Computability at zero temperature

TL;DR

This paper explores the limits of computability for thermodynamic invariants at zero temperature in one-dimensional subshifts, revealing which measures and entropies are computable or semi-computable.

Contribution

It characterizes the computability of zero-temperature measures and entropy for locally constant potentials, identifying conditions for their computational identification.

Findings

Residual entropy is upper semi-computable but not fully computable.

Zero-temperature measure can be identified as a periodic point measure for a dense set of potentials.

Methods do not extend to cases where the cylinder length k is unknown.

Abstract

In this paper, we investigate the computability of thermodynamic invariants at zero temperature for one-dimensional subshifts of finite type. In particular, we prove that the residual entropy (i.e., the joint ground state entropy) is an upper semi-computable function on the space of continuous potentials, but it is not computable. Next, we consider locally constant potentials for which the zero-temperature measure is known to exist. We characterize the computability of the zero-temperature measure and its entropy for potentials that are constant on cylinders of a given length k. In particular, we show the existence of an open and dense set of locally constant potentials for which the zero-temperature measure can be computationally identified as an elementary periodic point measure. Finally, we show that our methods do not generalize to treat the case when k is not given