Information Technology Reference
In-Depth Information
and continuous in space, and consists of N particles which are initially distributed
on a square plane of linear size L with periodic boundary conditions. Particles
share the same velocity modulus , whereas their initial directions of motion
i
(0)
are uniformly randomly distributed. At each time step, the position x
i
and the
direction of motion
i
of the ith particle are simultaneously updated according to
x
i
(t + t) = x
i
(t) + v(t)t ;
i
(t + t) =h
i
(t)i
r
+
(17.1)
whereh
i
(t)i
r
is the average direction of particles within the neighborhood of the
ith particle (which is considered as a circle of radius r) and t is the time step
width.
The average direction is given by the angle arctan(
hsin (t)i
r
hcos (t)i
r
). is a
random number acting as a noise term and chosen with uniform probability from
the interval [
2
;
2
].
The values of the noise parameter and particle density =
N
L
2
aect the
behavior of the system: for small values of density and noise the particles tend to
form groups moving toward random directions; increasing density and noise the
particles move randomly with some correlation; increasing density but maintaining
small the noise it is possible to observe an ordered motion of all the particles towards
the same spontaneously chosen direction. The last case is the most interesting
because it shows a kinetic phase transition due to the fact that particles are driven
with a constant absolute velocity. This transition was supposed to be of second
order in [Vicsek et al. (1995)]. Recently, contradicting results in favour of a rst-
order transition have been obtained in [Gregoire and Chate (2004)]. We will not
enter in the debate arised, which is beyond the aim of this work.
To characterize the ordered phase it is possible to dene an order parameter, the
average normalized velocity, which can be calculated with the following expression:
N
X
1
N
v
a
=
v
i
(17.2)
i=1
where is the constant module of the particle velocities. Clearly, v
a
= 0 indicates
that the agents are not coordinated (there is no net transport), while v
a
= 1 indicate
that all the agents move in the same direction (the net transport is maximum). In
fact, v
a
= 1 holds when
i
= for all i.
Vicsek's model is considered a cornerstone for the description of dynamical phe-
nomena concerning collective motion. In his work, Vicsek himself stated that there
could be many interesting variations of the model which could lead to additional
nontrivial eects. With the aim of emulating a wider range of collective phenomena,
relevant to the study of animal motion, Couzin and co-workers introduced a new
model able to take in account typical behaviors observed in animal world [Couzin
et al. (2002)]. In particular, Couzin's model take into account (i) the fact that an
individual tends to keep a vital space free of its neighbors, (ii) the existence of two
