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where
t
means Monte Carlo steps, i.e.,
t
denotes one simulation step when syn-
chronous updating is performed and
L
2
single spin flips in case when the asyn-
chronous updating is examined.
T
= 10 000 in all experiments. To avoid the
possibility that the examined state is attracted by some metastable state, we
perform
N
independent experiments, with
N
in the range 500
,...,
5500.
In case ofcontinuous phase transition on the lattice with a finite size we need
to observe the magnitude ofthe magnetization :
i
=1
,...,L
2
σ
i
(
t
)
1
T
1
L
2
|m|
L
=
(7)
t
=1
,...,T
and the
n
-th moments ofmagnetization,
n
=2
,
4:
n
1
T
1
L
2
m
L
=
i
=1
,...,L
2
σ
i
(
t
)
(8)
t
=1
,...,T
due to them we could study properties ofthe associated finite-size lattice sus-
ceptibility:
χ
L
=
L
2
(
m
L
−|m|
L
)
(9)
and the reduced fourth-order Binder cumulant [20]
m
L
3(
m
L
)
2
U
L
(
ε
)=1
−
(10)
In all systems considered: Toom or Glauber CA with synchronous or asyn-
chronous updating two qualitatively distinct regimes are easily observed: ofhigh
magnetization- the ferromagnetic phase, and zero magnetization — the para-
magnetic phase, see Fig.1. In each ofthe system studied the rapid change of
magnetization is observed. The huge value ofsusceptibility is recorded at the
same time. One can say that systems pass the ferromagnetic phase transition.
According to the scaling hypothesis, in the thermodynamic systems the sin-
gularities ofobservables: magnetization
m
, susceptibility
χ
, correlation length
ξ
have the power-law form, [2,5]:
m ∝
(
ε − ε
cr
)
β
for
ε → ε
cr
and
ε>ε
cr
,
χ ∝
(
ε − ε
cr
)
−γ
for
ε → ε
cr
,
(11)
ξ ∝
(
ε − ε
cr
)
−ν
for
ε → ε
cr
where
β
,
γ
and
ν
are the static critical exponents which determine the univer-
sality class ofa system. Due to the finite-size lattice theory reliable values or
β
,
γ
and
ν
are accessible [20,5]. For magnetization and susceptibility we obtain
the relations:
|m|
L
(
ε
cr
)
∝ L
−β/ν
,
χ
L
(
ε
cr
)
∝ L
γ/ν
.
(12)
