Consider the harmonic oscillators from exercise 4, connected to a reservoir of temperature \(T\) and chemical potential \(\mu\). Calculate (in the grand canonical ensemble) the thermodynamic potential \(\Omega\) and the heat capacity \(C_{V}\).

Data From Exercise 4:-

Consider \(N\) harmonic oscillators of frequencies \(\omega_{i}, i=1, \ldots, N\), connected to a reservoir of temperature. Calculate (in the canonical ensemble) the free energy and the heat capacity \(C_{V}\).