Constraints and relative entropies in nonextensive statistical mechanics

TL;DR

This paper investigates the role of different expectation value definitions in nonextensive statistical mechanics, demonstrating that the normalized q-expectation value aligns with the Shore-Johnson axioms, unlike the ordinary expectation.

Contribution

It clarifies the importance of the escort distribution in defining expectation values and establishes the normalized q-expectation as the physically correct choice within an axiomatic framework.

Findings

Escort distribution is essential for consistent relative entropy definitions.

The Shore-Johnson axioms support the normalized q-expectation formalism.

Ordinary expectation value is excluded by the axiomatic analysis.

Abstract

In nonextensive statistical mechanics, two kinds of definitions have been considered for expectation valu of a physical quantity: one is the ordinary definition and the other is the normalized q-expectation value employing the escort distribution. Since both of them lead to the maximum-Tsallis-entropy distributions of a similar type, it is of crucial importance to determine which the correct physical one is. A point is that the definition of expectation value is indivisibly connected to the form of generalized relative entropy. Studying the properties of the relative entropies associated with these two definitions, it is shown how the use of the escort distribution is essential. In particular, the axiomatic framework proposed by Shore and Johnson is found to support the formalism with the normalized q-expectation value and to exclude the ordinary expectation value in nonextensive statistical mechanics.