In quantum statistics, a branch of physics, Fermi–Dirac statistics describe a distribution of particles over energy states in systems consisting of many identical particles that obey the "Pauli exclusion principle". It is named after Enrico Fermi and Paul Dirac, each of whom discovered the method independently (although Fermi defined the statistics earlier than Dirac).^[1][2]

Fermi–Dirac (F–D) statistics apply to identical particles with half-integer spin in a system with thermodynamic equilibrium. Additionally, the particles in this system are assumed to have negligible mutual interaction. That allows the many-particle system to be described in terms of single-particle energy states. The result is the F–D distribution of particles over these states which includes the condition that no two particles can occupy the same state; this has a considerable effect on the properties of the system. Since F–D statistics apply to particles with half-integer spin, these particles have come to be called fermions. It is most commonly applied to electrons, which are fermions with spin 1/2. Fermi–Dirac statistics are a part of the more general field of statistical mechanics and use the principles of quantum mechanics.

The opposite of F–D statistics are the Bose–Einstein statistics, that apply to bosons (full integer spin or no spin, like the Higgs boson), particles that do not follow the Pauli exclusion principle, meaning that more than one boson can take up the same quantum configuration simultaneously.

Before the introduction of Fermi–Dirac statistics in 1926, understanding some aspects of electron behavior was difficult due to seemingly contradictory phenomena. For example, the electronic heat capacity of a metal at room temperature seemed to come from 100 times fewer electrons than were in the electric current.^[3] It was also difficult to understand why those emission currents generated by applying high electric fields to metals at room temperature were almost independent of temperature.

The difficulty encountered by the Drude model, the electronic theory of metals at that time, was due to considering that electrons were (according to classical statistics theory) all equivalent. In other words, it was believed that each electron contributed to the specific heat an amount on the order of the Boltzmann constant kB. This statistical problem remained unsolved until the discovery of F–D statistics.

F–D statistics was first published in 1926 by Enrico Fermi^[1] and Paul Dirac.^[2] According to Max Born, Pascual Jordan developed in 1925 the same statistics which he called Pauli statistics, but it was not published in a timely manner.^[4][5][6] According to Dirac, it was first studied by Fermi, and Dirac called it Fermi statistics and the corresponding particles fermions.^[7]

F–D statistics was applied in 1926 by Ralph Fowler to describe the collapse of a star to a white dwarf.^[8] In 1927 Arnold Sommerfeld applied it to electrons in metals and developed the free electron model,^[9] and in 1928 Fowler and Lothar Wolfgang Nordheim applied it to field electron emission from metals.^[10] Fermi–Dirac statistics continues to be an important part of physics.

For a system of identical fermions with thermodynamic equilibrium, the average number of fermions in a single-particle state i is given by a logistic function, or sigmoid function: the Fermi–Dirac (F–D) distribution,^[11]

where kB is Boltzmann's constant, T is the absolute temperature, εi is the energy of the single-particle state i, and μ is the total chemical potential.

At zero temperature, μ is equal to the Fermi energy plus the potential energy per electron. For the case of electrons in a semiconductor, μ, the point of symmetry, is typically called the Fermi level or electrochemical potential.^[12][13]

The above Fermi–Dirac distribution gives the distribution of identical fermions over single-particle energy states, where no more than one fermion can occupy a state. Using the F–D distribution, one can find the distribution of identical fermions over energy, where more than one fermion can have the same energy.^[17]

so that

where h is Planck's constant, and m is the mass of a particle.

Another example of a system that is not in the classical regime is the system that consists of the electrons of a star that has collapsed to a white dwarf. Although the white dwarf's temperature is high (typically T = 10000 K on its surface^[22]), its high electron concentration and the small mass of each electron precludes using a classical approximation, and again Fermi–Dirac statistics is required.^[8]

The Fermi–Dirac distribution, which applies only to a quantum system of non-interacting fermions, is easily derived from the grand canonical ensemble.^[23] In this ensemble, the system is able to exchange energy and exchange particles with a reservoir (temperature T and chemical potential μ fixed by the reservoir).

Due to the non-interacting quality, each available single-particle level (with energy level ϵ) forms a separate thermodynamic system in contact with the reservoir. In other words, each single-particle level is a separate, tiny grand canonical ensemble. By the Pauli exclusion principle, there are only two possible microstates for the single-particle level: no particle (energy E = 0), or one particle (energy E = ϵ). The resulting partition function for that single-particle level therefore has just two terms:

and the average particle number for that single-particle level substate is given by

This result applies for each single-particle level, and thus gives the Fermi–Dirac distribution for the entire state of the system.^[23]

The variance in particle number (due to thermal fluctuations) may also be derived (the particle number has a simple Bernoulli distribution):

Rearranging the summations,

Like the Maxwell–Boltzmann distribution and the Bose–Einstein distribution the Fermi–Dirac distribution can also be derived by the Darwin–Fowler method of mean values (see Müller-Kirsten^[29]).

A result can be achieved by directly analyzing the multiplicities of the system and using Lagrange multipliers.^[30]

Suppose we have a number of energy levels, labeled by index i, each level having energy εi  and containing a total of ni  particles. Suppose each level contains gi  distinct sublevels, all of which have the same energy, and which are distinguishable. For example, two particles may have different momenta (i.e. their momenta may be along different directions), in which case they are distinguishable from each other, yet they can still have the same energy. The value of gi  associated with level i is called the "degeneracy" of that energy level. The Pauli exclusion principle states that only one fermion can occupy any such sublevel.

The number of ways of distributing ni indistinguishable particles among the gisublevels of an energy level, with a maximum of one particle per sublevel, is given by the binomial coefficient, using its combinatorial interpretation

For example, distributing two particles in three sublevels will give population numbers of 110, 101, or 011 for a total of three ways which equals 3!/(2!1!).

The number of ways that a set of occupation numbers n**i can be realized is the product of the ways that each individual energy level can be populated:

Following the same procedure used in deriving the Maxwell–Boltzmann statistics, we wish to find the set of ni for which W is maximized, subject to the constraint that there be a fixed number of particles, and a fixed energy. We constrain our solution using Lagrange multipliers forming the function:

Using Stirling's approximation for the factorials, taking the derivative with respect to ni, setting the result to zero, and solving for ni yields the Fermi–Dirac population numbers:

The Fermi–Dirac distribution approaches the Maxwell–Boltzmann distribution in the limit of high temperature and low particle density, without the need for any ad hoc assumptions.