TY - JOUR

T1 - Phase space classification of an Ising cellular automaton

T2 - The Q2R model

AU - Montalva-Medel, Marco

AU - Rica, Sergio

AU - Urbina, Felipe

N1 - Publisher Copyright:
© 2020

PY - 2020/4

Y1 - 2020/4

N2 - An exact classification of the different dynamical behaviors that exhibits the phase space 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 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 types of cycles. Moreover, at a combinatorial level, we can determine the number of 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 and, in addition, we realize an exhaustive study of a small Ising system (4 × 4) which is thoroughly analyzed under this new framework, and where simple mathematical tools are introduced in order to have a more direct understanding of the Q2R dynamics and to rediscover known properties like the energy conservation.

AB - An exact classification of the different dynamical behaviors that exhibits the phase space 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 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 types of cycles. Moreover, at a combinatorial level, we can determine the number of 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 and, in addition, we realize an exhaustive study of a small Ising system (4 × 4) which is thoroughly analyzed under this new framework, and where simple mathematical tools are introduced in order to have a more direct understanding of the Q2R dynamics and to rediscover known properties like the energy conservation.

UR - http://www.scopus.com/inward/record.url?scp=85078697800&partnerID=8YFLogxK

U2 - 10.1016/j.chaos.2020.109618

DO - 10.1016/j.chaos.2020.109618

M3 - Article

AN - SCOPUS:85078697800

SN - 0960-0779

VL - 133

JO - Chaos, Solitons and Fractals

JF - Chaos, Solitons and Fractals

M1 - 109618

ER -