Phase Transitions in One-dimensional Translation Invariant Systems: a Ruelle Operator Approach

TL;DR

This paper investigates phase transitions in one-dimensional symbolic systems using Ruelle operators, revealing non-analytic pressure, multiple eigenprobabilities, and non-unique thermodynamic limits, with explicit critical point calculations and polynomial decay of correlations.

Contribution

It introduces a Ruelle operator approach to analyze phase transitions in Hofbauer-type potentials, including explicit critical point determination and non-uniqueness of thermodynamic limits.

Findings

Phase transitions characterized by non-analytic pressure and multiple eigenprobabilities.

Explicit calculation of critical points for phase transitions.

Polynomial decay of correlations in the studied systems.

Abstract

We consider a family of potentials f, derived from the Hofbauer potentials, on the symbolic space Omega=\{0,1\}^\mathbb{N} and the shift mapping $\sigma$ acting on it. A Ruelle operator framework is employed to show there is a phase transition when the temperature varies in the following senses: the pressure is not analytic, there are multiple eigenprobabilities for the dual of the Ruelle operator, the DLR-Gibbs measure is not unique and finally the Thermodynamic Limit is not unique. Additionally, we explicitly calculate the critical points for these phase transitions. Some examples which are not of Hofbauer type are also considered. The non-uniqueness of the Thermodynamic Limit is proved by considering a version of a Renewal Equation. We also show that the correlations decay polynomially and that each one of these Hofbauer potentials is a fixed point for a certain renormalization transformation.