Multiple phase transitions on compact symbolic systems

TL;DR

This paper constructs symbolic dynamical systems with potentials exhibiting multiple phase transitions at specified inverse temperature values, including countably infinite transitions, revealing complex thermodynamic behavior.

Contribution

It introduces a method to design potentials with prescribed multiple phase transitions in the pressure function of symbolic systems.

Findings

Constructed potentials with phase transitions at arbitrary sequences of inverse temperatures.

Demonstrated existence of potentials with countably infinite phase transitions.

Showed the precise control of phase transition locations in symbolic dynamical systems.

Abstract

Let $\phi:X\to \mathbb R$ be a continuous potential associated with a symbolic dynamical system $T:X\to X$ over a finite alphabet. Introducing a parameter $\beta>0$ (interpreted as the inverse temperature) we study the regularity of the pressure function $\beta\mapsto P_{\rm top}(\beta\phi)$ on an interval $[\alpha,\infty)$ with $\alpha>0$. We say that $\phi$ has a phase transition at $\beta_0$ if the pressure function $P_{\rm top}(\beta\phi)$ is not differentiable at $\beta_0$. This is equivalent to the condition that the potential $\beta_0\phi$ has two (ergodic) equilibrium states with distinct entropies. For any $\alpha>0$ and any increasing sequence of real numbers $(\beta_n)$ contained in $[\alpha,\infty)$, we construct a potential $\phi$ whose phase transitions in $[\alpha,\infty)$ occur precisely at the $\beta_n$'s. In particular, we obtain a potential which has a countably infinite set of phase transitions.