A system of $N$ identical spinless bosons of mass $m$ is in a box of volume $V=L^3$ at temperature $T>0$.
(a) Write a general expression for the number of particles, $n(E)$, having an energy between $\varepsilon$ and $\varepsilon+d \varepsilon$ in terms of their mass, the energy, the temperature, the chemical potential, the volume, and any other relevant quantities.
(b) Show that in the limit that the average distance, $d$, between the particles is very large compared to their de Broglie wavelength (i.e., $d \gg$ $\lambda)$ the distribution becomes equal to that calculated using the classical (Boltzmann) distribution function.
(c) Calculate the 1st order difference in average energy between a system of $N$ non-identical spinless particles and a system of $N$ identical spinless bosons when $d \gg \lambda$. For both systems the cubical box has volume $\dot{V}=L^3$ and the particles have mass $m$.