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authorPrefetch2023-04-02 16:57:12 +0200
committerPrefetch2023-04-02 16:57:12 +0200
commita8d31faecc733fa4d63fde58ab98a5e9d11029c2 (patch)
treeb8d039b13e026fb68f0bed439a2cb73397c35981 /source/know/concept/fermi-dirac-distribution
parent9b9346d5e54244f3e2859c3f80e47f2de345a2ad (diff)
Improve knowledge base
Diffstat (limited to 'source/know/concept/fermi-dirac-distribution')
-rw-r--r--source/know/concept/fermi-dirac-distribution/index.md41
1 files changed, 21 insertions, 20 deletions
diff --git a/source/know/concept/fermi-dirac-distribution/index.md b/source/know/concept/fermi-dirac-distribution/index.md
index 09a3e76..2a38eb3 100644
--- a/source/know/concept/fermi-dirac-distribution/index.md
+++ b/source/know/concept/fermi-dirac-distribution/index.md
@@ -11,67 +11,68 @@ 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.
+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 $$s$$:
+where $$\varepsilon$$ is the energy of $$s$$
+and $$\mu$$ is the chemical potential:
$$\begin{aligned}
\mathcal{Z}
- = \sum_{N = 0}^1 \exp(- \beta N (\varepsilon - \mu))
- = 1 + \exp(- \beta (\varepsilon - \mu))
+ = \sum_{N = 0}^1 \Big( e^{-\beta (\varepsilon - \mu)} \Big)^N
+ = 1 + e^{-\beta (\varepsilon - \mu)}
\end{aligned}$$
-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:
$$\begin{aligned}
\Omega
= - k T \ln{\mathcal{Z}}
- = - k T \ln\!\Big( 1 + \exp(- \beta (\varepsilon - \mu)) \Big)
+ = - k T \ln\!\Big( 1 + e^{-\beta (\varepsilon - \mu)} \Big)
\end{aligned}$$
-The average number of particles $$\Expval{N}$$
-in state $$s$$ is then found to be as follows:
+The average number of particles $$\expval{N}$$
+in $$s$$ is then found by taking a derivative of $$\Omega$$:
$$\begin{aligned}
- \Expval{N}
+ \expval{N}
= - \pdv{\Omega}{\mu}
= k T \pdv{\ln{\mathcal{Z}}}{\mu}
- = \frac{\exp(- \beta (\varepsilon - \mu))}{1 + \exp(- \beta (\varepsilon - \mu))}
+ = \frac{e^{-\beta (\varepsilon - \mu)}}{1 + e^{-\beta (\varepsilon - \mu)}}
\end{aligned}$$
-By multiplying both the numerator and the denominator by $$\exp(\beta (\varepsilon \!-\! \mu))$$,
+By multiplying both the numerator and the denominator by $$e^{\beta (\varepsilon \!-\! \mu)}$$,
we arrive at the standard form of
the **Fermi-Dirac distribution** or **Fermi function** $$f_F$$:
$$\begin{aligned}
\boxed{
- \Expval{N}
+ \expval{N}
= f_F(\varepsilon)
- = \frac{1}{\exp(\beta (\varepsilon - \mu)) + 1}
+ = \frac{1}{e^{\beta (\varepsilon - \mu)} + 1}
}
\end{aligned}$$
-This tells the expected occupation number $$\Expval{N}$$ of state $$s$$,
+This gives the expected occupation number $$\expval{N}$$
+of state $$s$$ with energy $$\varepsilon$$,
given a temperature $$T$$ and chemical potential $$\mu$$.
-The corresponding variance $$\sigma^2$$ of $$N$$ is found to be:
+
+{% comment %}
+The corresponding variance $$\sigma^2 \equiv \expval{N^2} - \expval{N}^2$$ is found to be:
$$\begin{aligned}
\boxed{
\sigma^2
- = k T \pdv{\Expval{N}}{\mu}
- = \Expval{N} \big(1 - \Expval{N}\big)
+ = k T \pdv{\expval{N}}{\mu}
+ = \expval{N} \big(1 - \expval{N}\big)
}
\end{aligned}$$
+{% endcomment %}