Fermi–Dirac statistics

In statistical thermodynamics, Fermi-Dirac statistics determines the statistical distribution of fermions over the energy states for a system in thermal equilibrium. Fermions are particles which are indistinguishable and obey the Pauli exclusion principle, i.e., that no two particles may occupy the same state at the same time. Statistical thermodynamics is used to describe the behaviour of large numbers of particles. A collection of non-interacting fermions is called a Fermi gas.

Fermi-Dirac (or F-D) statistics are closely related to Maxwell-Boltzmann statistics and Bose-Einstein statistics. While F-D statistics holds for fermions, B-E statistics plays the same role for bosons – the other type of particle found in nature. M-B statistics describes the velocity distribution of particles in a classical gas and represents the classical (high-temperature) limit of both F-D and B-E statistics. M-B statistics are particularly useful for studying gases, and B-E statistics are particularly useful when dealing with photons and other bosons. F-D statistics are most often used for the study of electrons in solids. As such, they form the basis of semiconductor device theory and electronics. The invention of quantum mechanics, when applied through F-D statistics, has made advances such as the transistor possible. For this reason, F-D statistics are well-known not only to physicists, but also to electrical engineers.

F-D statistics was introduced in 1926 by Enrico Fermi and Paul Dirac and applied in 1927 by Arnold Sommerfeld to electrons in metals.

Conceptual development

Say there are two fermions placed in a system with four levels. There are six possible arrangements of such a system, which are shown in the diagram below.

   ε1   ε2   ε3   ε4
A  *    *
B  *         *
C  *              *
D       *    *
E       *         *
F            *    *

Each of these arrangements is called a microstate of the system. It is a fundamental postulate of statistical physics that at thermal equilibrium, each of these microstates will be equally likely, subject to the constraints of known total energy and number of particles.

Depending on the values of the energy for each state, it may be that total energy for some of these six combinations is the same as others. Indeed, if we assume that the energies are multiples of some fixed value ε, the energies of each of the microstates become:

A: 3ε
B: 4ε
C: 5ε
D: 5ε
E: 6ε
F: 7ε

So if we know that the system has an energy of 5ε, we can conclude that it will be equally likely that it is in state C or state D. Note that if the particles were distinguishable (the classical case), there would be twelve microstates altogether, rather than six.

The Fermi-Dirac distribution function

Using arguments such as these, the distribution of fermions in any multi-level system can be calculated. Unfortunately, this distribution involves unwieldy factorials of very large numbers. However, by using Stirling's formula, we can produce a reasonable mathematical form with negligible loss of precision, at least on the scale at which Fermi-Dirac statistics are usually applied.

The formula is given here for reference only: a full explanation of the derivation, symbols and applications is beyond the scope of this article. Note that this is only one form of the function – it is often given as an integral over momentum space.

${\frac {N_{j}}{g_{j}}}={\frac {1}{\exp \left({\frac {\epsilon _{j}-\mu }{k_{B}T}}\right)+1}}$

where:

N_j is the number of particles in state j

g_j is the degeneracy of state j

exp is the exponential function

ε_j is the energy of state j

μ is the chemical potential. Sometimes the Fermi energy E_F is used instead, as a low-temperature approximation.

k_B is Boltzmann's constant

T is absolute temperature