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criticality is consistent with the Ising model in some cases and is different in
other cases [8]. There are propositions, initiated by Domany and Kinzel [9], to
represent the canonical ensemble [10,11]. At the microscopic level such cellular
automata and Ising thermodynamic systems are distinct because ofthe design
ofstochastic dynamics. The widely applied thermodynamic rule reerred to as
”Glauber dynamics” uses the following single spin-flip transition rate:
1
− σ
i
tanh
β
P
(
σ
i
→−σ
i
)=
1
2
σ
(
i
)
nn
(1)
nn
where
nn
denotes the list ofnearest neighbors of
σ
i
spin and
β
is proportional
to the inverse oftemperature. The corresponding probabilistic CA rule is the
following:
P
(
σ
i
→−σ
i
)=
1
2
σ
(
i
)
nn
1
− εσ
i
sign
(2)
nn
with
ε
replacing the parameter
β
. Notice that the stochastic noise perturbs the
execution ofa given deterministic rule (
ε
= 1 makes the rule fully deterministic)
while temperature
β
perturbs the influence ofthe neighboring spins.
In general, the evolution ofCA with the probabilistic rule (2) does not sat-
isfy the detailed balance condition [11]. Hence, these CA generate nonequilib-
rium stochastic systems. The question, whether the stationary state resulting
from this evolution is an equilibrium state, is not trivial [12]. It is widely be-
lieved that the universality hypothesis spreads to any non-equilibrium stochastic
system ifthe system has the same up-down symmetry as the Ising model [10,
13]. However, Marcq et al. [14] observe that in case ofcoupled map lattices the
synchronous updating leads to the correlation-lenght exponent
ν
=0
.
89
±
0
.
02
which is significantly lower than it is observed in the system with asynchronous
updating or in the Ising model for which
ν
Ising
= 1. The relation between cou-
pled map lattices and kinetic Ising models [15] as well as scaling properties of
other nonequilibrium stochastic systems, see e.g. [16], are vividly discussed.
The Toom CA is a system ofspins on a square lattice where interactions be-
tween three spins: North, East and a spin itself, are considered [10,17]. Extensive
numerical simulations indicate that the phase transition in these CA is contin-
uous [10,12]. Toom local interactions lead to the important global property of
dynamics called eroder property [17]. The eroder property means that any finite
island ofone phase surrounded by the sea ofthe other phase will decay in finite
number oftime steps. Because ofthis feature investigations are undertaken which
examine Gibbsianess ofstationary states ofToom systems [18]. Moreover, it is
doubtful whether the transition belongs to the Ising universality class [12]. The
novel aspect ofthis paper is that we consider the influence ofsynchronous and
asynchronous updating on critical properties ofCA systems. For better under-
standing ofthe role ofupdating we also consider the spin systems with Glauber
dynamics.
