Distribution of particles over energy

Fermi function ${\displaystyle F(\varepsilon )}$ with μ = 0.55 eV for various temperatures in the range 50 K ≤ T ≤ 375 K

From the Fermi–Dirac distribution of particles over states, one can find the distribution of particles over energy.^{[nb 3]} The average number of fermions with energy ${\displaystyle \varepsilon _{i}}$ can be found by multiplying the Fermi–Dirac distribution ${\displaystyle {\bar {n}}_{i}}$ by the degeneracy ${\displaystyle g_{i}}$ (i.e. the number of states with energy ${\displaystyle \varepsilon _{i}}$),^[19]

${\displaystyle {\begin{aligned}{\bar {n}}(\varepsilon _{i})&=g_{i}{\bar {n}}_{i}\\&={\frac {g_{i}}{e^{(\varepsilon _{i}-\mu )/k_{\rm {B}}T}+1}}.\end{aligned}}}$

When ${\displaystyle g_{i}\geq 2}$, it is possible that ${\displaystyle {\bar {n}}(\varepsilon _{i})>1}$, since there is more than one state that can be occupied by fermions with the same energy ${\displaystyle \varepsilon _{i}}$.

When a quasi-continuum of energies ${\displaystyle \varepsilon }$ has an associated density of states ${\displaystyle g(\varepsilon )}$ (i.e. the number of states per unit energy range per unit volume^[20]), the average number of fermions per unit energy range per unit volume is

${\displaystyle {\bar {\mathcal {N}}}(\varepsilon )=g(\varepsilon )F(\varepsilon ),}$

where ${\displaystyle F(\varepsilon )}$ is called the Fermi function and is the same function that is used for the Fermi–Dirac distribution ${\displaystyle {\bar {n}}_{i}}$,^[21]

${\displaystyle F(\varepsilon )={\frac {1}{e^{(\varepsilon -\mu )/k_{\rm {B}}T}+1}},}$

so that

${\displaystyle {\bar {\mathcal {N}}}(\varepsilon )={\frac {g(\varepsilon )}{e^{(\varepsilon -\mu )/k_{\rm {B}}T}+1}}.}$

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.^[24] 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:

${\displaystyle {\begin{aligned}{\mathcal {Z}}&=\exp {\big (}0(\mu -\varepsilon )/k_{\rm {B}}T{\big )}+\exp {\big (}1(\mu -\varepsilon )/k_{\rm {B}}T{\big )}\\&=1+\exp {\big (}(\mu -\varepsilon )/k_{\rm {B}}T{\big )},\end{aligned}}}$

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

${\displaystyle \langle N\rangle =k_{\rm {B}}T{\frac {1}{\mathcal {Z}}}\left({\frac {\partial {\mathcal {Z}}}{\partial \mu }}\right)_{V,T}={\frac {1}{\exp {\big (}(\varepsilon -\mu )/k_{\rm {B}}T{\big )}+1}}.}$

This result applies for each single-particle level, and thus gives the Fermi–Dirac distribution for the entire state of the system.^[24]

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

${\displaystyle {\big \langle }(\Delta N)^{2}{\big \rangle }=k_{\rm {B}}T\left({\frac {d\langle N\rangle }{d\mu }}\right)_{V,T}=\langle N\rangle {\big (}1-\langle N\rangle {\big )}.}$

This quantity is important in transport phenomena such as the Mott relations for electrical conductivity and thermoelectric coefficient for an electron gas,^[25] where the ability of an energy level to contribute to transport phenomena is proportional to ${\displaystyle {\big \langle }(\Delta N)^{2}{\big \rangle }}$.

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.^[15] Since there is negligible interaction between the fermions, the energy ${\displaystyle E_{R}}$ of a state ${\displaystyle R}$ of the many-particle system can be expressed as a sum of single-particle energies,

${\displaystyle E_{R}=\sum _{r}n_{r}\varepsilon _{r}}$

where ${\displaystyle n_{r}}$ is called the occupancy number and is the number of particles in the single-particle state ${\displaystyle r}$ with energy ${\displaystyle \varepsilon _{r}}$. The summation is over all possible single-particle states ${\displaystyle r}$.

The probability that the many-particle system is in the state ${\displaystyle R}$, is given by the normalized canonical distribution,^[26]

${\displaystyle P_{R}={\frac {e^{-\beta E_{R}}}{\displaystyle \sum _{R'}e^{-\beta E_{R'}}}}}$

where ${\displaystyle \beta =1/k_{\rm {B}}T}$, e^{${\displaystyle \scriptstyle -\beta E_{R}}$} is called the Boltzmann factor, and the summation is over all possible states ${\displaystyle R'}$ of the many-particle system.   The average value for an occupancy number ${\displaystyle n_{i}\;}$ is^[26]

${\displaystyle {\bar {n}}_{i}\ =\ \sum _{R}n_{i}\ P_{R}}$

Note that the state ${\displaystyle R}$ of the many-particle system can be specified by the particle occupancy of the single-particle states, i.e. by specifying ${\displaystyle n_{1},\,n_{2},\,\ldots \;,}$ so that

${\displaystyle P_{R}=P_{n_{1},n_{2},\ldots }={\frac {e^{-\beta (n_{1}\varepsilon _{1}+n_{2}\varepsilon _{2}+\cdots )}}{\displaystyle \sum _{{n_{1}}',{n_{2}}',\ldots }e^{-\beta ({n_{1}}'\varepsilon _{1}+{n_{2}}'\varepsilon _{2}+\cdots )}}}}$

and the equation for ${\displaystyle {\bar {n}}_{i}}$ becomes

${\displaystyle {\begin{alignedat}{2}{\bar {n}}_{i}&=\sum _{n_{1},n_{2},\dots }n_{i}\ P_{n_{1},n_{2},\dots }\\\\&={\frac {\displaystyle \sum _{n_{1},n_{2},\dots }n_{i}\ e^{-\beta (n_{1}\varepsilon _{1}+n_{2}\varepsilon _{2}+\cdots +n_{i}\varepsilon _{i}+\cdots )}}{\displaystyle \sum _{n_{1},n_{2},\dots }e^{-\beta (n_{1}\varepsilon _{1}+n_{2}\varepsilon _{2}+\cdots +n_{i}\varepsilon _{i}+\cdots )}}}\\\end{alignedat}}}$

where the summation is over all combinations of values of ${\displaystyle n_{1},n_{2},\ldots }$ which obey the Pauli exclusion principle, and ${\displaystyle n_{r}}$ = 0 or 1 for each ${\displaystyle r}$. Furthermore, each combination of values of ${\displaystyle n_{1},n_{2},\ldots }$ satisfies the constraint that the total number of particles is ${\displaystyle N}$,

${\displaystyle \sum _{r}n_{r}=N.}$

Rearranging the summations,

${\displaystyle {\bar {n}}_{i}={\frac {\displaystyle \sum _{n_{i}=0}^{1}n_{i}\ e^{-\beta (n_{i}\varepsilon _{i})}\quad \sideset {}{^{(i)}}\sum _{n_{1},n_{2},\dots }e^{-\beta (n_{1}\varepsilon _{1}+n_{2}\varepsilon _{2}+\cdots )}}{\displaystyle \sum _{n_{i}=0}^{1}e^{-\beta (n_{i}\varepsilon _{i})}\qquad \sideset {}{^{(i)}}\sum _{n_{1},n_{2},\dots }e^{-\beta (n_{1}\varepsilon _{1}+n_{2}\varepsilon _{2}+\cdots )}}}}$

where the ${\displaystyle ^{(i)}}$ on the summation sign indicates that the sum is not over ${\displaystyle n_{i}}$ and is subject to the constraint that the total number of particles associated with the summation is ${\displaystyle N_{i}=N-n_{i}}$. Note that ${\displaystyle \Sigma ^{(i)}}$ still depends on ${\displaystyle n_{i}}$ through the ${\displaystyle N_{i}}$ constraint, since in one case ${\displaystyle n_{i}=0}$ and ${\displaystyle \Sigma ^{(i)}}$ is evaluated with ${\displaystyle N_{i}=N,}$ while in the other case ${\displaystyle n_{i}=1}$ and ${\displaystyle \Sigma ^{(i)}}$ is evaluated with ${\displaystyle N_{i}=N-1.}$  To simplify the notation and to clearly indicate that ${\displaystyle \Sigma ^{(i)}}$ still depends on ${\displaystyle n_{i}}$ through ${\displaystyle N-n_{i}}$ , define

${\displaystyle Z_{i}(N-n_{i})\equiv \ \sideset {}{^{(i)}}\sum _{n_{1},n_{2},\ldots }e^{-\beta (n_{1}\varepsilon _{1}+n_{2}\varepsilon _{2}+\cdots )}\;}$

so that the previous expression for ${\displaystyle {\bar {n}}_{i}}$ can be rewritten and evaluated in terms of the ${\displaystyle Z_{i}}$,

${\displaystyle {\begin{alignedat}{3}{\bar {n}}_{i}\ &={\frac {\displaystyle \sum _{n_{i}=0}^{1}n_{i}\ e^{-\beta (n_{i}\varepsilon _{i})}\ \ Z_{i}(N-n_{i})}{\displaystyle \sum _{n_{i}=0}^{1}e^{-\beta (n_{i}\varepsilon _{i})}\qquad Z_{i}(N-n_{i})}}\\[8pt]&=\ {\frac {\quad 0\quad \;+e^{-\beta \varepsilon _{i}}\;Z_{i}(N-1)}{Z_{i}(N)+e^{-\beta \varepsilon _{i}}\;Z_{i}(N-1)}}\\[6pt]&=\ {\frac {1}{[Z_{i}(N)/Z_{i}(N-1)]\;e^{\beta \varepsilon _{i}}+1}}\quad .\end{alignedat}}}$

The following approximation^[27] will be used to find an expression to substitute for ${\displaystyle Z_{i}(N)/Z_{i}(N-1)}$ .

${\displaystyle {\begin{alignedat}{2}\ln Z_{i}(N-1)&\simeq \ln Z_{i}(N)-{\frac {\partial \ln Z_{i}(N)}{\partial N}}\\&=\ln Z_{i}(N)-\alpha _{i}\;\end{alignedat}}}$

where ${\displaystyle \alpha _{i}\equiv {\frac {\partial \ln Z_{i}(N)}{\partial N}}\ .}$

If the number of particles ${\displaystyle N}$ is large enough so that the change in the chemical potential ${\displaystyle \mu \;}$ is very small when a particle is added to the system, then ${\displaystyle \alpha _{i}\simeq -\mu /k_{\rm {B}}T\ .}$^[28]  Taking the base e antilog^[29] of both sides, substituting for ${\displaystyle \alpha _{i}\,}$, and rearranging,

${\displaystyle Z_{i}(N)/Z_{i}(N-1)=e^{-\mu /k_{\rm {B}}T}.}$

Substituting the above into the equation for ${\displaystyle {\bar {n}}_{i}}$, and using a previous definition of ${\displaystyle \beta \;}$ to substitute ${\displaystyle 1/k_{\rm {B}}T}$ for ${\displaystyle \beta \;}$, results in the Fermi–Dirac distribution.

${\displaystyle {\bar {n}}_{i}=\ {\frac {1}{e^{(\varepsilon _{i}-\mu )/k_{\rm {B}}T}+1}}}$

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.^[30]

Microcanonical ensemble

A result can be achieved by directly analyzing the multiplicities of the system and using Lagrange multipliers.^[31]

Suppose we have a number of energy levels, labeled by index i, each level having energy ε_i  and containing a total of n_i  particles. Suppose each level contains g_i  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_i  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_i indistinguishable particles among the g_i sublevels of an energy level, with a maximum of one particle per sublevel, is given by the binomial coefficient, using its combinatorial interpretation

${\displaystyle w(n_{i},g_{i})={\frac {g_{i}!}{n_{i}!(g_{i}-n_{i})!}}\ .}$

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:

${\displaystyle W=\prod _{i}w(n_{i},g_{i})=\prod _{i}{\frac {g_{i}!}{n_{i}!(g_{i}-n_{i})!}}.}$

Following the same procedure used in deriving the Maxwell–Boltzmann statistics, we wish to find the set of n_i 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:

${\displaystyle f(n_{i})=\ln(W)+\alpha \left(N-\sum n_{i}\right)+\beta \left(E-\sum n_{i}\varepsilon _{i}\right).}$

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

${\displaystyle n_{i}={\frac {g_{i}}{e^{\alpha +\beta \varepsilon _{i}}+1}}.}$

By a process similar to that outlined in the Maxwell–Boltzmann statistics article, it can be shown thermodynamically that ${\textstyle \beta ={\frac {1}{k_{\rm {B}}T}}}$ and ${\textstyle \alpha =-{\frac {\mu }{k_{\rm {B}}T}}}$, so that finally, the probability that a state will be occupied is:

${\displaystyle {\bar {n}}_{i}={\frac {n_{i}}{g_{i}}}={\frac {1}{e^{(\varepsilon _{i}-\mu )/k_{\rm {B}}T}+1}}.}$