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}

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

© Marcus R.A. Newman, a.k.a. "Prefetch". Available under CC BY-SA 4.0.