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X
E
(
X
)=
1
2
J
ij
η
i
η
j
,
(3)
<i,j>
where the sum runs over the
N
(
N −
1) pairs (
i, j
). At this stage it sounds
reasonable to assume each country chooses its coalition in order to minimize its
indivual cost. Accordingly to make two cooperating countries (
J
i,j
>
0) in the
same alliance, we put a minus sign in from of the expression of Eq. (3) to get,
X
H
=
−
1
2
J
ij
η
i
η
j
,
(4)
<i,j>
which is indeed the Hamiltonian of an Ising random bond magnetic system.
There exist by symmetry 2
N
/
2 distinct sets of alliances each country having
2 choices for coalition. Starting from any initial configuration, the dynamics of
the system is implemented by single country coalition flips. An country turns to
the competing coalition only if the flip decreases its local cost. The system has
reached its stable state once no more flip occurs. Given
{J
ij
}
, the
{η
i
}
are thus
obtained minimizing Eq. (4).
Since the system stable configuration minimizes the energy, we are from the
physical viewpoint, at the temperature
T
= 0. In practise for any finite system
the theory can tell which coalitions are possible. However, if several coalitions
have the same energy, the system is unstable and flips continuously from one
coalition set to another one at random and with no end.
For instance, in the case of three conflicting nations like Israel, Syria and
Iraq, any possible alliance configuration leaves always someone unsatisfied. Let
us label them respectively by 1, 2, 3 and consider equal and negative exchange
interactions
J
12
=
J
13
=
J
23
=
−J
with
J>
0asshown in Fig. (1). The
associated minimum of the energy is equal to
−J
. However this minimum value
is realized for several possible and equivalent coalitions which are respectively
(A, B, A), (B, A, A), (A, A, B), (B, A, B), (A, B, B), and (B, B, A). First
three are identical to last ones by symmetry since here what matters is which
countries are together within the same coalition. This peculiar property of a
degenerate ground state makes the system unstable. There exists no one single
stable configuration which is stable. Some dynamics is shown in Fig. (1). The
system jumps continuously and at random between (A, B, A), (B, A, A) and
(A, A, B).
To make the dynamics more explicit, consider a given site
i
. Interactions with
all others sites can be represented by a field,
N
X
h
i
=
J
ij
η
j
(5)
j
=1
resulting in an energy contribution
E
i
=
−η
i
h
i
,
(6)
