Information Shift Dynamics Described by Tsallis $q=3$ Entropy on a Compact Phase Space

TL;DR

This paper explores a novel information shift dynamics modeled by Chebyshev map-based Bernoulli shifts that maximize Tsallis q=3 entropy, revealing unique symmetry properties and potential cosmological implications.

Contribution

It introduces a new dynamical system framework using Tsallis q=3 entropy and analyzes its symmetry and physical implications, including the significance of the fine structure constant.

Findings

Chebyshev map dynamics maximize Tsallis q=3 entropy

Symmetry properties differ for even and odd N

The fine structure constant acts as a coupling constant in this model

Abstract

Recent mathematical investigations have shown that under very general conditions exponential mixing implies the Bernoulli property. As a concrete example of a statistical mechanics which is exponentially mixing we consider a Bernoulli shift dynamics by Chebyshev maps of arbitrary order $N\geq 2$, which maximizes Tsallis $q=3$ entropy rather than the ordinary $q=1$ Boltzmann-Gibbs entropy. Such an information shift dynamics may be relevant in a pre-universe before ordinary space-time is created. We discuss symmetry properties of the coupled Chebyshev systems, which are different for even and odd $N$. We show that the value of the fine structure constant $\alpha_{el}=1/137$ is distinguished as a coupling constant in this context, leading to uncorrelated behaviour in the spatial direction of the corresponding coupled map lattice for $N=3$.