## Fermi-Dirac Distribution Law

This statistics is obeyed by indistinguishable particles of half-integral spin that have anti-symmetric wave function and obey Pauli exclusion principle (e.g. electron, proton and neutron).

Consider a system having n distinguishable particles. Let these particles be divided into groups or lavels such that there are (n_{1}, n_{2}, n_{3},……n_{i}) number of particles in groups whose approximate constant energy (ϵ_{1}, ϵ_{2}, ϵ_{3},…… ϵ_{i}) respectively. Let g_{i}denote the degeneracy or statistical weight of *i*th level.

The number of distinguishable arrangements of (n_{i}) particles in (g_{i}) cells

`\frac{g_i!}{n_i!(g_i-n_i)!}` ........(1)

The total number of eigen states for the whole system is given by

`P=\π\frac{g_i!}{n_i!(g_i-n_i)!}` ........(2)

The * Fermi-Dirac* distribution law can now be obtained by determining the most probable distribution.

Taking log of eq. (2), we get

`\log P=\log\π\frac{g_i!}{n_i!(g_i-n_i)!}` ......(3)

The conditions of most probable distribution gives

`\delta(\log P)=-\sum_i\{\log\frac{g_i!}{(g_i-n_i)!}\}\delta n_i` .......(4)

For the equilibrium state, we must have δ(log P) = 0

`\sum_i\{\log\frac{g_i!}{(g_i-n_i)!}\}\delta n_i=0` ......(5)

Subject to the limitations

`\sum_i\delta n_i=0` ......(6)

`\sum_i\in_i\delta n_i=0` .......(7)

Applying the lagrange method of undetermined multiplier i.e., multiplying eq. (6) by α and eq. (7) by β and adding the resulting expression to eq. (5), We get

`\sum_i\{\log\frac{n_i}{g_i-n_i}+\alpha+\in_i\beta\}\delta n_i=0` ......(8)

As the variation (δn_{i}) are independent of each other,

`\log\frac{n_i}{(g_i-n_i)}+\alpha+\beta\in_i=0`

`\log\frac{n_i}{(g_i-n_i)}=-(\alpha+\beta\in_i)`

`\frac{n_i}{(g_i-n_i)}=e^{-(\alpha+\beta\in_i)}`

`\frac{(g_i-n_i)}{n_i}=e^{(\alpha+\beta\in_i)}`

`\frac{g_i}{n_i}-1=e^{(\alpha+\beta\in_i)}`

`\frac{g_i}{n_i}=e^{(\alpha+\beta\in_i)}+1`

`n_i=\frac{g_i}{e^{(\alpha+\beta\in_i)}+1}`

This is known as * Fermi-Dirac* distribution Law. This equation represents the most probable distribution of the particles among various energy levels for a system obeying

**statistics.**

*Fermi-Dirac*