A classical particle of mass $m$ moving non-relativistically in two-dimensional space is enclosed inside a circle of radius $R$ and attached by a spring with constant $\kappa$ to the centre of the circle. The particle thus moves in a potential

$$V(r)= \begin{cases}\frac{1}{2} \kappa r^{2} & \text { for } r<R \\ \infty & \text { for } r \geqslant R\end{cases}$$

where $r^{2}=x^{2}+y^{2}$. Let the particle be coupled to a heat reservoir at temperature $T$.

(i) Which of the ensembles of statistical physics should be used to model the system?

(ii) Calculate the partition function for the particle.

(iii) Calculate the average energy $\langle E\rangle$ and the average potential energy $\langle V\rangle$ of the particle.

(iv) What is the average energy in:

(a) the limit $\frac{1}{2} \kappa R^{2} \gg k_{\mathrm{B}} T$ (strong coupling)?

(b) the limit $\frac{1}{2} \kappa R^{2} \ll k_{\mathrm{B}} T$ (weak coupling)?

Compare the two results with the values expected from equipartition of energy.