Categories: Physics, Quantum mechanics, Statistics.

Fermi-Dirac distribution

Fermi-Dirac statistics describe how identical fermions, which obey the Pauli exclusion principle, distribute themselves across the available states in a system at equilibrium.

Consider one single-particle state ∣i⟩\ket{i}, which can contain 00 or 11 fermions. Because the occupation number nin_i is variable, we turn to the grand canonical ensemble, whose grand partition function Z\mathcal{Z} is as follows, where εi\varepsilon_i is the energy of ∣i⟩\ket{i} and μ\mu is the chemical potential:

Z=∑m=01(e−β(εi−μ))m=1+e−β(εi−μ)\begin{aligned} \mathcal{Z} = \sum_{m = 0}^1 \Big( e^{-\beta (\varepsilon_i - \mu)} \Big)^m = 1 + e^{-\beta (\varepsilon_i - \mu)} \end{aligned}

The corresponding thermodynamic potential is the Landau potential Ω\Omega, given by:

Ω=−kTln⁡Z=−kTln⁡ ⁣(1+e−β(εi−μ))\begin{aligned} \Omega = - k T \ln{\mathcal{Z}} = - k T \ln\!\Big( 1 + e^{-\beta (\varepsilon_i - \mu)} \Big) \end{aligned}

The average number of particles ⟨ni⟩\expval{n_i} in ∣i⟩\ket{i} is then found by taking a derivative of Ω\Omega:

⟨ni⟩=−∂Ω∂μ=kT∂ln⁡Z∂μ=e−β(εi−μ)1+e−β(εi−μ)\begin{aligned} \expval{n_i} = - \pdv{\Omega}{\mu} = k T \pdv{\ln{\mathcal{Z}}}{\mu} = \frac{e^{-\beta (\varepsilon_i - \mu)}}{1 + e^{-\beta (\varepsilon_i - \mu)}} \end{aligned}

By multiplying both the numerator and the denominator by eβ(εi−μ)e^{\beta (\varepsilon_i - \mu)}, we arrive at the standard form of the Fermi-Dirac distribution or Fermi function fFf_F:

⟨ni⟩=fF(εi)=1eβ(εi−μ)+1\begin{aligned} \boxed{ \expval{n_i} = f_F(\varepsilon_i) = \frac{1}{e^{\beta (\varepsilon_i - \mu)} + 1} } \end{aligned}

This gives the expected occupation number ⟨ni⟩\expval{n_i} of state ∣i⟩\ket{i} with energy εi\varepsilon_i, given a temperature TT and chemical potential μ\mu.

References

  1. H. Gould, J. Tobochnik, Statistical and thermal physics, 2nd edition, Princeton.