A critique of non-extensive q-entropy for thermal statistics

TL;DR

This paper critically examines the non-extensive q-entropy formalism, highlighting fundamental inconsistencies such as undefined joint entropy for systems with different q values and unphysical correlations, challenging its thermodynamic validity.

Contribution

It demonstrates that the non-extensive q-entropy framework has fundamental flaws, including undefined joint entropy and unphysical correlations, questioning its applicability in thermodynamics.

Findings

Joint entropy not defined for systems with different q values

Probability distribution does not factorize for weakly interacting systems

Unphysical consequences like non-extensive energy arise from the formalism

Abstract

During the past dozen years there have been numerous articles on a relation between entropy and probability which is non-additive and has a parameter $q$ that depends on the nature of the thermodynamic system under consideration. For $q=1$ this relation corresponds to the Boltzmann-Gibbs entropy, but for other values of $q$ it is claimed that it leads to a formalism which is consistent with the laws of thermodynamics. However, it is shown here that the joint entropy for systems having {\it different} values of $q$ is not defined in this formalism, and consequently fundamental thermodynamic concepts such as temperature and heat exchange cannot be considered for such systems. Moreover, for $q\ne 1$ the probability distribution for weakly interacting systems does not factor into the product of the probability distribution for the separate systems, leading to spurious correlations and other unphysical consequences, e.g. non-extensive energy, that have been ignored in various applications given in the literature.