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Universality Class of Probabilistic Cellular
Automata
Danuta Makowiec and Piotr Gnacinski
Institute of Theoretical Physics and Astrophysics, Gdansk University,
80-952 Gdansk, ul.Wita Stwosza 57, Poland
fizdm@univ.gda.pl
Abstract.
The Ising-like phase transition is considered in probabilis-
tic cellular automata (CA). The nonequilibrium CA with Toom rule are
compared to standard equilibrium lattice systems to verify influence of
synchronous vs asynchronous updating. It was observed by Marcq et al.
[Phys.Rev.E
55
(1997) 2606] that the mode of updating separates systems
of coupled map lattices into two distinct universality classes. The simi-
lar partition holds in case of CA. CA with Toom rule and synchronous
updating represent the weak universality class of the Ising model, while
Toom CA with asynchronous updating fall into the Ising universality
class.
1 Introduction
Undoubtedly, the Ising spin system [1] is one ofthe most undamental models
in statistical mechanics. The simplicity in designing together with richness of
cooperative phenomena observed here, generate the fruitful pool for verifying or
falsifying theories. One of them is the universality hypothesis. The universality
hypothesis claims that many local features of interactions are irrelevant when
a thermodynamic system is close to a phase transition [2]. In consequence, at
the critical point all equilibrium thermodynamic systems can be grouped in few
classes. The classes differ between each other by a set ofvalues, so-called scaling
exponents, which describe singularities ofbasic thermodynamic functions when
a system is approaching a critical point.
The Ising model leaves much freedom in the design of the microscopic in-
teractions. Therefore, many stochastic rules have been invented to mimic the
dynamics ofthe model [3] and new ones are still proposed, see , e.g. [4]. In gen-
eral, any stochastic reversible time evolution such that an elementary step means
a single spin flip, is acceptable [5]. The reversibility means that stochastic rule
satisfies the detailed balance conditions with respect to the Gibbs measure with
the Ising Hamiltonian.
Since the famous Wolfram paper [6] the finite-size Ising model has also been
considered as cellular automata (CA). The Q2R cellular automaton has been
proposed as an alternative, microcanonical method for studying the Ising model
[7]. However, these CA lead to the critical behavior which is unclear. For example,
