(a) What systems are described by microcanonical, canonical and grand canonical ensembles? Under what conditions is the choice of ensemble irrelevant?

(b) In a simple model a meson consists of two quarks bound in a linear potential, $U(\mathbf{r})=\alpha|\mathbf{r}|$, where $\mathbf{r}$ is the relative displacement of the two quarks and $\alpha$ is a positive constant. You are given that the classical (non-relativistic) Hamiltonian for the meson is

$$H(\mathbf{P}, \mathbf{R}, \mathbf{p}, \mathbf{r})=\frac{|\mathbf{P}|^{2}}{2 M}+\frac{|\mathbf{p}|^{2}}{2 \mu}+\alpha|\mathbf{r}|$$

where $M=2 m$ is the total mass, $\mu=m / 2$ is the reduced mass, $\mathbf{P}$ is the total momentum, $\mathbf{p}=\mu d \mathbf{r} / d t$ is the internal momentum, and $\mathbf{R}$ is the centre of mass position.

(i) Show that the partition function for a single meson in thermal equilibrium at temperature $T$ in a three-dimensional volume $V$ can be written as $Z_{1}=Z_{\text {trans }} Z_{\text {int }}$, where

$$Z_{\text {trans }}=\frac{V}{(2 \pi \hbar)^{3}} \int d^{3} P e^{-\beta|\mathbf{P}|^{2} /(2 M)}, \quad Z_{\text {int }}=\frac{1}{(2 \pi \hbar)^{3}} \int d^{3} r d^{3} p e^{-\beta|\mathbf{p}|^{2} /(2 \mu)} e^{-\beta \alpha|\mathbf{r}|}$$

and $\beta=1 /\left(k_{\mathrm{B}} T\right)$

Evaluate $Z_{\text {trans }}$ and evaluate $Z_{\text {int }}$ in the large-volume limit $\left(\beta \alpha V^{1 / 3} \gg 1\right)$.

What is the average separation of the quarks within the meson at temperature $T$ ?

$\left[\right.$ You may assume that $\int_{-\infty}^{\infty} e^{-c x^{2}} d x=\sqrt{\pi / c}$ for $c>0$ ]

(ii) Now consider an ideal gas of $N$ such mesons in a three-dimensional volume $V$.

Calculate the total partition function of the gas.

What is the heat capacity $C_{V} ?$