Editing Fermi–Dirac statistics (section)

==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>