Anomalous behavior of q-averages in nonextensive statistical mechanics

TL;DR

This paper investigates the stability of q-averages in nonextensive statistical mechanics, revealing that under certain physical deformations and norms, the q-average may behave unphysically and lack stability.

Contribution

It clarifies the physical relevance of deformations, demonstrates the instability of q-averages in various systems, and emphasizes the importance of appropriate norms for probability distributions.

Findings

q-averages can behave unphysically under certain deformations

q-averages are unstable in finite and infinite systems

A better norm than L^1 is needed for continuous systems

Abstract

A generalized definition of average, termed the q-average, is widely employed in the field of nonextensive statistical mechanics. Recently, it has however been pointed out that such an average value may behave unphysical under specific deformations of probability distributions. Here, the following three issues are discussed and clarified. Firstly, the deformations considered are physical and may experimentally be realized. Secondly, in view of thermostatistics, the q-average is unstable in both finite and infinite discrete systems. Thirdly, a naive generalization of the discussion to continuous systems misses a point, and a norm better than the $L^1$-norm should be employed for measuring the distance between two probability distributions. Consequently, stability of the q-average is shown not to be established in all the cases.