A set of N classical oscillators in one dimension is given by the Hamiltonian

A set of N classical oscillators in one dimension is given by the Hamiltonian

Using the formalism of the canonical ensemble in classical phase space, obtain expressions for the partition function, the energy per oscillator, the entropy per oscillator, and the specific heat. Compare with the results from the classical limit of the quantum oscillator. Calculate an expression for the quadratic deviation of the energy as a function of temperature.