Entropies for Coupled Harmonic Oscillators and Temperature

We study two entropies of a system composed of two coupled harmonic oscillators which is brought to a canonical thermal equilibrium with a heat-bath at temperature $T$. Using the purity function, we explicitly determine the R\'enyi and van Newmon entropies in terms of different physical parameters. We will numerically analyze these two entropies under suitable conditions and show their relevance.