Questioning the validity of non-extensive thermodynamics for classical Hamiltonian systems

TL;DR

This paper critically examines the non-extensive thermodynamics formalism for classical Hamiltonian systems, showing it is only consistent for certain parameter ranges and questioning its applicability to real physical systems.

Contribution

The study demonstrates fundamental limitations of non-extensive thermodynamics for Hamiltonian systems, especially regarding the range of the parameter q and the physical relevance of the resulting distributions.

Findings

Non-extensive thermodynamics should be questioned for explaining experimental distributions.

In the thermodynamic limit, the formalism is only consistent for 0 ≤ q ≤ 1.

Configurations with energy above a certain limit have zero probability in the formalism.

Abstract

We examine the non-extensive approach to the statistical mechanics of Hamiltonian systems with $H=T+V$ where $T$ is the classical kinetic energy. Our analysis starts from the basics of the formalism by applying the standard variational method for maximizing the entropy subject to the average energy and normalization constraints. The analytical results show (i) that the non-extensive thermodynamics formalism should be called into question to explain experimental results described by extended exponential distributions exhibiting long tails, i.e. $q$-exponentials with $q>1$, and (ii) that in the thermodynamic limit the theory is only consistent in the range $0\leq q\leq1$ where the distribution has finite support, thus implying that configurations with e.g. energy above some limit have zero probability, which is at variance with the physics of systems in contact with a heat reservoir. We also discuss the ($q$-dependent) thermodynamic temperature and the generalized specific heat.