Fermi-Dirac statistics — fermions, Fermi energy

WHY does Fermi-Dirac statistics exist?

WHAT problem are we solving? Classical Maxwell-Boltzmann statistics tells us how distinguishable particles spread over energy levels. But quantum particles are indistinguishable, and fermions have an extra rule: the Pauli exclusion principle. We need a counting rule that respects this.

WHY does the rule matter? Without exclusion, all electrons in a metal would collapse to the ground state at $T=0$ and metals would behave totally differently (no electron degeneracy pressure, no white dwarfs, no semiconductor band gaps mattering the way they do).

HOW we derive the Fermi-Dirac distribution (from scratch)

We use the grand canonical ensemble: the system exchanges particles and energy with a reservoir at temperature $T$ and chemical potential $\mu$.

Step 1 — Focus on a single quantum state of energy $\varepsilon$. Why? Because fermion states are independent in occupation — each single-particle state is its own tiny subsystem that can hold $n=0$ or $n=1$ particle (exclusion!).

Step 2 — Write the grand partition function for that one state. The grand canonical weight of a microstate with $n$ particles and energy $n\varepsilon$ is $e^{-(n\varepsilon-\mu n)/k_BT}$. $\mathcal{Z} = \sum_{n=0}^{1} e^{-n(\varepsilon-\mu)/k_BT} = 1 + e^{-(\varepsilon-\mu)/k_BT}$ Why only $n=0,1$? Pauli exclusion: a single state can't hold two identical fermions.

Step 3 — Average occupation number. $\bar n = \frac{\sum_n n\, e^{-n(\varepsilon-\mu)/k_BT}}{\mathcal{Z}} = \frac{0\cdot 1 + 1\cdot e^{-(\varepsilon-\mu)/k_BT}}{1 + e^{-(\varepsilon-\mu)/k_BT}}$

Divide top and bottom by $e^{-(\varepsilon-\mu)/k_BT}$:

The Fermi energy: the $T=0$ limit

WHAT is $E_F$? Define the Fermi energy $E_F \equiv \mu(T=0)$.

Take $T\to 0$ in $f(\varepsilon)$:

• For $\varepsilon < E_F$: exponent $\to e^{-\infty}=0 \Rightarrow f=1$.
• For $\varepsilon > E_F$: exponent $\to e^{+\infty}=\infty \Rightarrow f=0$.

Deriving $E_F$ for a free electron gas (3D)

Step 1 — Count states (density of states). In a box of volume $V$, allowed momenta are quantized; in $k$-space states fill uniformly with density $V/(2\pi)^3$. Including spin factor $g_s=2$: $\text{\# states with } |\mathbf k|<k_F = 2\cdot\frac{V}{(2\pi)^3}\cdot\frac{4}{3}\pi k_F^3$ Why the $\tfrac43\pi k_F^3$? Volume of a sphere of radius $k_F$ — at $T=0$ the filled states form a sphere (the Fermi sphere).

Step 2 — Set this equal to the number of electrons $N$. $N = \frac{V}{3\pi^2}k_F^3 \quad\Rightarrow\quad k_F = \left(3\pi^2 \frac{N}{V}\right)^{1/3}$

Step 3 — Convert to energy using $\varepsilon = \hbar^2 k^2/2m$:

We also define Fermi temperature $T_F = E_F/k_B$ and Fermi velocity $v_F = \hbar k_F/m$.

Worked Examples

Common Mistakes (Steel-manned)

Flashcards

Connections

• Pauli Exclusion Principle — the microscopic origin of the $+1$ and the step function.
• Maxwell-Boltzmann Statistics — recovered when $f\ll1$ (dilute / high-$T$ limit).
• Bose-Einstein Statistics — the $-1$ sibling; allows condensation.
• Density of States — needed to turn $f$ into actual particle counts.
• Grand Canonical Ensemble — the framework used to derive $f$.
• Electron Degeneracy Pressure & White Dwarfs — astrophysical consequence of the Fermi sea.
• Chemical Potential — $\mu$, the half-filling energy; equals $E_F$ at $T=0$.
• Sommerfeld Expansion — how $\mu(T)$ and heat capacity follow from $f$.

Concept Map

Hinglish (regional understanding)