Given that the free energy $F$ can be written in terms of the partition function $Z$ as $F=-k T \log Z$ show that the entropy $S$ and internal energy $E$ are given by

$$S=k \frac{\partial(T \log Z)}{\partial T}, \quad E=k T^{2} \frac{\partial \log Z}{\partial T}$$

A system of particles has Hamiltonian $H(\mathbf{p}, \mathbf{q})$ where $\mathbf{p}$ is the set of particle momenta and $\mathbf{q}$ the set of particle coordinates. Write down the expression for the classical partition function $Z_{C}$ for this system in equilibrium at temperature $T$. In a particular case $H$ is given by

$$H(\mathbf{p}, \mathbf{q})=p_{\alpha} A_{\alpha \beta}(\mathbf{q}) p_{\beta}+q_{\alpha} B_{\alpha \beta}(\mathbf{q}) q_{\beta}$$

Let $H$ be a homogeneous function in all the $p_{\alpha}, 1 \leq \alpha \leq N$, and in a subset of the $q_{\alpha}, 1 \leq \alpha \leq M(M \leq N)$. Derive the principle of equipartition for this system.

A system consists of $N$ weakly interacting harmonic oscillators each with Hamiltonian

$$h(p, q)=\frac{1}{2} p^{2}+\frac{1}{2} \omega^{2} q^{2} .$$

Using equipartition calculate the classical specific heat of the system, $C_{C}(T)$. Also calculate the classical entropy $S_{C}(T)$.

Write down the expression for the quantum partition function of the system and derive expressions for the specific heat $C(T)$ and the entropy $S(T)$ in terms of $N$ and the parameter $\theta=\hbar \omega / k T$. Show for $\theta \ll 1$ that

$$C(T)=C_{C}(T)+O(\theta), \quad S(T)=S_{C}(T)+S_{0}+O(\theta)$$

where $S_{0}$ should be calculated. Comment briefly on the physical significance of the constant $S_{0}$ and why it is non-zero.