On the energy translation invariance of probability distributions

TL;DR

This paper discusses the conditions under which probability distributions are invariant under energy translation, highlighting limitations and implications for nonextensive statistical mechanics and thermodynamics.

Contribution

It clarifies that energy translation invariance is not universal and depends on specific conditions, impacting the understanding of nonextensive statistical mechanics.

Findings

Invariance occurs only without long-term interactions or relativistic effects.

Energy translation invariance is not a universal property.

Invariance may affect the connection between nonextensive statistics and thermodynamics.

Abstract

We comment on the problem of energy translation invariance of probability distribution and present some observations. It is shown that a probability distribution can be invariant in the thermodynamic limit if there is no long term interaction or correlation and no relativistic effect. So this invariance should not be considered as a universal theoretical property. Some peculiarities within the invariant $q$-exponential distribution reveal that the connection of the current nonextensive statistical mechanics to thermodynamics might be disturbed by this invariance.