Editing Fermi–Dirac statistics

{{short description|Statistical description for the behavior of fermions}}
{{Use American English|date=January 2019}}{{Statistical mechanics|cTopic=[[Particle statistics|Particle Statistics]]}}

'''Fermi–Dirac statistics''' is a type of [[quantum statistics]] that applies to the [[physics]] of a [[physical system|system]] consisting of many non-interacting, [[identical particles]] that obey the [[Pauli exclusion principle]]. A result is the Fermi–Dirac distribution of particles over [[Energy level|energy states]]. It is named after [[Enrico Fermi]] and [[Paul Dirac]], each of whom derived the distribution independently in 1926.<ref name='Fermi1926' /><ref name='Dirac1926'/> Fermi–Dirac statistics is a part of the field of [[statistical mechanics]] and uses the principles of [[quantum mechanics]].

Fermi–Dirac statistics applies to identical and indistinguishable particles with [[half-integer]] [[Spin (physics)|spin]] (1/2, 3/2, etc.), called [[fermion]]s, in [[thermodynamic equilibrium]]. For the case of negligible interaction between particles, the system can be described in terms of single-particle [[Energy eigenstates|energy states]]. A result is the Fermi–Dirac distribution of particles over these states where no two particles can occupy the same state, which has a considerable effect on the properties of the system. Fermi–Dirac statistics is most commonly applied to [[electron]]s, a type of fermion with [[spin&nbsp;1/2]].

A counterpart to Fermi–Dirac statistics is [[Bose–Einstein statistics]], which applies to identical and indistinguishable particles with integer spin (0, 1, 2, etc.) called [[bosons]]. In classical physics, [[Maxwell–Boltzmann statistics]] is used to describe particles that are identical and treated as distinguishable. For both Bose–Einstein and Maxwell–Boltzmann statistics, more than one particle can occupy the same state, unlike Fermi–Dirac statistics.
[[File:Quantum and classical statistics.png|500px|thumb|right|Equilibrium thermal distributions for particles with integer spin (bosons, red), half integer spin (fermions, blue), and classical (spinless) particles (green). Average occupancy <math>\langle n\rangle</math> is shown versus energy <math>\epsilon</math> relative to the system chemical potential <math>\mu</math>, where <math>T</math> is the system temperature, and <math>k_B</math> is the Boltzmann constant.]]

==History==
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 [[electron]]s than were in the [[electric current]].<ref name='Kittel1971Cel249'>{{harv|Kittel|1971|pp=249–50}}</ref> It was also difficult to understand why the [[field electron emission|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]]&nbsp;''k''<sub>B</sub>.
This problem remained unsolved until the development of Fermi–Dirac statistics.

Fermi–Dirac statistics was first published in 1926 by [[Enrico Fermi]]<ref name='Fermi1926'>{{cite journal| title=Sulla quantizzazione del gas perfetto monoatomico| journal=Rendiconti Lincei| language=it| year=1926| first=Enrico| last=Fermi| volume=3| pages=145–9}}, translated as {{cite arXiv| title=On the Quantization of the Monoatomic Ideal Gas| eprint=cond-mat/9912229| date=1999-12-14 | last1=Zannoni| first1=Alberto}}</ref> and [[Paul Dirac]].<ref name='Dirac1926'>{{cite journal| title=On the Theory of Quantum Mechanics| journal=Proceedings of the Royal Society A| year=1926| first=Paul A. M.| last=Dirac| author-link=Paul Dirac| volume=112| issue=762| pages=661–77| jstor=94692| doi=10.1098/rspa.1926.0133 |bibcode = 1926RSPSA.112..661D | doi-access=free}}</ref> According to [[Max Born]], [[Pascual Jordan]] developed in 1925 the same statistics, which he called ''[[Wolfgang Pauli|Pauli]] statistics'', but it was not published in a timely manner.<ref name='Science-Week2000'>{{cite journal| title=History of Science: The Puzzle of the Bohr–Heisenberg Copenhagen Meeting| journal=[[ScienceWeek|Science-Week]]| date=2000-05-19| volume=4| issue=20| url=http://scienceweek.com/2000/sw000519.htm| oclc=43626035| access-date=2009-01-20| archive-url=https://web.archive.org/web/20090411105016/http://scienceweek.com/2000/sw000519.htm| archive-date=2009-04-11| url-status=dead}}</ref><ref>{{cite journal |last=Schücking |title=Jordan, Pauli, Politics, Brecht and a variable gravitational constant |journal=Physics Today |volume=52 |year=1999 |issue=10 |page=26 |doi=10.1063/1.882858 |doi-access=free |bibcode=1999PhT....52j..26S }}</ref><ref>{{cite journal |last1=Ehlers |last2=Schücking |title=Aber Jordan war der Erste |language=de |journal=Physik Journal |volume=1 |year=2002 |issue=11 |pages=71–72 |doi= |hdl=11858/00-001M-0000-0013-5513-D }}</ref> According to Dirac, it was first studied by Fermi, and Dirac called it "Fermi statistics" and the corresponding particles "fermions".<ref name='Dirac1958'>{{cite book | last=Dirac| first=Paul A. M.| author-link=Paul Dirac| title=Principles of Quantum Mechanics| edition=revised 4th| publisher=Oxford University Press| year=1967| location=London| pages=210–1| url=https://books.google.com/books?id=XehUpGiM6FIC&pg=PA210| isbn=978-0-19-852011-5 }}</ref>

Fermi–Dirac statistics was applied in 1926 by [[Ralph Fowler]] to describe the collapse of a [[star]] to a [[white dwarf]].<ref name='Fowler1926'>{{cite journal| title=On dense matter| journal=[[Monthly Notices of the Royal Astronomical Society]]|date=December 1926| first=Ralph H.| last=Fowler| author-link=Ralph Fowler| volume=87| issue=2| pages=114–22| bibcode=1926MNRAS..87..114F| doi=10.1093/mnras/87.2.114| doi-access=free}}</ref> In 1927 [[Arnold Sommerfeld]] applied it to electrons in metals and developed the [[free electron model]],<ref name='sommerfeld1927'>{{cite journal| title=Zur Elektronentheorie der Metalle|language=de|trans-title=On Electron Theory of Metals| journal=[[Naturwissenschaften]]| date=1927-10-14| first=Arnold| last=Sommerfeld| author-link=Arnold Sommerfeld| volume=15| issue=41| pages=824–32| doi=10.1007/BF01505083|bibcode = 1927NW.....15..825S |s2cid=39403393}}</ref> and in 1928 Fowler and [[Lothar Wolfgang Nordheim|Lothar Nordheim]] applied it to [[field electron emission]] from metals.<ref name='Fowler1928'>{{cite journal| title=Electron Emission in Intense Electric Fields| journal=Proceedings of the Royal Society A| date=1928-05-01| first1=Ralph H.| last1=Fowler| first2=Lothar W.| last2=Nordheim| volume=119| issue=781| pages=173–81| doi=10.1098/rspa.1928.0091| jstor=95023| author-link=Lothar Wolfgang Nordheim |bibcode = 1928RSPSA.119..173F | doi-access=free}}</ref> Fermi–Dirac statistics continue to be an important part of physics.

==Fermi–Dirac distribution==
For a system of identical fermions in thermodynamic equilibrium, the average number of fermions in a single-particle state {{mvar|i}} is given by the '''Fermi–Dirac (F–D) distribution''':<ref name='Reif1965dist341'>{{harvnb|Reif|1965|p=341}}.</ref>{{refn|group=nb|The F–D distribution is a type of mathematical function called a [[logistic function]] or [[sigmoid function]].}} 
{{Equation box 1
|indent = :
|equation = <math>\bar{n}_i = \frac{1}{e^{(\varepsilon_i - \mu) / k_\text{B} T} + 1},</math>
|cellpadding = 6
|border
|border color = #0070BF
|bgcolor = #FAFFFB
}}

where {{Math|''k''<sub>B</sub>}} is the [[Boltzmann constant]], {{mvar|T}} is the absolute [[temperature]], {{math|''ε<sub>i</sub>''}} is the energy of the single-particle state {{mvar|i}}, and {{mvar|μ}} is the [[total chemical potential]]. The distribution is normalized by the condition

: <math>\sum_i \bar n_i = N</math>

that can be used to express <math>\mu = \mu(T, N)</math> in that <math>\mu</math> can assume either a positive or negative value.<ref>Landau, L. D., Lifshitz, E. M. (2013). Statistical Physics: Volume 5 (Vol. 5). Elsevier.</ref>

At zero absolute temperature, {{mvar|μ}} is equal to the [[Fermi energy]] plus the potential energy per fermion, provided it is in a [[Neighbourhood (mathematics)|neighbourhood]] of positive spectral density. In the case of a spectral gap, such as for electrons in a semiconductor, the point of symmetry  {{mvar|μ}} is typically called the [[Fermi level]] or—for electrons—the [[electrochemical potential]], and will be located in the middle of the gap.<ref name=Blakemore2002p11>{{harvnb|Blakemore|2002|p=11}}.</ref><ref name=KittelKroemer1980>{{cite book |last1=Kittel |first1=Charles |author-link1=Charles Kittel |first2=Herbert |last2=Kroemer |title=Thermal Physics |edition=2nd |publisher=W. H. Freeman |year=1980 |location=San Francisco| pages=357 |url=https://books.google.com/books?id=c0R79nyOoNMC&pg=PA357 |isbn=978-0-7167-1088-2 |author2-link=Herbert Kroemer }}</ref>

The Fermi–Dirac distribution is only valid if the number of fermions in the system is large enough so that adding one more fermion to the system has negligible effect on {{mvar|μ}}.<ref name='Reif1965dist'>{{harvnb|Reif|1965|pp=340–342}}.</ref> Since the Fermi–Dirac distribution was derived using the [[Pauli exclusion principle]], which allows at most one fermion to occupy each possible state, a result is that <math>0 < \bar{n}_i < 1</math>.{{refn|group=nb|Note that <math> \bar{n}_i </math> is also the probability that the state <math>i</math> is occupied, since no more than one fermion can occupy the same state at the same time and <math>0 < \bar{n}_i < 1</math>.}}

<gallery class="center" widths="400" heights="300" caption="Fermi–Dirac distribution">
File:FD e mu.svg|'''Energy dependence.''' More gradual at higher ''T''. <math>\bar{n} = 0.5</math> when <math>\varepsilon = \mu</math>. Not shown is that <math>\mu</math> decreases for higher ''T''.<ref name="Kittel1971dist245">{{harvnb|Kittel|1971|p=245|loc=Figs. 4 and 5}}.</ref>
File:FD kT e.svg|'''Temperature dependence''' for <math>\varepsilon > \mu</math>.
</gallery>
The [[variance]] of the number of particles in state ''i'' can be calculated from the above expression for <math>\bar{n}_i</math>:<ref name="pears2">{{cite book |last1=Pearsall |first1=Thomas |title=Quantum Photonics, 2nd edition |publisher=Springer |date=2020 |doi=10.1007/978-3-030-47325-9 |url=https://www.springer.com/us/book/9783030473242 |isbn=978-3-030-47324-2 |series=Graduate Texts in Physics}}</ref><ref name='Reif1965dist351'>{{harv|Reif|1965|p=351}} Eq.&nbsp;9.7.7, where <math>\beta = 1/k_\text{B}T, \quad \alpha = -\mu/k_\text{B}T, \quad \frac{\partial \bar{n}_i}{\partial \epsilon_i} = - \frac{\partial \bar{n}_i}{\partial \mu}</math>.</ref>
: <math> V(n_i) = k_\text{B}T \frac{\partial}{\partial \mu} \bar{n}_i = \bar{n}_i(1 - \bar{n}_i).</math>

===Distribution of particles over energy===
[[File:Fermi-Dirac distribution.gif|thumb|500px|Fermi function <math>F(\epsilon)</math> with <math>\mu = 0.55~\text{eV}</math> for various temperatures in the range <math>2~\text{K} \leq T \leq 375~\text{K}</math>]]

From the Fermi–Dirac distribution of particles over states, one can find the distribution of particles over energy.{{refn|group=nb|These distributions over energies, rather than states, are sometimes called the Fermi–Dirac distribution too, but that terminology will not be used in this article.}} The average number of fermions with energy <math>\varepsilon_i</math> can be found by multiplying the Fermi–Dirac distribution <math>\bar{n}_i</math> by the [[degenerate energy level|degeneracy]] <math>g_i</math> (i.e. the number of states with energy <math>\varepsilon_i</math>),<ref name="Leighton1959">{{cite book |last=Leighton |first=Robert B. |author-link=Robert B. Leighton |title=Principles of Modern Physics |url=https://archive.org/details/principlesofmode00leig |url-access=registration |publisher=McGraw-Hill |year=1959 |page=[https://archive.org/details/principlesofmode00leig/page/340 340] |isbn=978-0-07-037130-9}} Note that in Eq.&nbsp;(1), <math>n(\varepsilon)</math> and <math>n_s</math> correspond respectively to <math>\bar{n}_i</math> and <math>\bar{n}(\varepsilon_i)</math> in this article. See also Eq.&nbsp;(32) on p.&nbsp;339.</ref>

: <math>\begin{align}
 \bar{n}(\varepsilon_i) &= g_i \bar{n}_i \\
      &= \frac{g_i}{e^{(\varepsilon_i - \mu) / k_\text{B} T} + 1}.
\end{align}</math>

When <math>g_i \ge 2</math>, it is possible that <math>\bar{n}(\varepsilon_i) > 1</math>, since there is more than one state that can be occupied by fermions with the same energy <math>\varepsilon_i</math>.

When a quasi-continuum of energies <math>\varepsilon</math> has an associated [[density of states]] <math>g(\varepsilon)</math> (i.e. the number of states per unit energy range per unit volume<ref name=Blakemore2002p8>{{harvnb|Blakemore|2002|p=8}}.</ref>), the average number of fermions per unit energy range per unit volume is

: <math>\bar{\mathcal{N}}(\varepsilon) = g(\varepsilon) F(\varepsilon),</math>

where <math>F(\varepsilon)</math> is called the '''Fermi function''' and is the same [[function (mathematics)|function]] that is used for the Fermi–Dirac distribution <math>\bar{n}_i</math>:<ref name='Reif1965FermiFnc'>{{harvnb|Reif|1965|p=389}}.</ref>

: <math>F(\varepsilon) = \frac{1}{e^{(\varepsilon - \mu) / k_\text{B}T} + 1},</math>

so that

: <math>\bar{\mathcal{N}}(\varepsilon) = \frac{g(\varepsilon)}{e^{(\varepsilon - \mu) / k_\text{B} T} + 1}.</math>

==Quantum and classical regimes==
The Fermi–Dirac distribution approaches the [[Maxwell–Boltzmann statistics#Applicability|Maxwell–Boltzmann distribution]] in the limit of high temperature and low particle density, without the need for any ad hoc assumptions:

* In the limit of low particle density, <math>
\bar{n}_i = \frac{1}{e^{(\varepsilon_i-\mu)/k_{\rm B}T}+1} \ll 1
</math>, therefore <math>
e^{(\varepsilon_i-\mu)/k_{\rm B}T}+1 \gg 1
</math> or equivalently <math>
e^{(\varepsilon_i-\mu)/k_{\rm B}T} \gg 1
</math>. In that case, <math>
\bar{n}_i \approx \frac{1}{e^{(\varepsilon_i-\mu)/k_{\rm B}T}}=\frac{N}{Z}e^{-\varepsilon_i/k_{\rm B}T}
</math>, which is the result from Maxwell-Boltzmann statistics.
* In the limit of high temperature, the particles are distributed over a large range of energy values, therefore the occupancy on each state (especially the high energy ones with <math>\varepsilon_i - \mu \gg k_{\rm B}T</math>) is again very small, <math>
\bar{n}_i = \frac{1}{e^{(\varepsilon_i-\mu)/k_{\rm B}T}+1} \ll 1
</math>. This again reduces to Maxwell-Boltzmann statistics.

The classical regime, where [[Maxwell–Boltzmann statistics]] can be used as an approximation to Fermi–Dirac statistics, is found by considering the situation that is far from the limit imposed by the [[Heisenberg uncertainty principle]] for a particle's position and [[momentum]]. For example, in physics of semiconductor, when the density of states of conduction band is much higher than the doping concentration, the energy gap between conduction band and fermi level could be calculated using Maxwell-Boltzmann statistics. Otherwise, if the doping concentration is not negligible compared to density of states of conduction band, the Fermi–Dirac distribution should be used instead for accurate calculation. It can then be shown that the classical situation prevails when the [[Quantum concentration|concentration]] of particles corresponds to an [[Mean inter-particle distance|average interparticle separation]] <math> \bar{R} </math> that is much greater than the average [[de Broglie wavelength]] <math> \bar{\lambda} </math> of the particles:<ref name="Reif1965classical">{{harv|Reif|1965|pp=246–8}}</ref>

:<math>\bar{R} \gg \bar{\lambda} \approx \frac{h}{\sqrt{3mk_{\rm B}T}}, </math>

where {{mvar|h}} is the [[Planck constant]], and {{mvar|m}} is the [[Elementary particle|mass of a particle]].

For the case of conduction electrons in a typical metal at {{mvar|T}} = 300&nbsp;[[Kelvin|K]] (i.e. approximately room temperature), the system is far from the classical regime because <math> \bar{R} \approx \bar{\lambda}/25 </math> . This is due to the small mass of the electron and the high concentration (i.e. small <math>\bar{R}</math>) of conduction electrons in the metal. Thus Fermi–Dirac statistics is needed for conduction electrons in a typical metal.<ref name='Reif1965classical' />

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 temperature of white dwarf is high (typically {{mvar|T}} = {{val|10000|u=K}} on its surface<ref name='Mukai1997'>{{cite web|url=http://imagine.gsfc.nasa.gov/docs/ask_astro/answers/970512e.html |title=Ask an Astrophysicist |last=Mukai |first=Koji |author2=Jim Lochner |year=1997 |work=NASA's Imagine the Universe |publisher=NASA Goddard Space Flight Center |archive-url=https://web.archive.org/web/20090118054305/http://imagine.gsfc.nasa.gov/docs/ask_astro/answers/970512e.html |archive-date=2009-01-18 |url-status=dead }}</ref>), its high electron concentration and the small mass of each electron precludes using a classical approximation, and again Fermi–Dirac statistics is required.<ref name='Fowler1926' />

==Derivations==

===Grand canonical ensemble===

The Fermi–Dirac distribution, which applies only to a quantum system of non-interacting fermions, is easily derived from the [[grand canonical ensemble]].<ref name="sriva">{{cite book
 |title=Statistical Mechanics
 |last1=Srivastava |first1=R. K.
 |last2=Ashok |first2=J.
 |year=2005
 |publisher=PHI Learning Pvt. Ltd.
 |isbn=9788120327825
 |location=[[New Delhi]]
|chapter=Chapter 6
}}</ref> 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 [[microstate (statistical mechanics)|microstate]]s for the single-particle level: no particle (energy ''E'' = 0), or one particle (energy ''E'' = ''ε''). The resulting [[partition function (statistical mechanics)|partition function]] for that single-particle level therefore has just two terms:
:<math> \begin{align}
 \mathcal Z &= \exp\big(0(\mu - \varepsilon)/k_{\rm B} T\big) + \exp\big(1(\mu - \varepsilon)/k_{\rm B} T\big) \\
  &= 1 + \exp\big((\mu - \varepsilon)/k_{\rm B} T\big),
\end{align}</math>
and the average particle number for that single-particle level substate is given by
:<math> \langle N\rangle = k_{\rm B} T \frac{1}{\mathcal Z} \left(\frac{\partial \mathcal Z}{\partial \mu}\right)_{V,T} = \frac{1}{\exp\big((\varepsilon-\mu)/k_{\rm B} T\big) + 1}. </math>
This result applies for each single-particle level, and thus gives the Fermi–Dirac distribution for the entire state of the system.<ref name="sriva"/>

The variance in particle number (due to [[thermal fluctuations]]) may also be derived (the particle number has a simple [[Bernoulli distribution]]):
:<math> \big\langle (\Delta N)^2 \big\rangle = k_{\rm B} T \left(\frac{d\langle N\rangle}{d\mu}\right)_{V,T} = \langle N\rangle \big(1 - \langle N\rangle\big). </math>

This quantity is important in transport phenomena such as the [[Seebeck coefficient|Mott relations]] for electrical conductivity and [[Thermoelectric effect#Charge carrier diffusion|thermoelectric coefficient]] for an [[electron gas]],<ref>{{Cite journal | last1 = Cutler | first1 = M. | last2 = Mott | first2 = N. | doi = 10.1103/PhysRev.181.1336 | title = Observation of Anderson Localization in an Electron Gas | journal = Physical Review | volume = 181 | issue = 3 | pages = 1336 | year = 1969 |bibcode = 1969PhRv..181.1336C }}</ref> where the ability of an energy level to contribute to transport phenomena is proportional to <math>\big\langle (\Delta N)^2 \big\rangle</math>.

===Canonical ensemble===

It is also possible to derive Fermi–Dirac statistics in the [[canonical ensemble]]. Consider a many-particle system composed of ''N'' identical fermions that have negligible mutual interaction and are in thermal equilibrium.<ref name="Reif1965dist"/> Since there is negligible interaction between the fermions, the energy <math>E_R</math> of a state <math>R</math> of the many-particle system can be expressed as a sum of single-particle energies:

: <math>E_R = \sum_{r} n_r \varepsilon_r,</math>

where <math>n_r</math> is called the occupancy number and is the number of particles in the single-particle state <math>r</math> with energy <math>\varepsilon_r</math>. The summation is over all possible single-particle states <math>r</math>.

The probability that the many-particle system is in the state <math>R</math> is given by the normalized [[canonical distribution]]:<ref name='Reif1965canonical'>{{harvnb|Reif|1965|pp=203–206}}.</ref>

: <math>P_R = \frac{e^{-\beta E_R}}{\displaystyle\sum_{R'} e^{-\beta E_{R'}}},</math>

where <math>\beta = 1/k_\text{B}T</math>, <math>e^{-\beta E_R}</math> is called the [[Boltzmann factor]], and the summation is over all possible states <math>R'</math> of the many-particle system. The average value for an occupancy number <math>n_i</math> is<ref name='Reif1965canonical'/>

: <math>\bar{n}_i = \sum_R n_i P_R.</math>

Note that the state <math>R</math> of the many-particle system can be specified by the particle occupancy of the single-particle states, i.e. by specifying <math>n_1, n_2, \ldots,</math> so that

: <math>P_R = P_{n_1, n_2, \ldots} = \frac{e^{-\beta(n_1 \varepsilon_1 + n_2 \varepsilon_2 + \cdots)}}
                                          {\displaystyle\sum_{{n_1}',{n_2}',\ldots} e^{-\beta(n_1' \varepsilon_1 + n_2' \varepsilon_2 + \cdots)}},</math>

and the equation for <math>\bar{n}_i</math> becomes

: <math>\begin{align}
 \bar{n}_i &= \sum_{n_1,n_2,\dots} n_i P_{n_1, n_2, \dots} \\
           &= \frac{\displaystyle\sum_{n_1,n_2,\dots} n_i e^{-\beta(n_1\varepsilon_1 + n_2\varepsilon_2 + \cdots + n_i\varepsilon_i + \cdots)}}
                   {\displaystyle\sum_{n_1,n_2,\dots} e^{-\beta(n_1\varepsilon_1 + n_2\varepsilon_2 + \cdots + n_i\varepsilon_i + \cdots)}},
\end{align}</math>

where the summation is over all combinations of values of <math>n_1, n_2, \ldots</math> which obey the Pauli exclusion principle, and <math>n_r = 0</math> = 0 or <math>1</math> for each <math>r</math>. Furthermore, each combination of values of <math>n_1, n_2, \ldots</math> satisfies the constraint that the total number of particles is <math>N</math>:

: <math>\sum_r n_r = N.</math>

Rearranging the summations,

: <math>\bar{n}_i = \frac
{\displaystyle\sum_{n_i=0}^1 n_i e^{-\beta(n_i\varepsilon_i)} \sideset{}{^{(i)}}\sum_{n_1,n_2,\dots} e^{-\beta(n_1\varepsilon_1+n_2\varepsilon_2+\cdots)}}
{\displaystyle\sum_{n_i=0}^1 e^{-\beta(n_i\varepsilon_i)} \sideset{}{^{(i)}}\sum_{n_1,n_2,\dots} e^{-\beta(n_1\varepsilon_1 + n_2\varepsilon_2 + \cdots)}},</math>

where the upper index <math>(i)</math> on the summation sign indicates that the sum is not over <math>n_i</math> and is subject to the constraint that the total number of particles associated with the summation is <math>N_i = N - n_i</math>. Note that <math>\textstyle\sum^{(i)}</math> still depends on <math>n_i</math> through the <math>N_i</math> constraint, since in one case <math>n_i = 0</math> and <math>\textstyle\sum^{(i)}</math> is evaluated with <math>N_i = N,</math> while in the other case <math>n_i = 1,</math> and <math>\textstyle\sum^{(i)}</math> is evaluated with <math>N_i = N - 1.</math> To simplify the notation and to clearly indicate that <math>\textstyle\sum^{(i)}</math> still depends on <math>n_i</math> through <math>N - n_i,</math> define

: <math>Z_i(N - n_i) \equiv \sideset{}{^{(i)}}\sum_{n_1,n_2,\ldots} e^{-\beta(n_1\varepsilon_1+n_2\varepsilon_2+\cdots)},</math>

so that the previous expression for <math>\bar{n}_i</math> can be rewritten and evaluated in terms of the <math>Z_i</math>:

: <math>\begin{align}
 \bar{n}_i &= \frac{\displaystyle\sum_{n_i=0}^1 n_i e^{-\beta(n_i\varepsilon_i)} \, Z_i(N - n_i)}
                   {\displaystyle\sum_{n_i=0}^1 e^{-\beta(n_i\varepsilon_i)} \, Z_i(N - n_i)} \\
           &= \frac{0 + e^{-\beta\varepsilon_i} \, Z_i(N - 1)}{Z_i(N) + e^{-\beta\varepsilon_i} \, Z_i(N - 1)} \\
           &= \frac{1}{[Z_i(N)/Z_i(N - 1)] \, e^{\beta\varepsilon_i}+1}.
\end{align}</math>

The following approximation<ref>See for example, {{slink|Derivative#Definition via difference quotients}}, which gives the approximation <math>f(a + h) \approx f(a) + f'(a) h.</math></ref> will be used to find an expression to substitute for <math>Z_i(N)/Z_i(N - 1)</math>:

: <math>\begin{align}
 \ln Z_i(N - 1) &\simeq \ln Z_i(N) - \frac{\partial \ln Z_i(N)}{\partial N } \\
                &= \ln Z_i(N) - \alpha_i,
\end{align}</math>
where <math>\alpha_i \equiv \frac{\partial \ln Z_i(N)}{\partial N}.</math>

If the number of particles <math>N</math> is large enough so that the change in the chemical potential <math>\mu</math> is very small when a particle is added to the system, then <math>\alpha_i \simeq - \mu / k_\text{B}T.</math><ref name='Reif1965ChemPot'>{{harvnb|Reif|1965|pp=341–342}}. See Eq.&nbsp;9.3.17 and ''Remark concerning the validity of the approximation''.</ref> Applying the exponential function to both sides, substituting for <math>\alpha_i</math> and rearranging,

: <math>Z_i(N) / Z_i(N - 1) =  e^{-\mu / k_\text{B}T}.</math>

Substituting the above into the equation for <math>\bar{n}_i</math> and using a previous definition of <math>\beta</math> to substitute <math>1/k_\text{B}T</math> for <math>\beta</math>, results in the Fermi–Dirac distribution:

: <math>\bar{n}_i = \frac{1}{e^{(\varepsilon_i - \mu)/k_\text{B}T} + 1}.</math>

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.<ref>{{cite book |first=H. J. W. |last=Müller-Kirsten |title=Basics of Statistical Physics |edition=2nd |publisher=World Scientific |year=2013 |isbn=978-981-4449-53-3 }}</ref>

===Microcanonical ensemble===

A result can be achieved by directly analyzing the multiplicities of the system and using [[Lagrange multipliers]].<ref name="Blakemore2002p343–5">{{harvnb|Blakemore|2002|pp=343–534}}.</ref>

Suppose we have a number of energy levels, labeled by index ''i'', each level having energy ε''<sub>i</sub>'' and containing a total of ''n<sub>i</sub>'' particles. Suppose each level contains ''g<sub>i</sub>'' 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 ''g<sub>i</sub>'' 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 ''n<sub>i</sub>'' indistinguishable particles among the ''g<sub>i</sub>'' sublevels of an energy level, with a maximum of one particle per sublevel, is given by the [[binomial coefficient]], using its [[Binomial coefficient#Combinatorial interpretation|combinatorial interpretation]]:
: <math>
 w(n_i, g_i) = \frac{g_i!}{n_i!(g_i - n_i)!}.
</math>

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&nbsp;3!/(2!1!).

The number of ways that a set of occupation numbers ''n''<sub>''i''</sub> can be realized is the product of the ways that each individual energy level can be populated:

: <math>
 W = \prod_i w(n_i, g_i) = \prod_i \frac{g_i!}{n_i!(g_i - n_i)!}.
</math>

Following the same procedure used in deriving the [[Maxwell–Boltzmann statistics]], we wish to find the set of ''n<sub>i</sub>'' 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:

: <math>
 f(n_i) = \ln W + \alpha\left(N - \sum n_i\right) + \beta\left(E - \sum n_i \varepsilon_i\right).
</math>

Using [[Stirling's approximation]] for the factorials, taking the derivative with respect to ''n<sub>i</sub>'', setting the result to zero, and solving for ''n<sub>i</sub>'' yields the Fermi–Dirac population numbers:

: <math>
 n_i = \frac{g_i}{e^{\alpha + \beta\varepsilon_i} + 1}.
</math>

By a process similar to that outlined in the [[Maxwell–Boltzmann statistics]] article, it can be shown thermodynamically that <math>\beta = \tfrac{1}{k_\text{B}T}</math> and <math>\alpha = -\tfrac{\mu}{k_\text{B}T}</math>, so that finally, the probability that a state will be occupied is

: <math>
 \bar{n}_i = \frac{n_i}{g_i} = \frac{1}{e^{(\varepsilon_i - \mu)/k_\text{B}T} + 1}.
</math>

==See also==
{{commons category|Fermi-Dirac distribution}}
{{cols|colwidth=20em}}
* [[Grand canonical ensemble]]
* [[Pauli exclusion principle]]
* [[Complete Fermi–Dirac integral|Complete Fermi-Dirac integral]]
* [[Fermi level]]
* [[Fermi gas]]
* [[Maxwell–Boltzmann statistics]]
* [[Bose–Einstein statistics]]
* [[Parastatistics]]
* [[Logistic function]]
* [[Sigmoid function]]
{{colend}}

==Notes==
{{reflist|group=nb}}

==References==
{{reflist|30em}}

==Further reading==
* {{cite book | last = Reif | first = F. | title = Fundamentals of Statistical and Thermal Physics | publisher = McGraw–Hill | year = 1965 | isbn = 978-0-07-051800-1 | url-access = registration | url = https://archive.org/details/fundamentalsofst00fred }}
* {{cite book | last = Blakemore | first = J. S. | title = Semiconductor Statistics | publisher = Dover | year = 2002 | url = https://books.google.com/books?id=cc4HE2YM1FIC&pg=front| isbn = 978-0-486-49502-6 }}
* {{cite book | last = Kittel | first=Charles | author-link = Charles Kittel | title = [[Introduction to Solid State Physics]]| edition=4th| publisher=John Wiley & Sons| year=1971|location=New York| oclc=300039591 | isbn = 978-0-471-14286-7 }}

{{Statistical mechanics topics}}
{{Authority control}}
<!-- Editors: Please do not add the probability distributions template here. The Fermi Dirac distribution is not a probability distribution. -->

{{DEFAULTSORT:Fermi-Dirac statistics}}
[[Category:Fermi–Dirac statistics| ]]
[[Category:Statistical mechanics]]