An overview of the field is given in the work by Prigogine *el al. * [1]
on kinetic theory of vehicular traffic.
The traffic flow can be understood by studying the distribution
function *f*(*x*,*v*,*t*), which describes the number of cars in the road
interval (*x*,*x*+*dx*) with a velocity in (*v*,*v*+*dv*) at a given time t.
A few measurements have been made to determine this function
(see Fig. 1).

**Figure 1:** Measured velocity distribution function f(v) versus velocity v.
(taken from [1]).

This function can also be expressed in terms of the two *fundamental*
variables, *concentration* and *flow* (or current). An understanding
of the relation of concentration and flow is the basis for describing the
traffic. The flow-concentration curve is called the *fundamental diagram*
of road traffic or the *equation of state* of traffic theory.
Two measurements of this function are shown in Fig. 2.

**Figure 2:** Fundamental diagram: Flow versus vehicle concentration is shown
in this *fundamental diagram*
for two different tunnels.
(taken from [1]).

At low densities, the flow increases linearly with concentration. Cars
move independently in the dilute traffic with small fluctuations about a
mean velocity. With increasing concentration, cars hinder each other more and
more, which leads to a reduction of the average velocity and a decreasing
slope in the fundamental diagram. These hindering effects become
dominant at high concentrations. At a *critical density*, the flow
exhibits a maximum. Then, an increase in density results
into an actual decrease of the total throughput, and
finally in a completely congested phase where no car can move and
the flow is zero.

The flow near the critical concentration is unstable. Small perturbations
that increase the density locally lead to a decrease in the flow,
which then amplifies the initial perturbation. The *
variance* in car velocities
is what triggers this process [6], which leads to the
formation of traffic jams. Traffic jams are locally congested phases, in which
cars travel at slow or zero velocity. Between jams, one finds cars
at low concentrations traveling with high velocities. These congested phases
are rather stable configurations with their own dynamics. Cars driving into
the jams stabilize and cars leaving from the front can reduce the length.
From this picture, it follows that jams move slowly in the opposite
direction of the cars.
Their velocity can be estimated in a very simple model where one
assumes a small constant velocity and a density
in the jam and a higher velocity and low density outside.
If cars accelerate and decelerate instantly, the jam moves with a velocity,

Since the density ratio is small, jams move slowly.
A much more sophisticated picture of jams is given by the theory of fluid
dynamics, in which jams are modeled in terms of density waves.

Sat May 9 11:34:31 CDT 1998