Phase Transition and Strong Predictability

TL;DR

This paper explores the phase transition in algorithmic information theory by analyzing the thermodynamic quantity Z(T) and its predictability, revealing critical behavioral differences at T=1.

Contribution

It introduces the concept of strong predictability for infinite sequences and applies it to analyze the phase transition in the thermodynamic quantity Z(T) in AIT.

Findings

Behavior of Z(T) changes critically at T=1

Strong predictability distinguishes between T<1 and T=1 cases

Phase transition corresponds to divergence of thermodynamic quantities

Abstract

The statistical mechanical interpretation of algorithmic information theory (AIT, for short) was introduced and developed in our former work [K. Tadaki, Local Proceedings of CiE 2008, pp.425-434, 2008], where we introduced the notion of thermodynamic quantities into AIT. These quantities are real functions of temperature T>0. The values of all the thermodynamic quantities diverge when T exceeds 1. This phenomenon corresponds to phase transition in statistical mechanics. In this paper we introduce the notion of strong predictability for an infinite binary sequence and then apply it to the partition function Z(T), which is one of the thermodynamic quantities in AIT. We then reveal a new computational aspect of the phase transition in AIT by showing the critical difference of the behavior of Z(T) between T=1 and T<1 in terms of the strong predictability for the base-two expansion of Z(T).