Consider an non-interacting gas of identical fermions (e.g. electrons \(e^-\)); this is called an ideal Fermi gas. Because the Pauli exclusion principle prohibits identical fermions from occupying the same quantum state, the grand canonical partition function \(\mathcal Z\) for an ideal Fermi gas is just:
\[\mathcal Z=\prod_{|k\rangle\in\mathcal H_0}\sum_{N_k=0,1}e^{-\beta(N_kE_k-\mu N_k)}=\prod_{|k\rangle\in\mathcal H_0}\left(1+e^{-\beta(E_k-\mu)}\right)\]
From which the grand canonical potential is:
\[\Phi=-\frac{1}{\beta}\ln\mathcal Z=-\frac{1}{\beta}\sum_{|k\rangle\in\mathcal H_0}\ln\left(1+e^{-\beta(E_k-\mu)}\right)\]
And the average number of fermions is:
\[\langle N\rangle=-\frac{\partial\Phi}{\partial\mu}=\sum_{|k\rangle\in\mathcal H_0}\frac{1}{e^{\beta(E_k-\mu)}+1}\]
from which one immediately reads off the Fermi-Dirac distribution of the Fermi occupation numbers of each of the single-fermion states \(|k\rangle\):
\[\langle N_k\rangle=\frac{1}{e^{\beta(E_k-\mu)}+1}\]
It is remarkable that a mere sign change in the denominator from the Bose-Einstein distribution is all that is needed to enforce the Pauli exclusion principle. Unlike for the ideal Bose gas where the chemical potential \(\mu<0\) had to be negative, for the Fermi-Dirac distribution \(\mu\in\textbf R\) can be anything.
Just as with the ideal Bose gas, for an ideal Fermi gas one would like to approximate the series with integrals (called the Thomas-Fermi approximation) \(\sum_{|k\rangle\in\mathcal H_0}\mapsto\int_0^{\infty}g(E)dE\). Taking the ideal Fermi gas to be non-relativistic, one has the density of states:
\[g(E)=\frac{g_sm^{3/2}V}{\sqrt{2}\pi^2\hbar^3}\sqrt{E}\]
where \(g_s=2s+1\) is a spin degeneracy factor (which has to be explicitly included for fermions by virtue of the spin-statistics theorem \(s=1/2,3/2,5/2,…\) and the fact that the free Hamiltonian \(H=T\) commutes with \(\textbf S^2\)). In the grand canonical ensemble, one thus has for an ideal Fermi gas:
\[\Phi=\frac{g_sV}{\beta\lambda^3}\text{Li}_{5/2}(-z)\]
\[\langle N\rangle=-\frac{g_sV}{\lambda^3}\text{Li}_{3/2}(-z)\]
\[\langle E\rangle=-\frac{3g_sV}{2\beta\lambda^3}\text{Li}_{5/2}(-z)\]
from which one obtains \(pV=\frac{2}{3}E\) for an ideal Fermi gas as was the case for the ideal Bose gas (and the ideal classical gas). In the high-temperature \(T\to\infty\) limit \(z\to 0\), one finds that, similar to the ideal Bose gas, the ideal Fermi gas looks like an ideal classical gas, at least to first order in the virial expansion (at second order, the quantum correction actually increases the pressure of the ideal Fermi gas whereas it was decreasing for the ideal Bose gas):
\[pV=NkT\left(1+\frac{\lambda^3N}{4\sqrt{2}g_sV}+O\left(\frac{N}{V}\right)^2\right)\]
In order to see more interesting, non-classical physics, it will as usual be necessary to look in the low-temperature limit \(T\to 0,z\to 1\). In fact, to start, one may as well look directly at the case of absolute zero \(T=0\). In this case, the ideal Fermi gas is said to be degenerate. At a glance, this is because the Fermi-Dirac distribution for the Fermi occupation numbers reduces to a top-hat filter:
\[N_k=\frac{1}{e^{\beta(E_k-\mu)}+1}=[E_k<\mu]\]
One can define the Fermi energy by \(E_F:=\mu(T=0)\) so that states \(|k\rangle\) with \(\hbar^2k^2/2m<E_F\) lying in the Fermi sea are fully occupied (i.e. have Fermi occupation number of \(N_k=1\)) while states \(|k\rangle\) with \(\hbar^2k^2/2m>E_F\) lying beyond the Fermi surface are completely empty. This definition of the Fermi energy \(E_F\) is strictly speaking a bit misleading since in the grand canonical ensemble \(\mu\) and \(T\) are independent and fixed while \(N\) fluctuates; in practice \(N\) is fixed and both \(\mu\) and \(T\) fluctuate in a way to keep \(N\) fixed so that working in the grand canonical ensemble is just a mathematical convenience. Therefore, it would make more sense to express/define \(E_F\) in terms of the fixed number \(N\) of fermions in the degenerate ideal Fermi gas:
\[N=\sum_{|k\rangle\in\mathcal H_0}N_k=\int_0^{\infty}[E<E_F]g(E)dE=\int_0^{E_F}g(E)dE\Rightarrow E_F=\frac{\hbar^2}{2m}\left(\frac{6\pi^2 N}{g_sV}\right)^{2/3}\]
This is of course related to the Fermi momentum and Fermi temperature by \(E_F=\hbar^2k_F^2/2m=kT_F\). The Fermi temperature \(T_F\) for the ideal Fermi gas determines whether the ideal Fermi gas is in the high-temperature \(T>T_F\) regime or the low-temperature \(T<T_F\) regime. For example, in a copper \(\text{Cu(s)}\) wire the number density of electrons \(e^-\) is \(N/V\approx 8.5\times 10^{28}\text{ m}^{-3}\), so the corresponding Fermi temperature is actually quite hot \(T_F\approx 8.2\times 10^4\text{ K}\) by everyday standards, and so in particular room temperature \(T\approx 300\text{ K}\ll T_F\) means that the electrons \(e^-\) in metals can be thought of to a good approximation as degenerate \(T=0\) Fermi gases.
Having computed the total number of fermions \(N=\langle N\rangle\), one can also compute the total energy \(E=\langle E\rangle\) in the grand canonical ensemble:
\[E=\sum_{|k\rangle\in\mathcal H_0}N_kE_k=\int_0^{\infty}[E<E_F]Eg(E)dE=\int_0^{E_F}Eg(E)dE=\frac{3}{5}NE_F\]
which is pretty intuitive, the factor of \(3/5\) essentially just coming from the average of \(k^2\) in a ball of radius \(k_F\), i.e. \(\frac{3}{4\pi k_F^3}\int_0^{k_F}k^24\pi k^2dk=\frac{3}{5}k_F^2\).
Finally, the “equation of state” \(pV=\frac{2}{3}E\) earlier yields the corresponding degeneracy pressure:
\[pV=\frac{2}{5}NE_F\]
For comparison, recall that below the critical temperature \(T<T_c\) the pressure \(p\sim T^{5/2}\) of a BEC approached \(p\to 0\) as \(T\to 0\); not so for an ideal Fermi gas. For both the ideal Bose and Fermi gases, \(pV=\frac{2}{3}E\) but because bosons can condense to the \(E=0\) ground state, their pressure \(p\) also drops to \(p\to 0\), however fermions cannot do this because of the Pauli exclusion principle (they are forced to fill out a Fermi sea instead), so their total energy \(E=\frac{3}{5}NE_F\) can never reach zero, and therefore their pressure \(p\) also cannot reach \(p\to 0\), leaving this residual \(T=0\) degeneracy pressure \(p=\frac{2}{5}\frac{N}{V}E_F>0\).
Finally, it is worth asking more generally just about the physics of an ideal Fermi gas not necessarily when it is degenerate at \(T=0\), but merely at some “low” temperature \(T\ll T_F\). Here, “physics” shall mean “low-temperature heat capacity” \(C_V=C_V(T)\).
In this case, the Fermi-Dirac distribution will be distorted from the degenerate \(T=0\) top-hat filter into a distribution that looks like:
The key observation is that only fermions close to the Fermi surface, specifically whose energy is within \(kT\) of the Fermi energy \(E_F\) can respond to any additional energy added to the ideal Fermi gas, and therefore contribute to the heat capacity \(C_V\) (since only they notice the non-degenerate temperature \(T>0\), the rest of the fermions being locked in the Fermi sea by the Pauli exclusion principle).
\[C_V=\frac{\partial E}{\partial T}=-\frac{3g_sV}{2}\frac{\partial}{\partial T}\left(\frac{1}{\beta\lambda^3}\text{Li}_{5/2}(-z)\right)\]
At this point, invoke the behavior of the polylogarithm as the fugacity \(z\to 1\) in the low-\(T\) limit (called the Sommerfeld expansion, essentially just a lot of binomial expansions):
\[-\text{Li}_{s}(-z)=\frac{(\ln z)^s}{\Gamma(s+1)}\left(1+\frac{\pi^2}{6}\frac{s(s-1)}{(\ln z)^2}+…\right)\]
where \(\ln z=\beta\mu\), so this simplifies to:
\[C_V\approx \frac{\sqrt{2}g_sm^{3/2}V}{5\pi^2\hbar^3}\frac{\partial}{\partial T}\left(\mu^{5/2}\left(1+\frac{5\pi^2}{8\beta^2\mu^2}\right)\right)\]
Here comes the subtle point; although \(\mu\) is supposed to be independent of \(T\) in the grand canonical ensemble with the expected total number of particles \(N\) determined through \(\mu\) and \(T\), in practice the number of fermions \(N\) is fixed so \(\mu=\mu(T)\) is implicitly a function of \(T\) in order to keep \(N\) fixed. This is why one cannot for instance just factor the \(\mu^{5/2}\) outside the \(\partial/\partial T\), etc. The resolution here is to write \(\mu\) in terms of \(N\) (or equivalently, in terms of the Fermi energy \(E_F\)) which would be fixed and therefore easy to deal with;
\[N\approx\frac{\sqrt{2}g_sm^{3/2}V}{3\pi^2\hbar^3}\mu^{3/2}\left(1+\frac{\pi^2}{8\beta^2\mu^2}\right)\]
This immediately implies:
\[E_F\approx \mu\left(1+\frac{\pi^2}{8\beta^2\mu^2}\right)^{2/3}\]
Or, isolating \(\mu\) to suitable order:
\[\mu=E_F\left(1+\frac{\pi^2}{8\beta^2\mu^2}\right)^{-2/3}\approx E_F\left(1-\frac{\pi^2}{12\beta^2\mu^2}\right)\approx E_F\left(1-\frac{\pi^2}{12\beta^2E_F^2}\right)\]
Finally, it is clear that one can re-express the heat capacity in terms of \(N\) and \(E_F\) (the fixed variables) as:
\[C_V=\frac{3N}{5}\frac{\partial}{\partial T}\left(\mu\frac{1+5\pi^2k^2T^2/8\mu^2}{1+\pi^2k^2T^2/8\mu^2}\right)\approx\frac{3NE_F}{5}\frac{\partial}{\partial T}\left(1-\left(\frac{5}{8}-\frac{1}{8}-\frac{1}{12}\right)\frac{\pi^2}{\beta^2E_F^2}\right)\]
leading to the linear heat capacity behavior of the low-\(T\) ideal Fermi gas:
\[C_V=\frac{\pi^2}{2}Nk\frac{T}{T_F}\]
Ignoring the \(\pi^2/2\) prefactor which came from the detailed Sommerfeld expansion of the polylogarithms, there is a simple intuitive way to understand this formula: the number of Fermi surface fermions living within \(kT\) of the Fermi energy \(E_F\) is \(g(E_F)kT\) and the energy of each fermion is of order \(kT\) so the total energy of all Fermi surface fermions is \(E\sim g(E_F)(kT)^2\). If one adds some energy \(dE\) into the ideal Fermi gas, then essentially all this energy has to go into the Fermi surface fermions so that one may legitimately equate \(dE\sim g(E_F)k^2TdT\) reproducing the linear heat capacity:
\[C_V\sim g(E_F)k^2T\sim E_F^{1/2}k^2T\sim N^{1/3}k^2T\sim Nk\frac{T}{T_F}\]
The theory of ideal Fermi gases has diverse applications, ranging from electrons \(e^-\) in a conductor (as justified by Landau’s Fermi liquid theory) to astrophysics (e.g. white dwarf stars are supported by electron degeneracy pressure, neutron stars are supported by neutron degeneracy pressure, thanks to the fact that both electrons \(e^-\) and neutrons \(n^0\) are fermions) to Pauli paramagnetism and Landau diamagnetism in condensed matter physics.