Show that, quite generally, the low-temperature behavior of the chemical potential, the specific heat, and the entropy of an ideal Fermi gas are given by

\[\mu \simeq \varepsilon_{F}\left[1-\frac{\pi^{2}}{6}\left(\frac{\partial \ln a(\varepsilon)}{\partial \ln \varepsilon}\right)_{\varepsilon=\varepsilon_{F}}\left(\frac{k T}{\varepsilon_{F}}\right)^{2}\right]\]

and

\[C_{V} \simeq S \simeq \frac{\pi^{2}}{3} k^{2} T a\left(\varepsilon_{F}\right)\]

where \(a(\varepsilon)\) is the density of (the single-particle) states in the system. Examine these results for a gas with energy spectrum \(\varepsilon \propto p^{s}\), confined to a space of \(n\) dimensions, and discuss the special cases: \(s=1\) and 2, with \(n=2\) and 3 .