The Fermi-Dirac distribution function, also called Fermi function, provides the probability of occupancy of energy levels by Fermions. Fermions are half-integer spin particles, which obey the Pauli exclusion principle. The Fermi-function is given by
$\mathrm{f}\left(\mathrm{E}\right)=\frac{1}{1+{\mathrm{e}}^{\frac{\left(\mathrm{E}-{\mathrm{E}}_{\mathrm{F}}\right)}{\mathrm{kT}}}}$
Where k is Boltmann constant, T is temperature in kelvin, f(E) is the probability that an energy level at energy 'E' in thermal equilibrium with a large system is occupied by a Fermion and E_F is Fermi energy given by
${\mathrm{E}}_{\mathrm{F}}=\frac{\mathrm{ħ}}{2{\mathrm{m}}_0}{\left(\frac{3{\mathrm{\pi }}^2{\mathrm{N}}_{\mathrm{f}}}{\mathrm{V}}\right)}^{\frac{2}{3}},$
Where N_f is number of fermions, m₀ the rest mass of each fermion, V is volume of system and h is reduced Planck constant $\left(\frac{\mathrm{h}}{2\mathrm{\pi }}\right)$
Towards the end of a star, depending on the mass of star, it can undergo a red giant phase or a supernova explosion. The first results in the formation of white dwarf star where helium fuses to form heavier element while the later results in the formation of neutron star (if mass of star > 1.5 times mass of sun)
In case of white dwarf, the average electron energy is $\frac{3}{5}{\mathrm{E}}_{\mathrm{F}},$ and therefore the total electron energy is $\frac{3}{5}{\mathrm{N}}_{\mathrm{f}}{\mathrm{E}}_{\mathrm{F}}.$ The total energy of the white dwarf, which we assume to be spherical, of uniform density and at a constant temperature, is
$\mathrm{E}=\frac{3}{5}{\mathrm{N}}_{\mathrm{f}}\frac{{\mathrm{ħ}}^2}{2{\mathrm{m}}_{\mathrm{e}}}{\left(\frac{3{\mathrm{\pi }}^2{\mathrm{N}}_{\mathrm{f}}}{\mathrm{V}}\right)}^{\frac{2}{3}}-\frac{3}{5}\frac{{\mathrm{GM}}^2}{\mathrm{R}}+\frac{3}{2}{\mathrm{N}}_{\mathrm{a}}\mathrm{kT}+{\mathrm{E}}_{\mathrm{rad}}$
The second term is the gravitational potential energy, the third term takes into account the thermal motions of the atoms (N_a = number of atoms), and the fourth term gives the energy radiated by the star. At equilibrium energy of the star must be minimum with respect to radius.