Entropy and Variational Principle for one-dimensional Lattice Systems with a general a-priori probability: positive and zero temperature

TL;DR

This paper extends classical Thermodynamic Formalism to general compact metric spaces, analyzing entropy, pressure, and eigenfunctions for shift systems with infinite coordinates, including zero temperature limits and applications to models like XY.

Contribution

It introduces a generalized framework for Thermodynamic Formalism on compact metric spaces with infinite coordinates, including new results on entropy, pressure, and zero temperature behavior.

Findings

Gibbs states can have arbitrary negative entropy values.

Zero entropy for infinite product measures on $S^1$.

Analysis of zero temperature limits and eigenfunction properties.

Abstract

We generalize several results of the classical theory of Thermodynamic Formalism by considering a compact metric space $M$ as the state space. We analyze the shift acting on $M^\mathbb{N}$ and consider a general a-priori probability for defining the Transfer (Ruelle) operator. We study potentials $A$ which can depend on the infinite set of coordinates in $M^\mathbb{N}.$ We define entropy and by its very nature it is always a nonpositive number. The concepts of entropy and transfer operator are linked. If M is not a finite set there exist Gibbs states with arbitrary negative value of entropy. Invariant probabilities with support in a fixed point will have entropy equal to minus infinity. In the case $M=S^1$, and the a-priori measure is Lebesgue $dx$, the infinite product of $dx$ on $(S^1)^\mathbb{N}$ will have zero entropy. We analyze the Pressure problem for a H\"older potential $A$ and its relation with eigenfunctions and eigenprobabilities of the Ruelle operator. Among other things we analyze the case where temperature goes to zero and we show some selection results. Our general setting can be adapted in order to analyze the Thermodynamic Formalism for the Bernoulli space with countable infinite symbols. Moreover, the so called XY model also fits under our setting. In this last case M is the unitary circle $S^1$. We explore the differentiable structure of $(S^1)^\mathbb{N}$ by considering potentials which are of class $C^1$ and we show some properties of the corresponding main eigenfunctions.