From 6ce0bb9a8f9fd7d169cbb414a9537d68c5290aae Mon Sep 17 00:00:00 2001 From: Prefetch Date: Fri, 14 Oct 2022 23:25:28 +0200 Subject: Initial commit after migration from Hugo --- .../know/concept/fermi-dirac-distribution/index.md | 80 ++++++++++++++++++++++ 1 file changed, 80 insertions(+) create mode 100644 source/know/concept/fermi-dirac-distribution/index.md (limited to 'source/know/concept/fermi-dirac-distribution') diff --git a/source/know/concept/fermi-dirac-distribution/index.md b/source/know/concept/fermi-dirac-distribution/index.md new file mode 100644 index 0000000..7fc4f0e --- /dev/null +++ b/source/know/concept/fermi-dirac-distribution/index.md @@ -0,0 +1,80 @@ +--- +title: "Fermi-Dirac distribution" +date: 2021-07-11 +categories: +- Physics +- Statistics +- Quantum mechanics +layout: "concept" +--- + +**Fermi-Dirac statistics** describe how identical **fermions**, +which obey the [Pauli exclusion principle](/know/concept/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 $N$ is variable, +we turn to the [grand canonical ensemble](/know/concept/grand-canonical-ensemble/), +whose grand partition function $\mathcal{Z}$ is as follows, +where we sum over all microstates of $s$: + +$$\begin{aligned} + \mathcal{Z} + = \sum_{N = 0}^1 \exp(- \beta N (\varepsilon - \mu)) + = 1 + \exp(- \beta (\varepsilon - \mu)) +\end{aligned}$$ + +Where $\mu$ is the chemical potential, +and $\varepsilon$ is the energy contribution per particle in $s$, +i.e. the total energy of all particles $E = \varepsilon N$. + +The corresponding [thermodynamic potential](/know/concept/thermodynamic-potential/) +is the Landau potential $\Omega$, given by: + +$$\begin{aligned} + \Omega + = - k T \ln{\mathcal{Z}} + = - k T \ln\!\Big( 1 + \exp(- \beta (\varepsilon - \mu)) \Big) +\end{aligned}$$ + +The average number of particles $\Expval{N}$ +in state $s$ is then found to be as follows: + +$$\begin{aligned} + \Expval{N} + = - \pdv{\Omega}{\mu} + = k T \pdv{\ln{\mathcal{Z}}}{\mu} + = \frac{\exp(- \beta (\varepsilon - \mu))}{1 + \exp(- \beta (\varepsilon - \mu))} +\end{aligned}$$ + +By multiplying both the numerator and the denominator by $\exp(\beta (\varepsilon \!-\! \mu))$, +we arrive at the standard form of +the **Fermi-Dirac distribution** or **Fermi function** $f_F$: + +$$\begin{aligned} + \boxed{ + \Expval{N} + = f_F(\varepsilon) + = \frac{1}{\exp(\beta (\varepsilon - \mu)) + 1} + } +\end{aligned}$$ + +This tells the expected occupation number $\Expval{N}$ of state $s$, +given a temperature $T$ and chemical potential $\mu$. +The corresponding variance $\sigma^2$ of $N$ is found to be: + +$$\begin{aligned} + \boxed{ + \sigma^2 + = k T \pdv{\Expval{N}}{\mu} + = \Expval{N} \big(1 - \Expval{N}\big) + } +\end{aligned}$$ + + + +## References +1. H. Gould, J. Tobochnik, + *Statistical and thermal physics*, 2nd edition, + Princeton. -- cgit v1.2.3