Modelling non-periodic systems by binary uniform substitution sequences

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Abstract

Substitution sequences are used for modelling non-homogeneous interactions between elements (atoms, spins, etc.) that form up neither periodic nor disordered lattice models whose physical properties may be different from those of homogeneous systems. In this work, we present a classification of substitution sequences generated by 2-letters uniform rules concerning periodicity with its demonstration as well as its relation with the concept of almost periodicity. We also discuss some physical consequences of our classification regarding spin models.

Introduction

In 1968, A. Lindenmayer introduced a mathematical formalism called L-system to model growth process in plants. L-systems are string rewriting machines in which the transition (production) rules are applied simultaneously to all symbols (letters) of the input string [1]. The branching structures found in many plants may be formally described with L-systems since the symbols represent different lengths of the branches and the rule indicates the angle and the finite word formed by symbols at the next step in such a way that, after some iterations, the plant architecture is complete (see Ref. [2]). Nowadays it is well known that L-systems is a powerful tool for modelling growth processes in general.

In this paper we investigate a special case of L-systems, called substitution systems [3] or D0L-systems for which the transition rule is deterministic (D) and do not depend on the symbols’ neighbors (zero(0)-sided interaction). So they are well suitable for describing the growth process of structures as, for example, of biological and physical systems, which are governed just by its elements. It is not the case for other processes for which the time evolution state is strongly related to its neighbors’ states like the growth of epidemics in a population.

In order for a substitution system to describe a growth process, it is necessary that the transition rule applied to, at least, one letter generates a finite word with more than one letter. After successive applications of such a given rule, an infinite word, the substitution sequence χ, is generated. For instance, the Thue–Morse rule (aab, bba), aababba generates the infinite word χ=abbabaabbaababbaba. In fact, self-similar substitution systems like the Thue–Morse system present a straight relation with fractal geometry since the finite word at step n corresponds to the initial part of the finite word at step n+1. It is easy to see that the Cantor set is generated by the rule aaba and bbbb, in which a represents the filled line and b, the blank space.

Substitution systems are object of investigation of mathematicians [4], [5], physicists [6], [7] and computation scientists with different approaches. For physicists, such an object became relevant after the discovery of the quasi-crystals [8] in the 1980s; since then, it is used for modelling inhomogeneous interactions between elements (atoms, spins, etc.) that generate neither periodic nor disordered lattice models. For example, an aperiodic Ising chain of two types of interactions, Ja and Jb, may be set up according to the order imposed by a non-periodic substitution sequence. The aperiodic character of the interactions may change the physical properties of the system in relation to the homogeneous one whose interactions are only of type Ja. For spin models, a heuristic criterion was proposed to establish in which cases the non-periodicity is relevant [9]. Therefore, the most fundamental question is which rules generate non-periodic sequences; that is the motivation of our work.

A classification of two-letter (a and b) uniform (both letters generate words with the same number of letters) substitution systems concerning periodicity is the object of this work. It is the basis for a general classification of uniform substitution systems on an alphabet of n letters [10]. It is useful for modelling non-periodic physical systems formed by two different interactions. In a previous work [11], we classified the binary uniform substitution systems concerning almost periodicity. In this paper, we exhibit a rigorous mathematical proof of the classification of binary uniform substitution systems concerning periodicity [12]. We also present a relation between the results concerning periodicity and almost periodicity and how our results are associated with the relevance/irrelevance criterion for spin systems [9].

Section snippets

Basic concepts and previous results

Let us introduce the fundamental definitions as well as some previous results that are necessary for our classification results.

Definition 1

An alphabet A is a finite set with η elements called letters. A* is the set of finite words generated by letters in A. AN is the set of (right) infinite words, i.e., sequences χ:NA.

In relation to finite words, we say that the finite word V occurs in the word W (VW) if V is a sub-word of W. The length of the finite word W is the number of letters that occur in W,

Mathematical results

Theorem 3.1

Given a uniform binary substitution system (1), the substitution sequence χ is periodic if, and only if, one of the following cases holds:

  • (i)

    ρ(a)=aλ;

  • (ii)

    ρ(a)=abλ-1 and ρ(b)=bλ;

  • (iii)

    ρ(a)=ρ(b) and ρ(a)aλ;

  • (iv)

    ρ(a)=ababa and ρ(b)=babab (alternated rule).

In cases (i), (iii) and (iv) |R|=0; in case (ii) R=a.

Proof.

Necessity. Lemmas III.1 and III.2 of Ref. [11] guarantee that substitutions (iii) and (iv) generate periodic sequences without radical: χ=aYaYaY (|T|>1) or χ=ababab (T=ab), respectively. Since a is

Physical consequences and perspectives

The fundamental character of our results makes them very suitable in applications to the modelling of non-periodic systems in general, whose interactions are governed by deterministic rules. In the context of statistical physics, there is a natural link between such results and the relevance/irrelevance criterion of the geometrical fluctuations associated with the non-periodic character of the interactions of a spin model [9]. For a one-dimensional spin model with two types of interactions, Ja

Acknowledgements

Pinho and Lobão thank their undergraduate students, Vinícius O.G. de Meneses and Marcelo S.P. Muniz, for fruitful discussions.

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Cited by (1)

  • An algorithm for periodicity and almost periodicity of uniform substitution sequences and its implications on aperiodic spin models

    2019, Computer Physics Communications
    Citation Excerpt :

    For example, according to our classification, Thue–Morse and Rudin–Shapiro sequences are weakly non periodic. Such a classification was much more simple for binary uniform substitution rules (such as the Thue–Morse rule), previously proposed by some of us [16], in contrast to non-binary ones (such as the Rudin–Shapiro rule). As a consequence, the application of Luck’s criterion [12] – which analyzes the geometrical fluctuations on the critical behavior of spin models on chains – is directly related to the elements of substitution matrix [17].

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