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Ta b l e 1 .
Different approximations, initial distribution binomially distributed with
p
(0)
Uni
0.3033
p
(
x
=1
;
0)
0.2
p
(
x
=1
;
50)
0.0000
p
(
x
=1
:
1)
p
(
x
2
0.0000
=1j
x
1
Bi
0.3033
0.2
0.1233
0.0000
0.50
Tri
0.3033
0.2
0.1737
0.1734
1.00
Markov 0.3033
0.2
0.1770
0.1770
1.00
Uni
0.3633
0.8
0.5381
0.0000
Bi
0.3633
0.8
0.6822
0.5000
0.74
Tri
0.3633
0.8
0.7657
0.93
Markov 0.3633
0.8
0.7756
0.7756
1.00
0
:
5400
For
we obtain
. This result can
easily be explained. For
there are two attractors only,
and
. The limit distribution consists of these two configurations
only. The approximations
and
give the mean-field result
,but the convergence is much slower for the bi-variate approximation. The
tri-variate approximation converges to
=1)
!
. This is a little
less than the exact value. The tri-variate approximation and the exact Markov
analysis give the same results for the conditional probabilities, indicating a high
correlation between 1's.
For
the mean-field analysis predicts convergence to 0.5 The
Markov analysis gives a different result, namely
. Unfortu-
nately the tri-variate approximation seems also to converge to 0.5, but it takes a
very long time.
We can now characterize the role which the point
=1)
!
0:7756
mathematically
plays. For
the value of
remains constant, for
the
value of
decreases up to a limit value for
. It increases
for
. There is not any kind of phase transition to be observed.
We now investigate the difficult case
there
often exist no limit distribution, but cycles. In table 2 the results are displayed.
All approximations reproduce the cycle of size 3 between the configurations
and
.For
,
,
.
we obtain after a very long time an
uninteresting limit distribution. In the limit all configurations are equally likely,
with the exception of
But if we change to
and
, which are less likely. In this case we can dis-
tinguish between a short term dynamics, where one still can observe the cycles,
and a long term dynamics, where the probabilities of the cycle configurations
decreases. Thus even the smallest disturbance has a dramatic influence for the
stationary distribution.
and
