(i) The pressure $P(r)$ and mass density $\rho(r)$, at distance $r$ from the centre of a spherically-symmetric star, obey the pressure-support equation

$$P^{\prime}=-\frac{G m \rho}{r^{2}}$$

where $m^{\prime}=4 \pi r^{2} \rho(r)$, and the prime indicates differentiation with respect to $r$. Let $V$ be the total volume of the star, and $\langle P\rangle$ its average pressure. Use the pressure-support equation to derive the "virial theorem"

$$\langle P\rangle V=-\frac{1}{3} E_{g r a v}$$

where $E_{g r a v}$ is the total gravitational potential energy [Hint: multiply by $4 \pi r^{3}$ ]. If a star is assumed to be a self-gravitating ball of a non-relativistic ideal gas then it can be shown that

$$\langle P\rangle V=\frac{2}{3} E_{k i n}$$

where $E_{k i n}$ is the total kinetic energy. Use this result to show that the total energy $U=E_{\text {grav }}+E_{k i n}$ is negative. When nuclear reactions have converted the hydrogen in a star's core to helium the core contracts until the helium is converted to heavier elements, thereby increasing the total energy $U$ of the star. Explain briefly why this converts the star into a "Red Giant". (ii) Write down the first law of thermodynamics for the change in energy $E$ of a system at temperature $T$, pressure $P$ and chemical potential $\mu$ as a result of small changes in the entropy $S$, volume $V$ and particle number $N$. Use this to show that

$$P=-\left(\frac{\partial E}{\partial V}\right)_{N, S}$$

The microcanonical ensemble is the set of all accessible microstates of a system at fixed $E, V, N$. Define the canonical and grand-canonical ensembles. Why are the properties of a macroscopic system independent of the choice of thermodynamic ensemble?

The Gibbs "grand potential" $\mathcal{G}(T, V, \mu)$ can be defined as

$$\mathcal{G}=E-T S-\mu N$$

Use the first law to find expressions for $S, P, N$ as partial derivatives of $\mathcal{G}$. A system with variable particle number $n$ has non-degenerate energy eigenstates labeled by $r^{(n)}$, for each $n$, with energy eigenvalues $E_{r}^{(n)}$. If the system is in equilibrium at temperature $T$ and chemical potential $\mu$ then the probability $p\left(r^{(n)}\right)$ that it will be found in a particular $n$-particle state $r^{(n)}$ is given by the Gibbs probability distribution

$$p\left(r^{(n)}\right)=\mathcal{Z}^{-1} e^{\left(\mu n-E_{r}^{(n)}\right) / k T}$$

where $k$ is Boltzmann's constant. Deduce an expression for the normalization factor $\mathcal{Z}$ as a function of $\mu$ and $\beta=1 / k T$, and hence find expressions for the partial derivatives

$$\frac{\partial \log \mathcal{Z}}{\partial \mu}, \quad \frac{\partial \log \mathcal{Z}}{\partial \beta}$$

in terms of $N, E, \mu, \beta$.

Why does $\mathcal{Z}$ also depend on the volume $V$ ? Given that a change in $V$ at fixed $N, S$ leaves unchanged the Gibbs probability distribution, deduce that

$$\left(\frac{\partial \log \mathcal{Z}}{\partial V}\right)_{\mu, \beta}=\beta P$$

Use your results to show that

$$\mathcal{G}=-k T \log \left(\mathcal{Z} / \mathcal{Z}_{0}\right)$$

for some constant $\mathcal{Z}_{0}$.