Statistical Physics of Coding for the Integers

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.

Figures (2)

• Figure 1: The entropy function $s(\epsilon)=\inf_{\beta>1}\{\beta\epsilon+\ln\zeta(\beta)\}$. Observe that for large $\epsilon$, the function becomes nearly linear in $\epsilon$.
• Figure 2: A graph of $\epsilon$ vs. $\theta$ for $R\log(e)=3$.