# What Is a Pattern in Statistical Mechanics? Formalizing Structure and Patterns in One-Dimensional Spin Lattice Models with Computational Mechanics

## TL;DR

This paper uses computational mechanics to define and analyze patterns in one-dimensional spin lattice models from statistical mechanics.

## Contribution

A novel derivation of the Boltzmann distribution and application of computational mechanics to quantify structure in spin-lattice models.

## Key findings

- The configurations determined by information measures and ϵ-machines align with typical Boltzmann configurations.
- Excess entropy and statistical complexity quantify structural properties of spin-lattice models.
- The paper provides a primer and code for implementing the methods.

## Abstract

This work formalizes the notions of structure and pattern for three distinct one-dimensional spin-lattice models (finite-range Ising, solid-on-solid, and three-body), using information- and computation-theoretic methods. We begin by presenting a novel derivation of the Boltzmann distribution for finite one-dimensional spin configurations embedded in infinite ones. We next recast this distribution as a stochastic process, thereby enabling us to analyze each spin-lattice model within the theory of computational mechanics. In this framework, the process’s structure is quantified by excess entropy E (predictable information) and statistical complexity Cμ (stored information), and the process’s structure-generating mechanism is specified by its ϵ-machine. To assess compatibility with statistical mechanics, we compare the configurations jointly determined by the information measures and ϵ-machines to typical configurations drawn from the Boltzmann distribution, and we find agreement. We also include a self-contained primer on computational mechanics and provide code implementing the information measures and spin-model distributions.