Average height of the particles in a monoatomic ideal gas when $T=0$ Consider a classical ideal monoatomic gas (made of $N$ particles in the canonical ensemble) confined in a prism with infinite height. Consider the hamiltonian of a particle as
$$H = \dfrac{p^2}{2m} + \Phi(z)\quad, $$
where $\Phi(z) = \Phi_0 \log \left( 1 + \dfrac{z}{z_0} \right)$ is a potential depending on the height of the particle, $z$, with $z > 0$. What happens to the average value of height for a single particle when we approach the absolute zero? Will the particles be at $z \approx 0$, as it's the state that minimizes the potential, having then $\langle z \rangle \approx 0$?
 A: 
Consider a classical ideal monoatomic gas... [snip stuff about bosons]... Consider the hamiltonian of a particle as
$$H = \dfrac{p^2}{2m} + \Phi(z)\quad, $$
where $\Phi(z) = \Phi_0 \log \left( 1 + \dfrac{z}{z_0} \right)$ is a potential depending on the height of the particle, $z$.


What happens to the average value of height for a single particle when we approach the absolute zero?

Starting from:
$$
\langle z \rangle =\frac{\int_0^\infty z e^{-\beta \Phi(z)}dz}{\int_0^\infty e^{-\beta \Phi(z)} dz} 
=
\frac{\int_0^\infty z\frac{1}{(1+\frac{z}{z_0})^{\beta \Phi_0}}dz}{\int_0^\infty \frac{1}{(1+\frac{z}{z_0})^{\beta \Phi_0}} dz}\;,
$$
where $\beta = \frac{1}{kT}$.
Assuming that $\frac{\Phi_0}{kT}$ is very large compared to 1, we have:
$$
\langle z \rangle\sim z_0\frac{kT}{\Phi_0}\to 0
$$

Will the particles be at $z \approx 0$, as it's the state that minimizes the potential, having then $\langle z \rangle \approx 0$?

Yes, assuming that we are limited to $z\ge 0$, we have $\langle z \rangle \to 0$.
