What is a pattern in statistical mechanics? Formalizing structure and patterns in one-dimensional spin lattice models with computational mechanics

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, which lets us analyze each spin-lattice model with the theory of computational mechanics. In this framework, the process' structure is quantified by excess entropy (predictable information) and statistical complexity (stored information), and the process' structure-generating mechanism is specified by its epsilon-machine. To assess compatibility with statistical mechanics, we compare the configurations jointly determined by the information measures and epsilon-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.