Flexibility of the Pressure Function

TL;DR

This paper demonstrates that in one-dimensional symbolic systems, the pressure function can be explicitly constructed to match any convex Lipschitz asymptotically linear shape, allowing for diverse phase transition behaviors.

Contribution

It introduces a method to realize any prescribed convex Lipschitz asymptotic pressure function in symbolic systems, revealing extensive flexibility in phase transition phenomena.

Findings

Pressure functions can be arbitrarily shaped within convex Lipschitz constraints.

Phase transitions can occur at a dense set of temperature values.

The number of ergodic equilibrium states can be controlled as a function of temperature.

Abstract

We study the flexibility of the pressure function of a continuous potential (observable) with respect to a parameter regarded as the inverse temperature. The points of non-differentiability of this function are of particular interest in statistical physics, since they correspond to phase transitions. It is well known that the pressure function is convex, Lipschitz, and has an asymptote at infinity. We prove that in a setting of one-dimensional compact symbolic systems these are the only restrictions. We present a method to explicitly construct a continuous potential whose pressure function coincides with \emph{any} prescribed convex Lipschitz asymptotically linear function starting at a given positive value of the parameter. In fact, we establish a multidimensional version of this result. As a consequence, we obtain that for a continuous observable the phase transitions can occur at a countable dense set of temperature values. We go further and show that one can vary the cardinality of the set of ergodic equilibrium states as a function of the parameter to be any number, finite or infinite.