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The Physics of Traffic Jams
Traffic phenomenon was
investigated through the comparison of fluids and granular media physics. It
was found that traffic is governed by a set of transitions between physical
states much like fluids. Macroscopically traffic was analysed as a set of waves
and this led to the presentation of evidence to suggest traffic is chaotic.
1.0 Introduction
It can be argued that traffic has existed ever since people have had the desire to move from one place to another. Today we associate traffic mostly with road networks and vehicular traffic but before the invention of the motorised vehicle, traffic most likely was present and still is in the form of pedestrian traffic. Traffic and its associated problems have thus evidently been around for quite some time. This given it is only until recently (1950’s) that research and thought on that matter has been given serious attention. The study of the interaction of traffic is though no mean feat when considering the complex human and physical elements that are involved. Any modelling must thus overcome the problem of identifying determining characteristics that can lead to a universal set of laws and relationships. As will be seen traffic is shown to be illustrative of many popular concepts of physics from granular flow to wave behaviour to even possibly chaos theory. The task at hand will thus be to attempt to describe the flow and interaction of traffic in such a way as to relate it to existing physical systems. It is not expected that this will necessarily lead to an improvement of any existing traffic models but will rather lead to a better understanding of the problem at hand.
2.0 Traffic understood as two states, free
and congested.
A traffic jam is a period whereby traffic flow between vehicles becomes synchronized. The synchronized traffic forms in such a way that the speed of each individual vehicle is limited by that of the vehicle in front. This is unlike the ‘normal’ state of traffic were vehicles move respective to one another predominately unsynchronized. In this ‘normal’ state the vehicles are able to travel at whatever speed their driver may select, analogous in many ways to an ideal gas with no potential. Such a state is often described as a free-state. The analogy of traffic to a fluid is one that will be visited regularly throughout this paper, but this analogy is not dissimilar to an analogy between traffic and electrical current. Current is comparable to the flow of a fluid [1] or the flow of traffic. Electrical resistance on the other hand is equivalent to the viscosity of a fluid or the interaction of traffic. In all cases, whether it is the viscosity of a fluid, the vibration of a lattice, or the interaction of traffic in question, there is a dissipation of energy that decreases the motion of the fluid, charge carrier or vehicle.
In order to understand the physics behind a system of events then a set
of distinguishing characteristics must be first identified. To this end, the
properties of utmost interest for the investigation of traffic jams will be the
traffic current flow and the density of traffic. Using these
variables as a means of investigation a set of traffic transitions and traffic
states will be identified. Firstly let the traffic current flow and the density
of traffic be quantitatively defined. Traffic current flow, 
as a function of
distance, x and time t, is expressed as
, (1)
where 
is the number of
vehicles passing through a section of road averaged over a given number of
lanes for a given distance. The denominator 
is the time taken for
the recording of the vehicle count sampling. Traffic density, 
is defined simply as
, (2)
where l is the length of the road sampled. Graphically, the relation between traffic current flow and traffic density has been shown by Nagatani [2] as is demonstrated in figure 2.0a.
Figure 2.0a Nagatani’s
schematic to show the current flow of vehicles as a function of vehicle number
density for a given road of unit length, averaged over several measurements [2].
In agreement with figure 2.0a monitoring
of traffic shows that whilst in a free traffic state although vehicles move largely
unsynchronized there is an overall current flow with a linear dependency on traffic
density. The curve intersects with the axis at the origin since quite obviously
if there is a zero density of traffic there is zero flow (consider equations 1 and
2). Conversely in the free traffic state the linear relation between density
and current flow is intuitive. This is since the higher the number of constituent
vehicles involved in the traffic the higher the current flow (see equation 1).
The linear relation holds valid in the free traffic state until the current
flow threshold for the given road is met. That is the linear relation holds
until the point whereby inter vehicle interactions occur [3]; traffic continues
with this linearity only when traffic is unsynchronised. It can be described
that the traffic becomes more ordered at the point where interactions between
vehicles dramatically increase. If comparing the
Once past the initial point of interaction a traffic system can be comparable with a liquid in many ways; traffic and liquids both for example under normal conditions have a finite density. Traffic density is limited to a maximum since ideally adjacent vehicles should avoid physical contact. Liquids are limited in a similar manner since electrostatic forces operate at a significant range to prevent molecules also coming into contact. The finite density for both systems means that compression is not possible beyond a finite point, and this can be idealized mathematically for both traffic and liquids as follows,
(3)
where 
is the maximum density (bumper to bumper). Let the properties
of these two systems hence be considered in tandem; in both traffic flow and
liquid flow when constituent entities are forced to interact there is a
dissipation of energy that results in a decrease of flow. In fluids this is
called viscosity and the forced interaction of molecules can be the result for
example of the introduction of a temperature gradient. The viscosity is thus
influenced by temperature since it controls the duration and frequency of
molecule interactions.
Interestingly in fluids when a pressure gradient
can be applied isothermally then there is little or no affect on the fluids
viscosity (except at extreme pressures) [4]. Let the analogy between free
traffic and a gas be revisited briefly to include viscosity. Air may be taken
as a common example of a gas, its density, 
is governed by the
following equation,
, (4)
where P is pressure, R is a constant and T is temperature. It can therefore be understood that if an isothermal process is considered then
. (5)
If viscosity (resistance to flow) is independent of pressure
therefore it must also be independent to fluctuations in density. This is in
agreement to figure 2.0a that shows whilst in the
In brief
summary comparison of traffic with fluids shows that there are two major phases
in traffic flow, one in which traffic is free and one were traffic is congested
(see figure 2.0b). In the
Figure 2.0b Nagatani’s
schematic to show the current flow of vehicles as a function of vehicle number
density for a given road of unit length, averaged over several measurements [2].
Added annotation to show the results of comparison of traffic to a fluid.
Synchronized flow as a granular material
Under discussing synchronized traffic as a
set of interacting entities this leads to the classification of traffic as a
granular material. This relation has been studied in detail by Helbing et al [5].
In general as described by ‘Wikipedia’ [6],
“a granular
material is a conglomeration of discrete solid, macroscopic particles
characterized by a loss of energy whenever the particles interact”
This classification can help to explain certain properties of traffic intuitively. The notion of a current flow threshold for example can be perceived in granular flow as a point of clogging. This has been defined by Reithmuller et al [7] as an area with high local density and low average velocity. The point of clogging is thus due to a high collision (interaction) rate between granular particles. This can be assumed since it is clear that a decrease in velocity is directly the result of increased collisions. Understandable since unlike vehicles, most granular particles operate without breaks. In account of such concurrent differences between vehicles and particles it must be assumed that the ambition of every vehicle is to travel at or close to the maximum speed limit. Under this assumption traffic can be considered similar for instance to a granular particle under the influence of gravity. One limitation of any comparison between granular materials and human behaviour can thus be identified; granular material analysis cannot easily account for drivers that may for some reason chose (without interaction) to travel at a speed lower than the flow of traffic.
The behaviour of granular particles under the force of gravity has been studied by Reithmuller et al [7]. In this paper the problem of sand particles flowing down a narrow vertical tube was considered. Figure 2.0c shows graphically that for the region considered the clogging of granular particles occurs at two separate points, A and B, similar to the jamming of vehicles in traffic flow. In between the two points of clogging it can be seen that the average velocity comparative with flow increases to a maximum amount. This comparison should stand if incremental points of flow are considered in space then flow is equivalent to average velocity at each point. It seems thus that in granular flow there is a flow threshold point that governs when clogging will occur, equivalent to the flow threshold point seen for traffic in figure 2.0a The main difference between the clogging of sand and the jamming of traffic is that with traffic the process occurs on a bigger scale of distance.
Figure 2.0c TAken from [7] Reithmuller et al shows the average velocity of sand grains falling down a narrow vertical pipe plotted as a function of distance down the pipe. (N.B maximum speed is very low compared to expected terminal velocity for a falling object in air)
Investigation of the maximum average speed shown in figure 2.0c (approx 0.9 meters per second) can be taken. My analysis shows that the clogging starts to occur well below the expected terminal velocity for a sand particle (probably of the order of 80 to 100 meters per second). Analysis thus shows that the clogging must be a result of interactions between grains and the tubing. The clogging is therefore not due to a change in the rate of acceleration of the leading grains. In relation to the problem at hand I feel this shows that traffic congestion is not necessarily the result of a leading vehicle having to slow down but simply the result of traffic meeting the current flow threshold. As not to devalue the earlier comparison between fluids and traffic the clogging phenomenon does not generally extend to fluids since they escape the classification of being a granular material; granular material physics cannot be applied to liquids since the constituent molecules are affected by thermal motion fluctuations. In a granular material it is generally assumed that the constituents are large enough to be unaffected by fluctuations of this type. This explains therefore why the clogging phenomenon does not occur in liquids. A result of this inherent difference between fluids and granular materials is therefore that granular media physics provides a much better basis for the formation of mathematical models for traffic. In their study Reithmuller et al puts forward several equations for the modelling of granular clogging. Unfortunately though these equations prove to be beyond the intended level of this paper, but would other wise be useful in the derivation of a model to predict exactly when traffic congestion will occur.
Figure 2.0d. Taken from [7] Reithmuller et al shows the average velocity of sand grains falling down a narrow vertical pipe plotted as a function of distance down the pipe. Plot redisplayed with annotation to summarise the results of analysis.
3.0 The
metastable state, explaining the transition from free to congested
In traffic the maximal current value can be understood as the point whereby the distance between vehicles is too short to allow for increased vehicle number whilst maintaining the same level of safe current flow. At this point vehicles must decrease speed thus reducing traffic current flow if they are to avoid collision. Figure 3.0a shows the probability of traffic making a transition from the free to congested state as a function of current flow.
Figure 3.0a Probability of free traffic breakdown plotted as a function of
current flow. There are four sets of data shown that were taken for four
different times but using the same time intervals. Experiment undertaken on a
freeway in
Figure 3.0a shows that the increasing probability of a free traffic breakdown increases almost exponentially with traffic flow [8]. Around the region of maximal current (see figure 2.0a) there is therefore a region of metastability since it is likely at high flow rates that the traffic will change state. In this case the metastable region is thus a short lived period whereby traffic density can either decrease to allow for current free flow to continue or traffic density can increase and thus result in a traffic jam. The region is best explained as a transition phase between the two states of free and congested traffic.
Figure 2.0a also shows that within the
metastable region there is a sharp change in the curve gradient from positive
to negative. This sharp change can be assumed as the point in the traffic jam
whereby there is an initial significant decrease in speed whilst the vehicles
become more closely packed and become synchronized. Density therefore increases
and current flow decreases since the vehicles are closer together and are
moving slower. After the abrupt change of gradient in figure 2.0a the current
flow once again becomes linear with the density since the slowing of vehicles
is more gradual as the traffic moves toward the maximum density capacity of the
road.
If the individual vehicles of a metastable traffic flow are compared to molecules in a liquid then it is possible to see how the changing of state from free to congested traffic is analogous to conventional physics theory; water for example is described to undergo a first order phase transition when it is heated to one hundred degrees Celsius. At this boiling point water violently changes state from liquid to gas similar to the traffic changing state from free to congested as the current threshold is exceeded.
A simple analysis of the individual free, metastable and congested traffic states taken as a whole shows that the relation of density and current flow are no longer described by a simple linear relationship. As can be seen from figure 2.0a at low current flow and low density (close to the origin) the relation is a simple single-valued problem. After this initial simple phase the system evolves into a more complicated multi-valued function. Once past this point of bifurcation, for each value of current flow there are two corresponding density possibilities, one low and one high. These two values of current flow thus exist in the metastable region until the current maximum is reached and at this point for an infinitely small increment of density the relation is single valued. After this point the multivalued relation thus remains until the point of maximum density is met. The maximum density is shown by the abrupt end to the function in figure 2.0a.
4.0
Traffic
waves and chaos theory
It is in and near to the congested state that traffic displays some of its most identifiable characteristics. One such phenomenon is the movement of traffic as longitudinal waves, another is the emergence of the phantom traffic jam. These two phenomena can now be discussed.
Figure 4.0a Position, x of several vehicles plotted as a function of time, t. Results obtained using aerial photography techniques to show the dynamics of a traffic jam [9].
Figure 4.0a shows an example of how the dynamics of a traffic jam can vary with time, were each individual line on the graph represents the unique trajectory of a vehicle [9]. It can be seen that the lower half of each vehicle path has an evident positive gradient, demonstrating that its position is advancing forward in time. After this initial period of constant velocity it can be seen that there is a deceleration for each vehicle shown by the gradient approaching unity. If the lines representing the vehicle paths are considered collectively it is shown that the central congested region of the traffic jam is shown by a series of darkened line segments. Most noticeable is the jam shown by the dark region extending for the full time scale, but also there are a number of smaller jams in the zero to eighty second region of the graph. Considering the path of the individual trajectories, each darkened segment represents a maximum nodal point in the traffic shockwave [10]. The classification of traffic as a wave allows for the introduction of the one dimensional wave equation that generally describes all kinds of waves considered for one dimension. The general form for the one dimensional wave equation is,
(6)
where u is the amplitude of the wave and v is the speed of the wave. The widths of the darkened line segments (figure 4.0a) represent a measure of position with time of the traffic jam wave fronts, which describes the duration of the jam. The wider segments thus represent longer jams following the shorter duration segments. Overall each darkened path has an apparent negative gradient. This confirms that although the vehicles are all moving with a positive direction the jam wave fronts are actually moving backwards in the opposite direction. This backward movement of the wave is common to traffic jams but the magnitude of its velocity can never be greater than the magnitude of the average velocity of the traffic [11]. This kind of wave in which the phase velocity is opposite to the energy velocity is also common in fluids with surface waves on water [12].
Notice also in figure 4.0a that around the sixty-second mark there emerges a region of few vehicle paths, this shows that for a short period the density of vehicles has decreased (see equation 2). Downstream of this decrease of density the darkened segments of the trajectories become narrower suggesting that the traffic fronts are closer together and that the duration of the jam has decreased. Conversely at the very beginning of the graph if the first vehicle paths are considered it is seen that the lines are very compact and close together. This of course suggests that at this point there was a high density of vehicles that will have led to the origin of the initial upstream jam (often called a bottleneck).
Overall there is a delayed reaction as the
effects of one region are shown further down the wave as it propagates through
space and time analogous to the so called ‘butterfly effect’. This effect
is known as sensitivity to initial conditions and is a hallmark of chaotic
systems. Chaos theory which is a relatively new field of physics may be defined
generally [13] as,
“the study of a forever changing complex system based on
mathematical concepts of recursion,
The relation between chaos and traffic has been studied by Frazier
and Kockelman [14]. Their paper investigates a series of traffic systems for
characteristics of chaotic behaviour. One common method of analysing a set of
data for chaos is with the study of the attractor [15]. The attractor of
a system can be described as the desirability of a system to reach a common
limit. The attractor is usually associated with a dissipation of energy. One example
is the movement of a damped pendulum that will always reach a state of rest. The
classification of a system as ‘normal’ or chaotic can be determined by
identifying whether the attractor of the system is normal or ‘strange’. A
normal attractor will be represented by a relatively simple motion type such as
the circle like curves that a swinging pendulum follows in approaching its rest
state. Strange attractors on the other hand are much greater in complexity and
are thus representative of chaotic behaviour (see [16] for examples).
The determination of the attractor by Frazier
and Kockelman is done through investigation of the correlation relation [17].
This relation describes an area of
interest represented by a sphere of radius r and the average number of
data points C(r) contained within the sphere. Numerically the average
number of data points (vehicles) contained in the sphere is assumed to change
at the rate
(7)
where k is a constant and DC is a deterministic quantity that shows
whether or not the attractor is strange. An integer value for DC
would show that the attractor for the given system is normal whilst a non
integer value suggests that it is strange and thus likely to be chaotic. In
order to investigate the attractor for a traffic data set Frazier
and Kockelman propose the following simple rearrangement of equation 7,
(8)
Frazier and Kockelman concentrated on traffic data collected from
Interstate 80 near
Figure 4.0b Logarithmic plot of the average number of vehicles C(r) contained in a sphere as a function of radius r/km of the sphere considered. The gradient of the dotted line region has a non integer gradient value of 5.73 that is indicative of a chaotic system. Due to the large collection period the average number of vehicles is low since at some times during the collection the number of vehicles would be zero. [14]
Figure 4.0b shows a logarithmic plot of
vehicle number verses the radius of the area considered. The gradient of the
graph is shown to be of non integer value thus suggesting that the attractor is
strange (consider equation 8). As earlier described the presence of a
strange attractor can suggest that the system is chaotic and would thus
validate traffic as being a deterministic system [15]. In other words a
system in which every reaction becomes the cause of subsequent reactions. This
perhaps helps to explain what popular science has branded as the phantom
traffic jam; the cause of traffic jams is not necessarily the result of an
obstruction but more the result of the interactions that occur downstream. This
also agrees with my previous analysis of figure 4.0a and in addition with my
analysis of the granular flow analogy. Frazier and Kockelman agree that the
evidence presented is by no means definitive in the characterisation of traffic
as an example of chaos theory. The possibility though opens up some interesting
ideas for the modelling of traffic since all dynamics of a traffic jam would
thus be able to be better accounted for and therefore more easily predicted.
5.0 Modelling
of traffic
Modelling can be applied to traffic in one of two ways, the traffic can either be considered microscopically or macroscopically. In the microscopic model the individual vehicle is considered as an interacting particle involved in a traffic flow within or close to the metastable state. Macroscopic modelling on the other hand considers the overall flow of the traffic, considering the individual vehicles on an aggregate level only.
5.1 Microscopic
modelling
Perhaps the most common microscopic model for traffic is the Gazis-Herman-Rothery (GHR) model [18] that assumes that the acceleration of a vehicle is proportional to its relative speed with the vehicle ahead. Numerically the GHR model can be expressed as follows,
(9)
where
, is the acceleration of vehicle n at time t, 
, is the velocity of vehicle n at time t, 
is the relative
velocity of vehicle n to the vehicle ahead of it at time t minus the
driver reaction time, T, and c, l and m are constants
to be determined by road and environment conditions. The relation was tested
experimentally at the General Motors Research laboratories in Detroit by
Chandler et al (1958) and in Japan by Kometani and Sasaki (1958) [18]. This research
led experimentalists to the conclusion that the dominating factor in the GHR
model was indeed the relative velocity of one car to its predecessor rather
than the distance apart. This of course would be the intuitive result since
otherwise the following car would need to increase acceleration when the
distance between cars became shorter (consider equation 9). The experimentation
thus validated the GHR as a useful model for traffic flow although there was
much debate over suitable l and m values as can be seen from
table 5.1a.
Table 5.1a Summary of alterations to the parameters of the GHR model between 1958 and 1993. Key: dcn/acn: deceleration/acceleration; brk/no brk: deceleration with and without the use of brakes; uncgd/cgd: uncongested / congested; ss: steady state. [18]
Table 1 shows that since the implementation of the GHR model for traffic there have been much contradictory reporting on the issue of what values for l and m should be used. In addition Panwai and Dia [19] suggest that these parameters also need to be calibrated for each network investigated. The need for calibration for each model thus limits the application of the GHR especially in comparison between different networks.
In addition
to the GHR model many other microscopic models have been put forward. The
Collision Avoidance Model (CA) that is reported as easier to calibrate than the
GHR model is one other popular choice for the practitioner. It was first
presented in 1959 by Kometani and Sasaki [19] and operates on the basis of
describing the following distance of a vehicle required to avoid collision.
Expectedly the distance increases as a function of speed of the vehicles since
at higher speeds a larger breaking distance is required. Algebraically the safe
following distance, 
can be expressed as,
(10)
where vn is the speed of the nth
vehicle, 
is the speed of the (n-1)th
vehicle, t is time and T is the driver reaction time. The
calibration constants for the model are represented by 
and 
. Variations of this model are widely used due to its ease of
calibration, one such variation is the Gipps Model (1981) [19]. Panwai and Dia
demonstrate the success of the Gipps Model in comparing calculated profile
headway distances to actual field measurements of the distance between vehicles
(see equation 10). This is shown in figure 5.1a where the
distance between vehicles () is abbreviated to M and is plotted as a function of
time. The close correlation between the actual measurements and the Gipps Model
calculated values show that the Gipps model is largely efficient model in
practice.
Figure 5.1a Taken from reference [19], plot to show the following distance, M of one vehicle to its predecessor as a function of time. Two relations are plotted; one in red that represents the field measurements and one in green that represents the values calculated using the Gipps Model. Results are taken from readings recorded at the AINSUM microscopic traffic simulator. [18]
5.2 Macroscopic
modelling
Macroscopically, traffic flow may be
investigated using the Lighthill Whitham-Richards (LWR) model. As
described by Kerner [20] the traffic flow is assumed to be a function only of
traffic density:
(11)
This
reasoning is in agreement to earlier discussions of the metastable state of
figure 2.0a. In addition to this the continuity equation may also
be used since the principal of conservation of mass (The Lomonosov-Lavoisier
law) applies to traffic. The continuity equation in its differential
form can be described as
(12)
where Ve is the fluid (traffic) equilibrium velocity. Since the problem of traffic may be considered as generally one dimensional then equation (7) may be simplified to the continuity equation for one dimension as follows.
(13)
The fluid velocity term in this equation for traffic will be a function only of density since at high density traffic is forced to approach zero velocity (see figure 2.0a) and at low density traffic is generally only limited by legality. For a given road therefore with a constant speed limit the traffic velocity may be assumed simply as [11],
(14)
where
vmax is the maximum velocity of vehicles considered. This
assumption thus allows for the simplified continuity equation [21], that describes
how the change in flow 
with time plus the
change in density 
with position is equal
to zero,
(15)
Since change in flow is assumed to be a function of density only then there is only one autonomous variable in the equation, that being density. It is this variable that controls the traffic wave as it mediates through space and time. The LWR continuity equation thus describes that if an increase in density occurs then a decrease in vehicle current flow must occur. The equation holds providing that traffic density is not too low but is not able to describe situations such as stop and start traffic. This restriction is a result of the equation relying on the equilibrium for traffic velocity [21].
5.3 Chaos
Modelling
After the
previously discussed analysis of whether or not traffic is illustrative of a
chaotic system, Frazeir and Kockelman provide the results of modelling that
takes into account the chaos theory. They investigated the flow of vehicles as
a function of time for
Figure 5.3a Plot to show the flow of traffic sampled
as a function of time. The data
was collected on Interstate 80 at
6.0 Conclusion
This paper describes how traffic can be divided into three different clear states, free, metastable and congested. The transition between these states has been compared to that associated with fluid transitions between the states of liquid and gas. This analogy has been shown possible since the nature of the state is determined primarily due to the frequency and duration of interactions between vehicles.
The motivation for this investigation was due to the hypothesis that traffic is a complex physical system controlled by the actions of human behaviour and other physical elements. The aim of the paper has thus been to identify a comparable system to simplify the problem of traffic. The failure of the paper has thus been in finding one single system that can adequately compare to the problem of traffic. In actuality it has been found that traffic is best described as a marrying of fluid flow ideas with granular material physics.
Granular material physics has provided facilitation to describe the actual jamming point of traffic as analogous to the clogging of granular materials such as sand in a tube. This analysis would prove useful to the practitioner in deriving formulae to state the maximum flow of a given network and thus in working toward designing a network with maximum efficiency. Indeed as is often reported the problem of traffic jam physics is usually in the lack of sufficient data. It is proposed therefore that granular particle clogging be investigated further to allow for more detailed comparison to traffic networks. This may thus lead to a more cost and/or time effective method of data collection.
In agreement with the initial hypothesis of traffic being complex this has been strengthened further with the presentation of evidence in traffic for chaotic behaviour. Whilst it is agreed that the evidence presented is by no means definitive it could prove to be an interesting area to research by means of further analytical techniques. If it is in fact the case that traffic is entirely chaotic then this is not necessarily a undesirable result. It is reported that although models currently in use such as the GHR, Gipps or LWR are useful in most cases they are certainly not exact in predicting traffic patterns. Chaos theory could thus prove a powerful method of investigation to transportation analysts.
References
[1] Koehler, K, (2003) Fluid Flow, http://www.rwc.uc.edu/koehler/biophys.2ed/analogy.html
[2] Nagatani, T, (2002) The
physics of traffic jams,
[3] Schadschneider, A, (2000) Traffic flow modelling, http://www.thp.uni-koeln.de/~as/Mypage/traffic.html
[4] Elert, G, (2005) The Physics Hypertextbook – Viscosity, http://hypertextbook.com/physics/matter/viscosity/
[5] Helbing, D, et al, (1999) Traffic and granular flow, Springer.
[6] Wikipedia, (200