Phase space classification of an Ising cellular automaton: The Q2R model

Abstract

An exact characterization of the different dynamical behavior that exhibit the space phase of a reversible and conservative cellular automaton, the so called Q2R model, is shown in this paper. Q2R is a cellular automaton which is a dynamical variation of the Ising model in statistical physics and whose space of configurations grows exponentially with the system size. As a consequence of the intrinsic reversibility of the model, the phase space is composed only by configurations that belong to a fixed point or a limit cycle. In this work we classify them in four types accordingly to well differentiated topological characteristics. Three of them -which we call of type S-I, S-II and S-III-share a symmetry property, while the fourth, which we call of type AS, does not. Specifically, we prove that any configuration of Q2R belongs to one of the four previous limit cycles. Moreover, at a combinatorial level, we are able to determine the number of limit cycles for some small periods which are almost always present in the Q2R. Finally, we provide a general overview of the resulting decomposition of the arbitrary size Q2R phase space, in addition, we realize an exhaustive study of a small Ising system (4 × 4) which is fully analyzed under this new framework.

Order-disorder on cluster dynamics in the Q2R-Potts cellular automaton

“…This system has energy E = −9, and under the rule section 2.2, the dynamics is shown in figure 4. First, this system has period T = 4, and due to theorem 4.10 [16], is classified as an asymmetric cycle, e.i., it has always different configurations.…”

“…Due to the reversibility of this model, Hermann et al [13] demonstrated that the model dynamics can have periods of 'infinite' length on lattices of 'infinite' size. In this context, Montalva et al [16] classified the entire space of 2 32 ≈ 4.3 × 10 9 configurations by identifying all the fixed points and the orbits for each energy's value. Moreover, they presented a theorem that classifies the different types of cycles: symmetric and asymmetric.…”

Information-theoretic analysis of complex systems modeled by two dimensional pairwise Ising models

“…In this study, we assume a second-order neighborhood system, that is, ∆ = 8, defining the Moore neighborhood. In each site s i we observe a random variable x i that assume values in {−1, +1}, representing the spin of a particle [28]. A realization X = (x k ) k∈S is an assignment of a spin value to each s i , for i = 1, 2, ..., n. For any pair of adjacent sites s i , s j ∈ S there is an interaction J ij .…”

On the complexity of the generalized Q2R automaton

“…Several studies have carried out concerning the Q2R [1,5,6,7,8,9,10,11,12]. Remarkably, the Q2R dynamic preserves an Ising-like energy [5], appealing the analogy with the continuous dynamics of Hamiltonian systems.…”

“…Recently, in an attempt to establish general mathematical properties, a full characterization and combinatorial results of the attractors associated to the Q2R model were proposed in [12]. In tune with this mathematical approach, in this paper we tackle an analytical study of the dynamics and complexity of Q2R.…”