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In-Depth Information
1
d
= 0
d
= 1
d
= 10
Ideal
0.9
0.8
0.7
0.6
0.5
1
0.4
0.8
0.3
0.6
d
= 10
d
= 6
d
= 1
d
= 0
0.2
0.4
0.2
0.1
0
200
400
600
800
t (s)
0
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
h
Fig. 17.4. Order parameter v
a
vs. . Curves refer to dierent values of . Other parameters are
N = 100, L = 10, = 1, r = 1, = 0:03, t = 1. Simulation time is 200 step beyond the time
necessary to reach convergence in the case = 0. Inset: time evolution of the average velocity for
the case N = 100, = 1, = 1:0 for dierent values of .
These considerations allow us to conclude that in the case of = 4 the system
shows similar values of v
a
independently of the value of , but there is a small
improvement of the time necessary to reach the steady speed. The behavior of the
system has been further investigated with respect to dierent values of the density,
leading to the conclusion that long-range connections introduce clear advantages for
low density values. This is a very important point since, as shown in [Vicsek et al.
(1995)], v
a
decreases with . At low densities, coordination performance is poorer
than at high densities. Therefore, it is at low density levels that the introduction
of long-range connections leads to the greatest benets.
The case of low density ( = 1) is dealt with in Fig. 17.4, where it is clearly
shown that the improvement is more signicant. Also in the case of low density,
long-range connections decrease the time needed to reach an ordered behavior. The
inset of Fig. 17.4 shows the trend of the average velocity v
a
for N = 100 and = 1.
Three curves corresponding to three values of are shown.
In [Vicsek et al. (1995)] the emergence of a kinetic transition between ordered
and disordered phase is described in terms of the power law which expresses the
relationship between v
a
and the critical noise
c
L (i.e. the value of noise beyond
which the phase transition occurs) as follows:
v
a
((
c
(L))=
c
)
:
(17.12)
The improvement due to long-range connections does not modify this relation-
ship, but leads to a decreased exponent in the power law. We have numerically
veried this result as shown in Fig. 17.5.
