On the equivalence between stochastic baker's maps and two-dimensional spin systems

TL;DR

This paper establishes a mathematical equivalence between certain stochastic baker's maps and two-dimensional spin systems, linking their invariant measures and entropy properties, and demonstrates this with the Ising model.

Contribution

It introduces a class of stochastic baker's transformations that are equivalent to equilibrium states of 2D spin systems, providing a new perspective on their relationship.

Findings

Invariant measures of baker's maps match spin system distributions.

Entropy of spin systems correlates with entropy production in baker's maps.

Numerical calculations confirm the free energy agreement with Ising model results.

Abstract

We show that there is a class of stochastic baker's transformations that is equivalent to the class of equilibrium solutions of two-dimensional spin systems with finite interaction. The construction is such that the equilibrium distribution of the spin lattice is identical to the invariant measure in the corresponding baker's transformation. We also find that the entropy of the spin system is up to a constant equal to the rate of entropy production in the corresponding stochastic baker's transformation. We illustrate the equivalence by deriving two stochastic baker's maps representing the Ising model at a temperature above and below the critical temperature, respectively. We calculate the invariant measure of the stochastic baker's transformation numerically. The equivalence is demonstrated by finding that the free energy in the baker system is in agreement with analytic results of the two-dimensional Ising model.