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.