To understand how electrons arrange themselves in a material, we can't just toss them into energy levels randomly. Electrons are fermions, which means they are governed by the Pauli Exclusion Principle: no two electrons can occupy the same quantum state. They are also indistinguishable. You can't label one electron 'Alice' and another 'Bob' and track them. If you swap two electrons, the system is fundamentally unchanged.

These two rules—single occupancy and indistinguishability—are the keys to their statistical behaviour. Imagine you have a handful of available energy states, like seats in a tiny cinema, and a group of electrons, the audience. How many ways can you seat them? Unlike classical particles (which are distinguishable, like people with assigned tickets), with electrons, it only matters which seats are taken, not who is in them.

Let's say we have $g_i$ available states at a certain energy level $E_i$, and we want to place $n_i$ electrons into them. Because of the exclusion principle, we must have $n_i \le g_i$. The number of ways to do this is a classic combinatorics problem: choosing $n_i$ states to be occupied from a total of $g_i$ available states. The formula for this is "$g_i$ choose $n_i$".

To find the total number of ways $\Omega$ to arrange all the electrons across all energy levels, we multiply the possibilities for each level.

In statistical mechanics, the most probable distribution of particles is the one with the largest number of possible arrangements, or microstates ($\,\Omega\,$). It's easier to work with the logarithm of $\Omega$, because it turns the products into sums and allows us to use Stirling's approximation for the factorials, since we're dealing with very large numbers of particles and states.

Applying Stirling's approximation to $\ln(\Omega)$ gives us:

We want to find the set of occupation numbers $n_i$ that maximises this expression. However, we have two constraints:

1. The total number of particles is constant: $\sum_i n_i = N$.
2. The total energy of the system is constant: $\sum_i n_i E_i = U$.

To handle this constrained maximisation, we use the method of Lagrange multipliers introducing multipliers $\alpha$ and $\beta$. We want to maximise the following function $F$:

We find the maximum by taking the partial derivative of $F$ with respect to each $n_i$ and setting it to zero: $\frac{\partial F}{\partial n_i} = 0$. After some algebra, this leads to a simple relationship for the occupation number $n_i$ for each state $g_i$.

Solving the maximisation problem yields the most probable occupation number $n_i$ for an energy level $E_i$. We define the average occupancy of a single state at that energy level, $f(E_i)$, as the ratio $n_i/g_i$. This gives us the final form of the Fermi-Dirac distribution:

Through deeper thermodynamic connections, these multipliers can be identified. The parameter $\beta$ is related to temperature, where $\beta = 1/(k_B T)$. The parameter $\alpha$ is related to the chemical potential $\mu$, where $\alpha = -\mu/(k_B T)$. The chemical potential represents the energy required to add one more particle to the system while keeping temperature and volume constant.

Substituting these in gives the familiar form of the Fermi-Dirac function:

This function has a distinct shape. At absolute zero ($T=0$ K), it's a perfect step function: all states with energy below the chemical potential are 100% occupied ($f(E)=1$), and all states above it are 100% empty ($f(E)=0$). As temperature increases, the step smooths out into a curve. States near the chemical potential have a chance of being either occupied or empty, creating a 'thermal tail' of excited electrons.

How does this differ from classical Maxwell-Boltzmann statistics? Maxwell-Boltzmann applies to identical but distinguishable particles where any number of particles can occupy a single state. Its distribution function is:

Notice that when the energy $E$ is much larger than the chemical potential $\mu$, the $+1$ in the denominator of the Fermi-Dirac function becomes negligible. In this high-energy limit, the Fermi-Dirac distribution simplifies to look just like the Maxwell-Boltzmann distribution. This makes sense: at very high energies, states are so sparsely occupied that the chance of two electrons wanting the same state is almost zero. The Pauli Exclusion Principle becomes less relevant, and the fermions start to behave more like classical particles.

Time to check your understanding of the Fermi-Dirac distribution.

This derivation is the mathematical foundation for understanding the behaviour of electrons in solids, which is essential for designing everything from transistors to solar cells.