Fermi-Dirac statistics

(statistical mechanics) The statistics of an assembly of identical half-integer spin particles; such particles have wave functions antisymmetrical with respect to particle interchange and satisfy the Pauli exclusion principle.

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The statistical description of particles or systems of particles that satisfy the Pauli exclusion principle. This description was first given by E. Fermi, who applied the Pauli exclusion principle to the translational energy levels of a system of electrons. It was later shown by P. A. M. Dirac that this form of statistics is also obtained when the total wave function of the system is antisymmetrical. See also Exclusion principle.

Such a system is described by a set of occupation numbers {n_i} which specify the number of particles in energy levels ε_i. It is important to keep in mind that ε_i represents a finite range of energies, which in general contains a number, say g_i, of nondegenerate quantum states. In the Fermi statistics, at most one particle is allowed in a nondegenerate state. (If spin is taken into account, two particles may be contained in such a state.) This is simply a restatement of the Pauli exclusion principle, and means that n_i ≤ g_i. The probability of having a set {n_i} distributed over the levels ε_i, which contain g_i nondegenerate levels, is described by Eq. (1), which gives just the number
1.
of ways that n_i can be picked out of g_i, which is intuitively what one expects for such a probability. The equilibrium state which actually exists is the set of n's that makes W a maximum, under the auxiliary conditions given in Eqs. (2a) and (2b).
2a.

2b.
These conditions express the fact that the total energy E and the total number of particles N are given. Equation (3)
3.
holds for this most probable distribution. Here A and β are parameters, to be determined from Eq. (3); in fact, β = 1/kT, where k is Boltzmann's constant and T is the absolute temperature. When the 1 in the denominator may be neglected, Eq. (3) goes over into the Boltzmann distribution.

Classical conditions pertain when the volume per particle is much larger than the volume associated with the de Broglie wavelength λ of a particle. For electrons in a metal at 300 K, the ratio of the volume per particle to λ³ has the value 10⁻⁴, showing that classical statistics fail altogether. When the classical distribution fails, a degenerate Fermi distribution results. A somewhat lengthy calculation yields the result that in this case the contribution of the electrons to the specific heat is negligible. This resolves an old paradox, for, according to the classical equipartition law, the electronic specific heat C should be (3/2)Nk, whereas in reality it is very small. See also Bose-Einstein statistics; Kinetic theory of matter; Quantum statistics; Statistical mechanics.

The noun has one meaning:

Meaning #1: (physics) law obeyed by a systems of particles whose wave function changes when two particles are interchanged (the Pauli exclusion principle applies)