In this section, a criterion is established that distinguishes
homogeneous from coexistence states. As pointed out before,
coexistence states, for example at and 
in
our model, see Figs. 1(c) (ii) and 2(a),
are characterized by the coexistence of laminar and jammed traffic.
Deep inside the coexistence regime, one would expect that the phases
coagulate, leading to one large laminar and one large jammed section
in the system. As one has seen in Fig. 2(a) for
, this is essentially correct, except that small
additional mini-jams always lead to the detection of a small number of
additional jams. When approaching the boundaries of the coexistence
regime, this characterization will become less clear-cut, and it may
be possible to have more than one jam. Typically, there will be one
major jam and many small ones, and for many measurement criteria this
will cause enough problems to no longer be able to differentiate
between the coexistence and a homogeneous state. This is particularly
true for criteria that attempt a binary classification into
homogeneous or not. In contrast, our criterion will show a gradual
decrease in differentiating power.
The criterion is defined as follows: Partition the road into segments
of length for simplicity let
Lremainder). For each segment the local density
_be computed as the number of cars in that segment
divided by . An interesting value is the variance of the
local density:
| (20) |
What this value picks up is how much each individual measurement segment of
length 
deviates, in terms of its density, from the average
density. Assume a system consisting of jammed and laminar traffic.
If there is a jam in one segment, then the segment's density will be
much higher than the average density. Conversely, if there is only
laminar traffic in a segment, then the segment's density will be much
lower than the average density. takes the average over the
square of these deviations.
Fig. 2(b) shows this value as a function of the
global density 
and the noise parameter .
Each gridpoint is the result of a computer simulation. The simulations
run until the average number of jams over the last 100'000 time steps
is (almost) equal for a system started with a big jam and a system
started with laminar flow (see
Section 3.4). Over these last times the variance
of the local density is averaged.
Look at Fig. 2(b) for fixed , say . One sees that at densities up to , the value of is close to zero, indicating a homogeneous state, which is in this case the laminar state. Similarly, for densities higher than , is again close to zero, indicating a homogeneous state, which is in this case the jammed state. In between, for , the value of is significantly larger than zero, indicating a coexistence state.
Now slowly increase . We see that the laminar regime ends
at smaller and smaller densities, while the jammed regime starts at
smaller and smaller densities (see Fig. 2(b)
bottom). The latter means that for large , the jammed phase
has many relatively small holes, which reduce the density, but do not
break the jam. At
, the coexistence phase
completely goes away; for larger , we do not pick up any
inhomogeneity at any density (look at the bottom plot in order
to get information about behavior not visible in the 3d plot).
Compare this to the theoretical expectation in Fig. 1(b),
where for increasing 
the two densities eventually merge and thus
the different phases go away. Note that close to the transition the
system still looks like it possesses different phases (see
Fig. 1(c)(iv) and locate the corresponding
and in Fig. 2(b)). These structures do
however exist on small scales only. This means that for system
size and measurement interval
(but
), all intervals of size will eventually return the
same density value. A segment length of , as used for
Fig. 2(b), is already sufficient in order to not
measure any inhomogeneity for the state in Fig. 1(c)(iv).
This will not be the case for coexistence states: In coexistence
state, there will always be segments with different densities, unless
. This is because droplets will coagulate so that they
will eventually show up on all possible length scales .
Remember again that is a model parameter while is a
traffic observable. That is, once one has settled for an ,
the model behavior is fixed, and one has decided if one can encounter
a second phase or not. If one can encounter a second phase, it
will come into existence through changing traffic demand throughout
the day - traffic can move from the laminar into the coexistence and
potentially into the jammed state and back.
As a side remark, let us note that there is also another 1-phase regime for . Albeit potentially interesting, this is outside the scope of this paper.
In summary, one obtains, for the traffic model, a phase diagram as in
Fig. 1(b), which is the schematic phase diagram for a
gas-liquid transition in fluids. Again, the important feature of this
phase diagram is that there are three states for low temperatures
(small or small): gas/laminar; coexistence;
liquid/jammed. For higher temperatures, the coexistence range becomes
more and more narrow, while the density of the gas phase and the
density of the liquid phase in the coexistence state approach each
other. Eventually, these densities become equal, and the coexistence
state dies out. The only important difference is that for our traffic
model the phase diagram is bent to the left with increasing
.
There are other criteria which can be used to understand these types of phase transitions. In particular, one can look at the gap distribution between jams, and one would expect a fractal structure at the point where the 2-phase and the 1-phase model meet, i.e. and . This is indeed the case but goes beyond the scope of this paper; see (30) for further information.
|
[]
[]
|