arXiv:2604.00858 · v2

Statistical Physics of Coding for the Integers

首次提交 2026-04-01 21:09

版本更新 2026-11-08 23:28

雷达观测 2026-11-10 12:16

摘要

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.

作者备注：41 pages, 2 figures, accepted to Journal of Statistical Physics Theory and Experiment (JSTAT)