# Phase space classification of an Ising Cellular Automaton: the Q2R model

## TL;DR

This paper provides an exact classification of the dynamical behaviors in the phase space of the reversible Q2R cellular automaton, revealing four types of limit cycles and analyzing their properties in small Ising systems.

## Contribution

It introduces a novel classification scheme for the phase space of the Q2R automaton, identifying four distinct cycle types and analyzing their topological and combinatorial properties.

## Key findings

- Four types of limit cycles classified by topological characteristics
- Explicit enumeration of limit cycles for small periods
- Complete analysis of a 4x4 Ising system under the new framework

## 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 4x4 which is fully analyzed under this new framework.