Categories: Physics, Quantum mechanics, Statistics.

Fermi-Dirac distribution

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

Consider one single-particle state \(s\), which can contain \(0\) or \(1\) fermions. Because the occupation number \(N_s\) is variable, we turn to the grand canonical ensemble, whose grand partition function \(\mathcal{Z}_s\) is as follows, where we sum over all microstates of \(s\):

\[\begin{aligned} \mathcal{Z}_s = \sum_{N_s = 0}^1 \exp\!(- \beta N_s (\varepsilon_s - \mu)) = 1 + \exp\!(- \beta (\varepsilon_s - \mu)) \end{aligned}\]

Where \(\mu\) is the chemical potential, and \(\varepsilon_s\) is the energy contribution per particle in \(s\), i.e. the total energy of all particles \(E_s = \varepsilon_s N_s\).

The corresponding thermodynamic potential is the Landau potential \(\Omega_s\), given by:

\[\begin{aligned} \Omega_s = - k T \ln{\mathcal{Z}_s} = - k T \ln\!\Big( 1 + \exp\!(- \beta (\varepsilon_s - \mu)) \Big) \end{aligned}\]

The average number of particles \(\expval{N_s}\) in state \(s\) is then found to be as follows:

\[\begin{aligned} \expval{N_s} = - \pdv{\Omega_s}{\mu} = k T \pdv{\ln{\mathcal{Z}_s}}{\mu} = \frac{\exp\!(- \beta (\varepsilon_s - \mu))}{1 + \exp\!(- \beta (\varepsilon_s - \mu))} \end{aligned}\]

By multiplying both the numerator and the denominator by \(\exp\!(\beta (\varepsilon_s \!-\! \mu))\), we arrive at the standard form of the Fermi-Dirac distribution or Fermi function \(f_F\):

\[\begin{aligned} \boxed{ \expval{N_s} = f_F(\varepsilon_s) = \frac{1}{\exp\!(\beta (\varepsilon_s - \mu)) + 1} } \end{aligned}\]

This tells the expected occupation number \(\expval{N_s}\) of state \(s\), given a temperature \(T\) and chemical potential \(\mu\). The corresponding variance \(\sigma_s^2\) of \(N_s\) is found to be:

\[\begin{aligned} \boxed{ \sigma_s^2 = k T \pdv{\expval{N_s}}{\mu} = \expval{N_s} \big(1 - \expval{N_s}\big) } \end{aligned}\]

References

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

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