Invertible mappings and the large deviation theory for the $q$-maximum entropy principle

TL;DR

This paper explores the connection between the $q$-maximum entropy principle and large deviation theory, establishing an invertible mapping that enables implicit reconciliation between generalized and classical statistical frameworks.

Contribution

It introduces an invertible mapping linking $q^*$-maximum entropy distributions with Shannon-Jaynes entropy, bridging generalized and classical statistical mechanics.

Findings

The $q^*$-maximum entropy principle does not satisfy large deviation properties.

An invertible mapping between $q^*$-entropy and Shannon entropy ensembles is established.

Numerical examples demonstrate the practical application of the theoretical results.

Abstract

The possibility of reconciliation between canonical probability distributions obtained from the $q$-maximum entropy principle with predictions from the law of large numbers when empirical samples are held to the same constraints, is investigated into. Canonical probability distributions are constrained by both: $(i)$ the additive duality of generalized statistics and $(ii)$ normal averages expectations. Necessary conditions to establish such a reconciliation are derived by appealing to a result concerning large deviation properties of conditional measures. The (dual) $q^*$-maximum entropy principle is shown {\bf not} to adhere to the large deviation theory. However, the necessary conditions are proven to constitute an invertible mapping between: $(i)$ a canonical ensemble satisfying the $q^*$-maximum entropy principle for energy-eigenvalues $\varepsilon_i^*$, and, $(ii)$ a canonical ensemble satisfying the Shannon-Jaynes maximum entropy theory for energy-eigenvalues $\varepsilon_i$. Such an invertible mapping is demonstrated to facilitate an \emph{implicit} reconciliation between the $q^*$-maximum entropy principle and the large deviation theory. Numerical examples for exemplary cases are provided.