks”, Trans-
portation Research B, 1995 (to be published).
[7] M. Hilliges, R. Reiner, a nd W. Weidlich, in Modelli ng an d S i mulation 19 93, edited by
A. Pave (Society for Computer Simulation International, Ghent, Belgium, 1 993).
[8] B. S. Kerner a nd P. K onh¨auser, Phys. Rev. E 48, 2335 (1993).
[9] B. S. Kerner a nd P. K onh¨auser, Phys. Rev. E 50, 54 (1994).
[10] M. J. Lighthill and G. B. Whitham, Proc. Roy. Soc. A 229, 317 (1 955).
[11] K. Nagel and M. Schreckenberg, J. Phys. I France 2, 2221 (1992).
[12] J. A. Cuesta, F. C. Mart´ınez, J. M. Molera, and A. S´anchez, Phys. Rev. E. 48, 4175
(1993).
[13] T. Nagatani, Phys. Rev. E 48, 3290 (1993).
[14] O. Biham, A. A. Middleton, and D. Levine, Phys. Rev. A 46, 6124 (1992).
[15] P. I. Richards, Operations Research 4, 42 (1956).
[16] A. Reuschel,
¨
Osterr. Ing.-Archiv 4, 193 (1950).
37

[17] A. Reuschel, Zeits. d.
¨
Osterr. Ing. u. Arch. Vereines 95, 59 and 73 (1950).
[18] L. A. Pipes, J. Ap. Phys. 24, 274 (1953).
[19] R. E. Chandler, R. Herman, and E. W. Montroll, Operations Research 6, 165 (1958).
[20] R. Herman, E. W. Montroll, R. B. Potts, and R. W. Rothery, Operations R esearch 7,
86 (1959).
[21] D. C. Gazis, R. Herman, and R. B. Potts, Operations Research 7, 499 (1959) .
[22] G. F. Newell, Operations Research 9, 209 (1961).
[23] D. C. Gazis, R. Herman, and R. W. Rothery, Operations Research 9, 545 (1961).
[24] I. Prigogine and F. C. Andrews, Operations Research 8, 789 (1960 ) .
[25] I. Prigogine, in Theory of Traffic Flow, edited by R. Herman (Elsevier, Amsterdam,
1961).
[26] E. Alberti and G. Belli, Transportation Research 12, 33 (1978).
[27] D. R. P. Gibson, in Special Report 194, edited by B. J. Vumbaco (National Academy o f
Sciences, Washington, 1981).
[28] A. D. May, in Special Report 194, edited by B. J. Vumbaco (National Academy of
Sciences, Washington, 1981).
[29] H. J. Payne, Tr anspor t ation Research Record 722, 68 (1979).
[30] M. Cremer and A. D. May, An Extended Traffic Model fo r Freeway Control (Research
Report UCB-ITS-RR-85-7, Institute of Transportation Studies, University of California,
Berkeley).
[31] R. D. K¨uhne and M. B. R¨odiger, in Proceedings of the 1991 Winter Simulation Con-
ference, edited by B. L. Nelson, W. D. Kelton and G. M. Clark (Society for Computer
Simulation International, San Diego, CA, 1991).
38

[32] B. S. Kerner and P. Konh¨auser, in Simulation von Verkehrssystemen, edited by K.-H.
M¨unch (ASIM Arbeitskreis, Braunschweig, Germany, 1993).
[33] A. Alvarez, J. J. Brey, and J. M. Casado, Transportation Research B 24, 193 (1990).
[34] A. K. Rathi and A. J. Santiago, Traffic Engineering and Control 31, 317 and 351 (1990).
[35] R. Wiedemann, Simulation de s Straßenverkehrsflusses (Institut f¨ur Verkehrswesen,
Karlsruhe, Germany, 1974), Vol. 8.
[36] D. A. Wicks and E. B. Lieberman, Development and Testing of INTRAS, A Microscopic
Freeway Simulation Mod el (Federal Highway Administration, Washington, D C, 1977),
Vols. 1–4.
[37] M. Cremer and J. Ludwig, Math. Comput. Simulation 28, 297 (1 986).
[38] K. Nagel and A. Schleicher, Parallel Computing 20, 125 (1994).
[39] K. Nagel and S. Rasmussen, in Proceedings of the Alife 4 meeting, edited by R. Brooks
and P. Maes (MIT press, Cambridge, MA, 1994).
[40] R. Wiedemann and T. Schwerdtfeger, Straßenbau und Straßenverkehrstechnik (Bun-
desminister f¨ur Verkehr, Abteilung Straßenbau, Bonn, Germany, 1987), Vol. 500.
[41] H. J. Payne, in Mathematical Models of Public Systems, edited by G. A. Bekey (Simu-
lation Council, La Jolla, CA, 1971), Vol. 1.
[42] M. Papageorgiou, Applications of Automatic Control C oncepts to Traffic Flow Modeling
and Control (Springer, Heidelberg, Germany, 1983).
[43] R. D. K¨uhne, in Proceedings of the 9th International S ymposium on Transportation and
Traffic Theory, edited by I. Volmuller and R. Hamerslag (VNU Science Press, Utrecht,
The Netherlands, 1984).
[44] S. A. Smulders, in Proceedin gs of the 10th International Symposium on Transportation
39

and Traffic Theory, edited by N. H. Gartner and N. H. M. Wilson (Elsevier, New York,
1987).
[45] H. J. Payne, in Research Directions in Computer Control of Urban Traffic Systems,
edited by W. S. Levine, E. Lieberman, and J. J. Fearnsides (American Society of Civil
Engineers, New York, 1979).
[46] E. Hauer and V. F. Hurdle, Transportat io n R esearch Record 722, 75 (1 979).
[47] P. S. Babcock IV, D. M. Auslander, M. Tomizuka, and A. D. May, Transportation
Research Record 971, 80 (1 984).
[48] A. K. Rat hi, E. B. Lieberman, and M. Yedlin, Transportation Research Record 1112,
61 (1987).
[49] P. Ross, Transportation Research B 22, 421 (1988).
[50] G. F. Newell, Transportat io n Research B 23, 386 (1989).
[51] C. F. Daganzo, “Requiem for second order fluid approximations of traffic flow”, Trans-
portation Research B, 1995 (to be published).
[52] J. J¨ackle, Einf¨uhrung in di e Transporttheorie (Vieweg, Braunschweig, Germany, 1978).
[53] A. Rieckers and H. Stumpf, Thermodynamik (Vieweg, Braunschweig, Germany, 1977),
Vol. 2.
[54] K. Huang, S tatistical Mechanics (Wiley, New York, 1987), 2nd edition.
[55] R. L. Lib off, Kinetic Th eory ( Prentice-Hall, Lo ndon, 1990).
[56] S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases
(Cambridge University Press, New York, 1970 ) , 3rd edition.
[57] D. Enskog, The K i netic Theory of Phenomena in Fairly Rare Gases (Dissertation, Up-
psala, 1917).
40

[58] Often the terms ‘Euler equation’ and ‘Navier-Stokes equation’ denote the respective
velocity equation only. Since it is easier to name the whole set of connected equations
(containing the respective density equation, velocity equation, and variance equation)
by one single term, the notation of Ref. [55] has been applied, here.
[59] D. Helbing, Complex Systems 6, 391 (1992).
[60] F. Pampel, Ein Beitrag zur Berechnung der Leistungsf¨ahigkeit von Straßen.
(Kirschbaum, Bielefeld, Germany, 1955).
[61] P. K. Munjal and L. A. Pipes, Transportation Research 5, 241 (1971).
[62] Phillips [3] fits his traffic data not to a Gaussian distribution but to a more complicated
one which he derived from his gas-kinetic model. However, Phillips’ figures clearly show
that, in the region of stable traffic, the asymmetric distribution function favoured by
him does not fit the data as well as a Gaussian distribution would.
[63] G. B. Whitham, Linear and Nonlinear Waves (Wiley, New York, 1974).
[64] T. Musha and H. Higuchi, Japanese Journal of Applied Physics 17, 811 (1978).
[65] B. D. Greenshields, in Proceedings of the Fourteenth Annual Meeting of the Highway
Research Board, edited by R. W. Crum (National Research Council, Washington, DC,
1935), Part I.
[66] J. M. Burgers, Adv. Appl. Mech. 1, 171 (1948).
[67] This derivation is not really consistent since the relaxation time τ ≈ 10 s in (8) is not
of the or der of magnitude which τ has in the follow-the-leader models where it has the
meaning of a reaction time (≈ 1.3 s [41,23]) [45]. Another criticism was pointed out by
Daganzo [51].
[68] Payne’s simulation model FREFLO, in contrast to his macroscopic model (1), (8),
predicts unstable traffic flow above a critical density that dep ends on the spatial dis-
41

cretization ∆r [45].
[69] C. J. Leo, R. L. Pretty, Transportatio n Research B 26, 207 (1992).
[70] R. D. K¨uhne, in Proceedings of the 10th I nternational Symposium on Transportation
and Traffic Theory, edited by N. H. Gartner and N. H. M. Wilson (Elsevier, New York,
1987).
[71] W. F. Phillips, Transportation Planning and Technology 5, 131 (1979).
[72] R. D. K¨uhne, Transportation Research Record 1320, 251 (1991).
[73] R. D. K¨uhne and R. Beckschulte, in Proceedings of the 12th International Symposium
on the Theory of Traffic Flow and Transportation, edited by C. F. D aganzo (Elsevier,
Amsterdam, 1993).
[74] J. Keizer, Statistical Thermodynamics of Nonequilibrium Processes (Springer, New York,
1987).
[75] D. Helbing, Working Paper (unpublished).
[76] Y. L. Klimontovich, Statistical Physics (Harwood Academic, Chur, Swizzerland, 1982 ) .
[77] Probably Paveri-Fontana’s equation (27) should, on its right-hand side, be extended by
the additional adaptation term (∂ ˆρ/∂t)
ad
:= ˜ρ(r, v, t)[P
′
0
(v
0
; r, t) − P
0
(v
0
; r, t)]/T , where
T ≈ 1 s is about the reac tion time. This term is intended to describe an adaptation of
the actual d i s tribution of desired velocities P
0
(v
0
; r, t) to the reasonabl e distribution of
desired velocities P
′
0
(v
0
; r, t) ≈ exp{−[v
0
−V
′
0
(r, t)]
2
/[2Θ
′
0
(r, t)]}/
q
2πΘ
′
0
(r, t). The mean
va lue V
′
0
(r, t) and variance Θ
′
0
of P
′
0
(v
0
; r, t) mainly depend on speed limits and road
conditions (gradient of the road; fog, rain, snow, or ice on the road). Therefore, they are
constant almost everywhere. Applying the method of a d iabatic elimi nation [78], which
is applicable due to the smallness of T , we find P
0
(v
0
; r, t) ≈ P
′
0
(v
0
; r, t). This implies
that the average desired v elocity V
0
(r, t) :=
R
dv
R
dv
0
v
0
ˆρ(r, v, v
0
, t)/ρ(r, t) and the vari-
42

ance of desired velocities Θ
0
(r, t) :=
R
dv
R
dv
0
(v
0
− V
0
)
2
ˆρ(r, v, v
0
, t)/ρ(r, t) are given by
the external traffic conditions that determine V
′
0
(r, t) and Θ
′
0
(r, t): V
0
(r, t) ≈ V
′
0
(r, t),
Θ
0
(r, t) ≈ Θ
′
0
(r, t). Interestingly enough, the adaptation term do es not bring about
modifications of the density equation, the velocity equation, the variance equation, or
the covariance equation derived later on. Therefore, it was not included in the main
discussion.
[78] H. Haken, Advanced Synergetics (Springer, Berlin, 1983).
[79] D. Helbing, “High- fidelity macroscopic traffic equations”, Physica A, 1995 (to be pub-
lished).
[80] E. Barone, Transport Theory and Statistical Physics 9, 59 (1981) .
[81] R. Semenzato, Transport Theory and Statistical Physics 9, 83 (1981).
[82] R. Semenzato, Transport Theory and Statistical Physics 9, 95 (1981).
[83] G. E. Uhlenbeck and G. W. Ford, Lectures in Statistical Mechanics (American Math.
Soc., Providence, RI, 1963).
[84] C. W. Wang-Chang and G. E. Uhlenbeck, in Studies in Statistical Mechanics, edited by
J. deBoer and G. E. Uhlenbeck (Nort h-Holland, Amsterdam, 1970).
[85] H. Grad, Phys. Fluids 6, 147 (1963) .
[86] H. Grad, in Rarefied Gas Dynamics Symposium, edited by J. Laurmann (Academic
Press, New York, 1963), Vol. I.
[87] To illustrate this, assume that we are confronted with a linear differential equation
d~x/dt = −L
~x with a linear operato r (or matrix) L. This equation has the general
solution ~x(t) =
P
µ
a
µ
~x
µ
e
−t/τ
µ
where t he eigenvalues 1/τ
µ
are the solutions of the
characteristic equation det(L
− 1/τ
µ
) = 0 and the eigenvectors ~x
µ
satisfy the equa-
tion L~x
µ
= ~x
µ
/τ
µ
. If 1/τ
µ
≫ 1/τ
0
for µ 6= 0 we have ~x(t) ≈ a
0
~x
0
e
−t/τ
0
after a
43

very short time t > 3 max{τ
µ
: µ 6= 0} (adiabatic approximation). This implies
−L
~x(t) = d~x(t)/dt ≈ −~x(t)/τ
0
.
[88] D. Helbing, “Modeling multi-lane traffic flow with queuing effects”, Transportation
Research B, 1995 (to be published).
[89] Here, the term “characteristic speeds” r efers to the slopes of the characteristics in the
(r, t)-plane which can be calculated for a hyperbolic system of partial differential equa-
tions [63].
[90] However, not e that the traffic model simulated in Ref. [5] differs in some details from
the Navier-Stokes-like model derived in this paper.
[91] If one assumes τ(ρ, V, Θ) ≡ τ(ρ) and p(ρ, V, Θ) ≡ p(ρ) like Prigogine [4 ], Phillips [3], and
Paveri-Fontana [2] did, the unknown functional relation for the product τ(ρ)[1 − p(ρ)]
can be determined from the empirical velocity-density relation V
e
(ρ
e
) by inserting (82)
into (80).
44