Fermi-Dirac Statistics

Fermi-Dirac statistics are one of two kinds of statistics exhibited by!identical quantum particles, the other being !Bose-Einstein statistics. Such particles are called fermions and bosons respectively (the terminology is due to Dirac [1902-1984] [1]). In the light of the !spin-statistics theorem, and consistent with observation, fermions are invariably spinors (of half-integral spin), whilst bosons are invariably scalar or vector particles (of integral spin). See !spin. In general, in quantum mechanics, the available states of a homogeneous many-particle system in thermal equilibrium, for given total energy, are counted as equiprobable. For systems of exactly similar (‘identical’) fermions or bosons, states which di¤er only in the permutation of two or more particles are not only counted as equiprobable –they are identi…ed (call this permutivity). Fermions di¤er from bosons in that no two fermions can be in exactly the same 1-particle state. This further restriction follows from the!Pauli exclusion principle. The thermodynamic properties of gases of such particles were …rst worked out by Fermi [1901-1954] in 1925 [2], and, independently, by Dirac in 1926 [3]. To understand the consequences of these two restrictions, consider a system of N weakly-interacting identical particles, with states given by the various 1particle energies s together with their degeneracies–the number Cs of distinct 1-particle states of each energy s. From permutivity, the total state of a gas is fully speci…ed by giving the number of particles with energy s in each of the Cs possible states, i.e. by giving the occupation numbers nk for each s, k = 0; 1; ; ; ; Cs. We suppose all possible states of the same total energy E and, supposing particle number is conserved, of the same total number N; are available to the N particles when in thermal equilibrium, i.e. all sets of occupations numbers that satisfy: