Information Technology Reference
In-Depth Information
It worth to notice that field terms
b
i
i
η
i
account for the difference in en-
ergy cost in breaking a pair proper relationship for respectively a large and a
small country. Consider for instance two countries
i
and
j
with
b
i
=2
b
j
=2
b
0
.
Associated pair energy is
H
ij
≡−C
ij
i
η
i
j
η
j
−
2
b
0
i
η
i
− b
0
j
η
j
.
(10)
Conditions
η
i
=
i
and
η
j
=
j
give the minimum energy,
H
m
ij
=
−J
ij
−
2
b
0
− b
0
.
(11)
From Eq. (11) it is easily seen that in case
j
breaks proper alignment shifting to
η
j
=
−
j
the cost in energy is 2
J
ij
+2
b
0
. In parallel when
i
shifts to
η
i
=
−
i
the
cost is higher with 2
J
ij
+4
b
0
. Therfore the cost in energy is lower for a breaking
from proper alignment by the small country (
b
j
=
b
0
) than by the large country
(
b
j
=2
b
0
). In the real world, it is clearly not the same for instance for the US
to be against Argentina than to Argentina to be against the US.
We now consider the uncoherent case in which cultural and economical trends
may go along opposite coalitions, i.e.,
β
i
6
=
i
. Using above variable change
τ ≡
i
η
i
, the Hamiltonian becomes,
X
N
X
H
2
=
−
1
2
J
ij
τ
i
τ
j
−
δ
i
b
i
τ
i
,
(12)
<i,j>
i
=1
where
δ
i
≡ β
i
i
is given and equal to
±
1.
H
2
is formally identical to the ferro-
magnetic Ising Hamiltonian in random fields
±b
i
.
The local field term
δ
i
b
i
τ
i
modifies the country field
h
i
in Eq. (9) to
h
i
+
δ
i
b
i
which now can happen to be zero. This change is qualitative since now there
exists the possibility to have “unstability”, i.e., zero local effective field coupled
to the individual choice. Moreover countries which have opposite cultural and
economical trends may now follow their economical interest against their cultural
interest or vice versa. Two qualitatively different situations may occur.
-
Unbalanced economical power: in this case we have
P
N
i
δ
i
b
i
6
=0.
The symmetry is now broken in favor of one of the coalition. But still there
exists only one minimum.
-
Balanced economical power: in this case we have
P
N
i
δ
i
b
i
=0.
Symmetry is preserved and
H
2
is identical to the ferromagnetic Ising Hamil-
tonian in random fields which has one unique minimum.
5UniqueWorldLeader
Very recently the Eastern block has disappeared. However it the Western block
is still active as before. In this model, within our notations, denoting A the
Western alignment, we have still
i
= +1for countries which had
i
=+1. On
the opposite, countries which had
i
=
−
1havenow turned to either
i
=+1if
joining Nato or to
i
= 0 otherwise.
