The Fermi Distribution
The graph below shows the occupation index F(E), as a function of the energy in terms of multiples of the (Fermi Energy - rest mass energy), ie. the kinetic energy corresponding to the Fermi Energy. The slider controls the "degeneracy parameter" - the ratio Ef'/kT = (Ef-mc^2)/kT. Notice how the distribution approaches the T=0 rectangular, completely degenerate case as the degeneracy parameter increases. In practice, even for degeneracy parameters of ~100, there are still some Fermions above the Fermi energy (where x=1) and some "gaps" below. A solid bar above the curve shows how the width of the "roll-off" region is of order kT in size. A gas of Fermions can be made degenerate either by increasing the Fermi Energy (by increasing the number density of Fermions), or by decreasing the temperature.
The plots are only valid for a degeneracy parameter quite a bit bigger than 1 and use the "Sommerfeld Expansion" to evaluate the chemical potential μ . F(E) = 1/(exp(E-μ)/kT + 1), Where in this version (appropriate for astrophysics), the energy E includes the rest mass energy.