Mathematical treatment of the canonical finite state machine for the Ising model: $\epsilon$-machine

TL;DR

This paper provides a corrected mathematical framework for constructing the $psilon$-machine of the one-dimensional Ising model, enabling detailed complexity and entropy analysis of specific instances.

Contribution

It introduces a novel inverse approach to derive the stochastic matrix from local characteristics, correcting previous inaccuracies in the $psilon$-machine construction for the Ising model.

Findings

Derived explicit expressions for the stochastic matrix from transfer matrix data.

Applied the framework to analyze complexity and entropy in specific Ising models.

Demonstrated the approach on examples including nearest and next-nearest neighbor models.

Abstract

The complete framework for the $\epsilon$-machine construction of the one dimensional Ising model is presented correcting previous mistakes on the subject. The approach follows the known treatment of the Ising model as a Markov random field, where usually the local characteristic are obtained from the stochastic matrix, the problem at hand needs the inverse relation, or how to obtain the stochastic matrix from the local characteristics, which are given via the transfer matrix treatment. The obtained expressions allow to perform complexity-entropy analysis of particular instance of the Ising model. Three examples are discussed: the 1/2-spin nearest neighbor and next nearest neighbor Ising model, and the persistent biased random walk.