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In-Depth Information
2 The Models
Toom (TCA) and Glauber (GCA ) systems are defined on a square lattice. Each
site ofthe lattice is occupied by a spin
σ
i
which points either
up
or
down
what is
coded by +1 or
−
1, respectively. The future state of a spin
σ
i
is determined by
present states ofits three nearest neighbors, named
N
i
,E
i
,C
i
chosen as follows:
| | |
− . − N
i
− . − .
| | |
− . − C
i
=
σ
i
− E
i
− .
|
(3)
|
|
Thus, dealing with three-spin interaction on a square lattice we mimic the type
ofthe Ising model on a triangular lattice in which only one type oftriangles is
considered.
Let
Σ
i
=
N
i
+
E
i
+
C
i
. The deterministic dynamic rule is the same in both
systems but the stochastic perturbation acts differently, namely:
— in Toom system:
sgn
Σ
i
with prob
.
1
2
(1 +
ε
)
σ
i
(
t
+1)=
(4)
1
−
sgn
Σ
i
with prob
.
2
(1
− ε
)
— in Glauber system:
sgn
Σ
i
with prob
.
1
2
(1 + tanh
ε |Σ
i
|
)
σ
i
(
t
+1)=
(5)
1
−
sgn
Σ
i
with prob
.
2
(1
−
tanh
ε |Σ
i
|
)
3 Monte Carlo Simulations
We use the standard importance sampling technique to simulate models intro-
duced in the last section. We consider square lattices oflinear size
L
with values
of
L
ranging from
L
=20to
L
= 100 and we apply periodic boundary conditions.
The computer experiments start with all spins aligned. A new configuration is
generated from the old one by the following Markov process: for a given
ε
the
evolution rule either (4) or (5) is employed to each spin in case ofsynchronous
updating, or to a randomly chosen spin when asynchronous updating case is ex-
amined. The evolving system is given 100
L
time steps to reach the steady state.
Such time interval is su
I
cient to find all systems studied in stationary ergodic
states [19].
When a system is in the stationary state then an expectation value ofmag-
netization
m
is computed according to a sequence ofstates
{σ
i
(
t
)
}
i
=1
,...,L
2
:
i
=1
,...,L
2
σ
i
(
t
)
,
1
T
1
L
2
m
L
=
(6)
t
=1
,...,T
