On the extended Kolmogorov-Nagumo information-entropy theory, the q -> 1/q duality and its possible implications for a non-extensive two dimensional Ising model

TL;DR

This paper explores the q -> 1/q duality in generalized entropy theories and its implications for a non-extensive 2D Ising model, revealing invariance properties and dualities across different entropy measures.

Contribution

It demonstrates the form invariance of all q-generalized entropy functionals under the q -> 1/q duality using a unified exponential-logarithm representation.

Findings

Generalized partition function is invariant under q -> 1/q duality.

Duality links systems with different long-range interactions and temperatures.

The invariance applies to all stationary distributions with q-exponential Boltzmann factors.

Abstract

The aim of this paper is to investigate the q -> 1/q duality in an information-entropy theory of all q-generalized entropy functionals (Tsallis, Renyi and Sharma-Mittal measures) in the light of a representation based on generalized exponential and logarithm functions subjected to Kolmogorov's and Nagumo's averaging. We show that it is precisely in this representation that the form invariance of all entropy functionals is maintained under the action of this duality. The generalized partition function also results to be a scalar invariant under the q -> 1/q transformation which can be interpreted as a non-extensive two dimensional Ising model duality between systems governed by two different power law long-range interactions and temperatures. This does not hold only for Tsallis statistics, but is a characteristic feature of all stationary distributions described by q-exponential Boltzmann factors.