Write down the partition function for a single classical non-relativistic particle of mass $m$ moving in three dimensions in a potential $U(\mathbf{x})$ and in equilibrium with a heat bath at temperature $T$.

A system of $N$ non-interacting classical non-relativistic particles, in equilibrium at temperature $T$, is placed in a potential

$$U(\mathbf{x})=\frac{\left(x^{2}+y^{2}+z^{2}\right)^{n}}{V^{2 n / 3}}$$

where $n$ is a positive integer. Using the partition function, show that the free energy is

$$F=-N k_{B} T\left(\log V+\frac{3}{2} \frac{n+1}{n} \log k_{B} T+\log I_{n}+\text { const }\right)$$

where

$$I_{n}=\left(\frac{m}{2 \pi \hbar^{2}}\right)^{3 / 2} \int_{0}^{\infty} 4 \pi u^{2} \exp \left(-u^{2 n}\right) d u$$

Explain the physical relevance of the constant term in the expression $(*)$.

Viewing $V$ as an external parameter, akin to volume, compute the conjugate pressure $p$ and show that the equation of state coincides with that of an ideal gas.

Compute the energy $E$, heat capacity $C_{V}$ and entropy $S$ of the gas. Determine the local particle number density as a function of $|\mathbf{x}|$.