Chaos and Turing Machines on Bidimensional Models at Zero Temperature

TL;DR

This paper constructs a two-dimensional model demonstrating non-convergence of equilibrium measures at zero temperature, extending previous work on chaotic behavior in thermodynamic systems with complex phase transitions.

Contribution

It introduces a novel bidimensional example with a locally constant potential where equilibrium measures do not converge as temperature approaches zero, building on and improving prior constructions.

Findings

Existence of a subsequence where measures do not converge at zero temperature

Extension of non-convergence results to two-dimensional models

Use of advanced construction techniques to demonstrate chaos

Abstract

In equilibrium statistical mechanics or thermodynamics formalism one of the main objectives is to describe the behavior of families of equilibrium measures for a potential parametrized by the inverse temperature $\beta$. Here we consider equilibrium measures as the shift invariant measures that maximizes the pressure. Other constructions already prove the chaotic behavior of these measures when the system freezes, that is, when $\beta\rightarrow+\infty$. One of the most important examples was given by Chazottes and Hochman where they prove the non-convergence of the equilibrium measures for a locally constant potential when the dimension is bigger then 3. In this work we present a construction of a bidimensional example described by a finite alphabet and a locally constant potential in which there exists a subsequence $(\beta_k)_{k\geq 0}$ where the non-convergence occurs for any sequence of equilibrium measures at inverse of temperature $\beta_k$ when $\beta_k\rightarrow+\infty$. In order to describe such an example, we use the construction described by Aubrun and Sablik which improves the result of Hochman used in the construction of Chazottes and Hochman.