Statistical Physics of Coding for the Integers

TL;DR

This paper explores a statistical physics approach to coding natural numbers, linking entropy, phase transitions, and coding efficiency through the zeta distribution and prime number logarithms.

Contribution

It introduces a novel statistical-mechanical interpretation of coding for natural numbers and derives key entropy and large deviations results.

Findings

Develops a simple coding scheme for the zeta distribution with near-optimal length.

Derives the micro-canonical entropy function showing Hagedorn system behavior.

Identifies a phase transition causing partial ensemble equivalence.

Abstract

We study a paradigm of coding for compression of the natural numbers via the zeta distribution and develop a statistical-mechanical interpretation, both in terms of Hagedorn systems and a Bose gas with energy levels given by logarithms of prime numbers. We also propose a simple coding scheme for the zeta distribution that nearly achieves the ideal code length. For block coding of vectors of natural numbers, we derive the micro-canonical entropy function and demonstrate its asymptotic linearity implying that its behavior is analogous to that of a Hagedorn system. We also derive the large deviations rate function, and provide a formula for the best coding parameter in the large deviations sense. We show that due the Hagedorn-type phase transition there is only partial equivalence of ensembles, due to the degeneration of the domain of the partition function.