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 ss, which can contain 00 or 11 fermions. Because the occupation number NN is variable, we turn to the grand canonical ensemble, whose grand partition function Z\mathcal{Z} is as follows, where ε\varepsilon is the energy of ss and μ\mu is the chemical potential:

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

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

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

The average number of particles ⟨N⟩\expval{N} in ss is then found by taking a derivative of Ω\Omega:

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

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

⟨N⟩=fF(ε)=1eβ(ε−μ)+1\begin{aligned} \boxed{ \expval{N} = f_F(\varepsilon) = \frac{1}{e^{\beta (\varepsilon - \mu)} + 1} } \end{aligned}

This gives the expected occupation number ⟨N⟩\expval{N} of state ss with energy ε\varepsilon, given a temperature TT and chemical potential μ\mu.

References

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