Degenerate Fermi gas (ideal Fermi gas at low temperature)

Ground-state thermodynamics and low-temperature expansions of the ideal Fermi gas: Fermi energy, pressure, and heat capacity.

Degenerate Fermi gas (ideal Fermi gas at low temperature)

Example: ideal Fermi gas in $d=3$

Consider $N$ noninteracting spin- $1/2$ fermions of mass $m$ in volume $V$ at number density $n=N/V$ . The system is naturally treated in the canonical ensemble ; at low temperature it is “degenerate,” meaning $T\ll T_F$ (defined below).

Zero-temperature (ground-state) formulas

The Fermi momentum and Fermi energy are

k_F=(3\pi^2 n)^{1/3}, \qquad \varepsilon_F=\frac{\hbar^2 k_F^2}{2m} =\frac{\hbar^2}{2m}(3\pi^2 n)^{2/3}.

Define the Fermi temperature $T_F=\varepsilon_F/k_B$ .

At $T=0$ (filled Fermi sea),

\frac{U_0}{N}=\frac{3}{5}\varepsilon_F, \qquad P_0=\frac{2}{5}n\,\varepsilon_F.

This finite pressure at $T=0$ is the “degeneracy pressure.”

Low-temperature corrections (Sommerfeld behavior)

For $T\ll T_F$ , the leading temperature corrections are quadratic in $T$ :

\frac{U}{N}=\frac{3}{5}\varepsilon_F\left[1+\frac{5\pi^2}{12}\left(\frac{T}{T_F}\right)^2+o\!\left(\frac{T}{T_F}\right)^2\right],

and the constant-volume heat capacity is linear in $T$ :

C_V \sim \frac{\pi^2}{2}N k_B \frac{T}{T_F}.

Thermodynamic interpretation

The pressure and internal energy follow from derivatives of the pressure (log-partition density) / free energy density in the thermodynamic limit.
The low- $T$ linear behavior of $C_V$ is a characteristic signature of Fermi statistics.

Example: ideal Fermi gas in d=3d=3d=3

Zero-temperature (ground-state) formulas

Low-temperature corrections (Sommerfeld behavior)

Thermodynamic interpretation

Prerequisites / cross-links

Example: ideal Fermi gas in $d=3$