Purity temperature dependency for coupled harmonic oscillator

We consider the thermal aspect of a system composed of two coupled harmonic oscillators and study the corresponding purity. We initially consider a situation where the system is brought to a canonical thermal equilibrium with a heat-bath at temperature T. We adopt the path integral approach and introduce the evolution operator to calculate the density matrix and subsequently the reduced matrix density. It is used to explicitly determine the purity in terms of different physical quantities and therefore study some limiting cases related to temperature as well as other parameters. Different numerical results are reported and discussed in terms of the involved parameters of our system. (paper: quantum statistical physics, condensed matter, integrable systems)