A statistical mechanical interpretation of algorithmic information theory

TL;DR

This paper introduces a novel interpretation of algorithmic information theory through statistical mechanics, defining thermodynamic quantities like free energy and entropy in relation to program complexity and randomness.

Contribution

It presents a new framework linking thermodynamic concepts with algorithmic information theory, including fixed point theorems on compression rate.

Findings

Thermodynamic quantities are defined within algorithmic information theory.

Temperature acts as a compression rate for thermodynamic quantities.

Fixed point theorems on compression rate are established.

Abstract

We develop a statistical mechanical interpretation of algorithmic information theory by introducing the notion of thermodynamic quantities, such as free energy, energy, statistical mechanical entropy, and specific heat, into algorithmic information theory. We investigate the properties of these quantities by means of program-size complexity from the point of view of algorithmic randomness. It is then discovered that, in the interpretation, the temperature plays a role as the compression rate of the values of all these thermodynamic quantities, which include the temperature itself. Reflecting this self-referential nature of the compression rate of the temperature, we obtain fixed point theorems on compression rate.