Thermodynamic quantities of independent harmonic oscillators in microcanonical and canonical ensembles in the Tsallis statistics

TL;DR

This paper analyzes the thermodynamic properties of independent harmonic oscillators within Tsallis statistics, comparing microcanonical and canonical ensembles, and deriving conditions under which their entropies differ.

Contribution

It provides explicit expressions for energy and entropy in both ensembles in Tsallis statistics, highlighting the ensemble differences based on the parameters N and q.

Findings

Entropy in the canonical ensemble depends on q, unlike in microcanonical ensemble.

The condition N(q-1)<1 determines the similarity of entropies between ensembles.

For N(q-1)<1, entropy differences are minimal.

Abstract

We study the energy and entropies for $N$ independent harmonic oscillators in the microcanonical and the canonical ensembles in the Tsallis classical and the Tsallis quantum statistics of entropic parameter $q$, where $N$ is the number of the oscillators and the value of $q$ is larger than one. The energy and entropies are represented with the physical temperature, and the well-known expressions are obtained for the energy and R\'enyi entropy. The difference between the microcanonical and the canonical ensembles is the existence of the condition for $N$ and $q$ in the canonical ensemble: $N(q-1)<1$. The condition does not appear in the microcanonical ensemble. The entropies are $q$-dependent in the canonical ensemble, and are not $q$-dependent in the microcanonical ensemble. For $N(q-1)<1$, this difference in entropy is quite small, and the entropy in the canonical ensemble does not differ from the entropy in the microcanonical ensemble substantially.