diff options
Diffstat (limited to 'source/know/concept/fermi-dirac-distribution')
| -rw-r--r-- | source/know/concept/fermi-dirac-distribution/index.md | 32 |
1 files changed, 16 insertions, 16 deletions
diff --git a/source/know/concept/fermi-dirac-distribution/index.md b/source/know/concept/fermi-dirac-distribution/index.md index ea1f8b8..09a3e76 100644 --- a/source/know/concept/fermi-dirac-distribution/index.md +++ b/source/know/concept/fermi-dirac-distribution/index.md @@ -13,12 +13,12 @@ layout: "concept" 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, +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 $s$: +whose grand partition function $$\mathcal{Z}$$ is as follows, +where we sum over all microstates of $$s$$: $$\begin{aligned} \mathcal{Z} @@ -26,12 +26,12 @@ $$\begin{aligned} = 1 + \exp(- \beta (\varepsilon - \mu)) \end{aligned}$$ -Where $\mu$ is the chemical potential, -and $\varepsilon$ is the energy contribution per particle in $s$, -i.e. the total energy of all particles $E = \varepsilon N$. +Where $$\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$$. The corresponding [thermodynamic potential](/know/concept/thermodynamic-potential/) -is the Landau potential $\Omega$, given by: +is the Landau potential $$\Omega$$, given by: $$\begin{aligned} \Omega @@ -39,8 +39,8 @@ $$\begin{aligned} = - k T \ln\!\Big( 1 + \exp(- \beta (\varepsilon - \mu)) \Big) \end{aligned}$$ -The average number of particles $\Expval{N}$ -in state $s$ is then found to be as follows: +The average number of particles $$\Expval{N}$$ +in state $$s$$ is then found to be as follows: $$\begin{aligned} \Expval{N} @@ -49,9 +49,9 @@ $$\begin{aligned} = \frac{\exp(- \beta (\varepsilon - \mu))}{1 + \exp(- \beta (\varepsilon - \mu))} \end{aligned}$$ -By multiplying both the numerator and the denominator by $\exp(\beta (\varepsilon \!-\! \mu))$, +By multiplying both the numerator and the denominator by $$\exp(\beta (\varepsilon \!-\! \mu))$$, we arrive at the standard form of -the **Fermi-Dirac distribution** or **Fermi function** $f_F$: +the **Fermi-Dirac distribution** or **Fermi function** $$f_F$$: $$\begin{aligned} \boxed{ @@ -61,9 +61,9 @@ $$\begin{aligned} } \end{aligned}$$ -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: +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{ |