Information Technology Reference
In-Depth Information
One could think to
H
as the television influence, and
J
as educational effects.
H
pushes towards one opinion or the other, and people educated towards con-
formism will have
J>
0, while non-conformists will have
J<
0. In the statistical
mechanics lingo, all parameters are rescaled to include the temperature.
The hypothesis of alignment to overwhelming local majority is represented
by a parameter
q
, indicating the critical size of local majority. If
s<q
(
m<
2
q −
2
r −
1), then
x
t
+1
i
=
−
1, and if
s>
2
r
+1
− q
(
m>
2
r
+1
−
2
q
), then
x
t
+1
i
=1.
In summary, the local transition probabilities of the model are
0
if
s<q
;
AB
s
1+
AB
s
if
q ≤ s ≤
2
r
+1
− q
;
1
p
s
=
(1)
if
s>
2
r
+1
− q
;
where
A
= exp[2
H −
2
J
(2
r
+ 1)] and
B
= exp(4
J
).
For
q
= 0 the model reduces to an Ising spin system. For all
q>
0we
have two absorbing homogeneous states,
x
=
−
1
(
c
= 0) and
x
=
1
(
c
=1)
corresponding to infinite coupling (or zero temperature) in the statistical me-
chanical sense. With these assumptions, the model reduces to a one-dimensional,
one-component, totalistic cellular automaton with two absorbing states.
The order parameter is the fraction
c
of people sharing opinion 1.
3
It is
zero or one in the two absorbing states, and assumes other values in the active
phase. The model is symmetric since the two absorbing states have the same
importance.
3 A Simple Case
Let us study the model for the simplest, nontrivial case: one dimensional lattice,
r
= 1 and
q
= 1. We have a probabilistic cellular automaton with three inputs,
and two free control parameters,
p
1
and
p
2
, while
p
0
≡
0 and
p
3
≡
1, according
to Eq. (1).
One can also invert the relation between (
H,J
) and (
p
1
,p
2
):
p
2
(1
− p
1
)
p
1
(1
− p
2
)
H
=
1
p
1
p
2
(1
− p
1
)(1
− p
2
)
J
=
1
4
log
,
4
log
.
The diagonal
p
2
=
p
1
corresponds to
J
= 0 and the diagonal
p
2
=1
− p
1
to
H
=0.
At the boundaries of probability intervals, we have four deterministic (ele-
mentary) cellular automata, that we denote T3, T23, T13 and T123, where the
digits indicate the values of
s
for which
p
s
= 1 [8].
Rule T3 (T123) brings the system into the
c
=0(
c
= 1) absorbing state
except for the special case of the initial configuration homogeneously composed
by the opposite opinion.
3
The usual order parameter for magnetic system is the magnetization
M
=2
c −
1.
