Fermi-Dirac Distribution

Video Explaining Fermi-Dirac Statistics

Topics

• Fermi Energy

$f(E)=\frac{1}{1+e^{(E-E_f)/kT}}$

• Probability of find a particle at a certain energy $E$
• $E_f$ is the fermi level
• $T$ is the temperature of the gas. For $T=0$, the function approaches a step function, indicating you’ll have 0 probability of finding particles above $E_f$
  
  Desmos Demo

Statistical Mechanics

$f(\epsilon )\equiv ⟨N(\epsilon )⟩=\frac{1}{e^{\beta (\epsilon -\mu )}+1}$

• Denotes the average occupancy
• Same as above equation, but with

$\mathbb{Z}=e^{\beta (\mu -\epsilon )}+1=\lambda e^{-\beta \epsilon }+1$

• $\mathbb{Z}$ is the grand partition function
• $\lambda$ is the absolute activity

$⟨N⟩=\frac{1}{\mathbb{Z}}$