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

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.