Editing Fermi–Dirac statistics (section)

===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>.