Derivation of the Fermi-Dirac and Bose-Einstein distributions

Below we derive in standard textbook fashion the Fermi-Dirac and Bose-Einstein distributions from the perspective of the grand canonical ensemble, which in a way is the natural ensemble to consider.

Consider a system of {N} identical quantum particles. Let {n_j} be the number of particles occupying the single-particle states {|j\rangle}. Let {\epsilon_j} be the energy for the state {|j\rangle}. The occupation number {n_j} is limited to {n_j=0,~1} for fermions due to the Pauli exclusion principle. For bosons the occupation numbers can be arbitrarily large.

The total energy is thus given by

\displaystyle E= \sum_{j} \epsilon _j n_j ~, \ \ \ \ \ (1)

and

\displaystyle N= \sum_j n_j ~. \ \ \ \ \ (2)

Our goal is to calculate the mean occupation number {\overline{ n_j}} for fermions and bosons when they are in thermodynamic equilibrium, at temperature {T} and with chemical potential {\mu}.

1. The grand canonical partition function

For fixed chemical potential {\mu}, temperature parameter {\beta} and volume {V}, the grand canonical partition function is given by

\displaystyle \Xi(\beta, \mu, V) = \sum _{\rm all~states} \exp [-\beta (E-\mu N)] ~. \ \ \ \ \ (3)

In the situation previously described, we can write it as

\displaystyle \begin{array}{rcl} \Xi &=& \sum _{n_1} \sum _{n_2} \sum _{n_3} \ldots \exp \bigg[ \sum_j -\beta (n_j \epsilon_j - \mu n_j)\bigg] \\ &=& \sum _{n_1} \sum _{n_2} \sum _{n_3} \ldots \prod_j \exp [ -\beta (n_j \epsilon_j - \mu n_j) ] \\ &=& \bigg[\sum _{n_1} \exp [ -\beta (n_1 \epsilon_1 - \mu n_1) ] \bigg] \bigg[\sum _{n_2} \exp [ -\beta (n_2 \epsilon_1 - \mu n_2) ] \bigg] \ldots \end{array}

Hence,

\displaystyle \Xi = \prod_j ~\sum _{n_j} \exp [-\beta n_j (\epsilon_j - \mu)] ~. \ \ \ \ \ (4)

In other words, {\Xi} can be factored as product over a sum over occupation numbers for the {j}-th single-particle state. The latter sum is easy to calculate for bosons and even easier for fermions.

The mean occupation numbers can then be calculated as a weighted average for the grand canonical ensemble:

\displaystyle \begin{array}{rcl} \overline{n_j} &=& \frac{ \sum _{n_1,n_2\ldots} n_j \exp \bigg[ \sum_j -\beta (n_j \epsilon_j - \mu n_j)\bigg] } { \sum _{n_1,n_2\ldots} \exp \bigg[ \sum_j -\beta (n_j \epsilon_j - \mu n_j)\bigg] } \\ &=& \frac {1}{\Xi} \frac{\partial \Xi}{\partial(- \beta \epsilon_j)} \end{array}

so that

\displaystyle \overline{n_j} = -\frac{1 }{ \beta } \frac{\partial \log \Xi}{\partial\epsilon_j} ~. \ \ \ \ \ (5)

Substituting for {\Xi} from (4) we get

\displaystyle \begin{array}{rcl} \overline{n_j} &=& -\frac{1 }{ \beta } \frac{\partial}{\partial\epsilon_j} \log \left[ \prod_k ~\sum _{n_k} \exp [-\beta n_k (\epsilon_k - \mu)] \right] \\ &=& -\frac{1 }{ \beta } \frac{\partial}{\partial\epsilon_j} \sum_k \log \left[ \sum _{n_k} \exp [-\beta n_k (\epsilon_k - \mu)] \right] \\ &=& \sum_k -\frac{1 }{ \beta } \frac{\partial}{\partial\epsilon_j} \log \left[ \sum _{n_k} \exp [-\beta n_k (\epsilon_k - \mu)] \right]~. \end{array}

Unless {k=j} the derivative vanishes, so that

\displaystyle \overline{n_j} = -\frac{1 }{ \beta } \frac{\partial }{\partial\epsilon_j} \log \left[ \sum _{n_j} \exp [-\beta n_j (\epsilon_j - \mu)] \right] ~. \ \ \ \ \ (6)

2. Fermi-Dirac statistics

For fermions {n_j} cannot be greater than 1. So

\displaystyle \sum _{n_j} \exp [-\beta n_j (\epsilon_j - \mu)] = 1+ \exp [-\beta (\epsilon_j - \mu)]~. \ \ \ \ \ (7)

Hence,

\displaystyle \begin{array}{rcl} \overline{n_j} &=& -\frac{1 }{ \beta } \frac{\partial \log }{\partial\epsilon_j} (1+ \exp [-\beta (\epsilon_j - \mu)])\\ &=& -\frac{1 }{ \beta } \frac{ -\beta \exp [-\beta (\epsilon_j - \mu)]} {(1+ \exp [-\beta (\epsilon_j - \mu)])}\\ &=& \frac{ \exp [-\beta (\epsilon_j - \mu)]} {(1+ \exp [-\beta (\epsilon_j - \mu)])} ~. \end{array}

Finally we get

\displaystyle \overline{n_j} = \frac{1 } {\exp [\beta (\epsilon_j - \mu)]+1} ~. \ \ \ \ \ (8)

3. Bose-Einstein Statistics

For bosons we have

\displaystyle \sum _{n_j} \exp [-\beta n_j (\epsilon_j - \mu)] = \sum _{n_j} (\exp [-\beta (\epsilon_j - \mu)])^{n_j} \ \ \ \ \ (9)

which is a geometric series.

Recall that for {|r|<1} we can write

\displaystyle (1-r) \sum_{n=0}^\infty r^n = 1~, \ \ \ \ \ (10)

so that

\displaystyle \sum_{n=0}^\infty r^n = {1 \over 1-r}~, \ \ \ \ \ (11)

from which we get

\displaystyle \sum _{n_j} \exp [-\beta n_j (\epsilon_j - \mu)] = {1 \over 1 - \exp [-\beta (\epsilon_j - \mu)]}~. \ \ \ \ \ (12)

Substituting, we get

\displaystyle \begin{array}{rcl} \overline{n_j} &=& -\frac{1 }{ \beta } \frac{\partial \log }{\partial\epsilon_j} \left[ {1 \over 1 - \exp [-\beta (\epsilon_j - \mu)]} \right] \\ &=& -\frac{1 }{ \beta } \frac{\partial \log }{\partial\epsilon_j} \big[ 1 - \exp [-\beta (\epsilon_j - \mu)] \big]^{-1} \\ &=& \frac{1 }{ \beta } \frac{\partial \log }{\partial\epsilon_j} \big[ 1 - \exp [-\beta (\epsilon_j - \mu)] \big] \\ &=& \frac{1 }{ \beta } \bigg[ { \beta \exp [-\beta (\epsilon_j - \mu)] \over 1 - \exp [-\beta (\epsilon_j - \mu)]} \bigg] \\ &=& { \exp [-\beta (\epsilon_j - \mu)] \over 1 - \exp [-\beta (\epsilon_j - \mu)]} . \end{array}

Finally we get

\displaystyle \overline{n_j} = \frac{1 } {\exp [\beta (\epsilon_j - \mu)]-1} ~. \ \ \ \ \ (13)

4. Summary

The Fermi-Dirac and Bose-Einstein distributions can be written as

\displaystyle \overline{n(\epsilon)}= \frac{1 } {\exp [\beta (\epsilon - \mu)] \pm 1} ~, \ \ \ \ \ (14)

with the plus sign for Fermi-Dirac statistics and the minus sign for Bose-Einstein statistics.