Many of the arguments regarding the nature of a stochastic and
possibly critical phase transition
(32,35,33,31,34)
have been made using so-called cellular automata (CA) models. CA
models use coarse spatial, temporal, and state space resolution. For
traffic, a standard way is to segment a 1-lane road into cells of
length , where
is the length a vehicle occupies
in the average in a jam, i.e.tex2html_verbatim_mark>#math437#.
As with the Krauss model, the time step for the CA models is best
selected similar to the reaction time; a time step of 1 second works well
in practice. Taking the time step together with , one finds
that a speed of 135 km/h corresponds to five cells per time step; this
is often taken as maximum velocity .
Stochastic CA models contain a noise parameter that introduces randomness into the driving rules: With a probability , the deterministically calculated velocity gets reduced by one. This is the same idea as Eq. (9) in the model by Krauss. One can make dependent on the velocity (36); the resulting models are sometimes called models with ``velocity-dependent randomization (VDR)''. Often one uses just two probablilities: when the car is standing and when the car is driving. Standard values are and . This models that drivers, once stopped, are a bit sloppy in restarting again.
With this family of models, one can again plot the density variance
(Fig. 2(c)). Instead of the noise amplitude
, the parameter is used. means
deterministic driving except when accelerating from zero; increasing
means increasingly more randomness when moving. In this plot,
one finds a behavior similar to Fig. 2(b): For
small , the system displays three states (laminar,
coexistence, and jammed). For , the system is a
1-phase system.
Close to and still at , the system
displays a lot of jam formation and structure which vanishes only when
observed at very large scales (very large ).
In consequence, it is now clear why there was so much discussion about
possible fractals for the original model (37) where
: It is indeed close to a critical point, and
therefore fractal behavior up to a certain cut-off length scale should
be expected.