diff options
Diffstat (limited to 'source/know/concept/bose-einstein-distribution')
| -rw-r--r-- | source/know/concept/bose-einstein-distribution/index.md | 26 |
1 files changed, 13 insertions, 13 deletions
diff --git a/source/know/concept/bose-einstein-distribution/index.md b/source/know/concept/bose-einstein-distribution/index.md index 594d6e0..e420d7c 100644 --- a/source/know/concept/bose-einstein-distribution/index.md +++ b/source/know/concept/bose-einstein-distribution/index.md @@ -14,13 +14,13 @@ which do not obey the [Pauli exclusion principle](/know/concept/pauli-exclusion- will distribute themselves across the available states in a system at equilibrium. -Consider a single-particle state $s$, +Consider a single-particle state $$s$$, which can contain any number of bosons. -Since the occupation number $N$ is variable, +Since 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 $\varepsilon$ is the energy per particle, -and $\mu$ is the chemical potential: +whose grand partition function $$\mathcal{Z}$$ is as follows, +where $$\varepsilon$$ is the energy per particle, +and $$\mu$$ is the chemical potential: $$\begin{aligned} \mathcal{Z} @@ -29,7 +29,7 @@ $$\begin{aligned} \end{aligned}$$ The corresponding [thermodynamic potential](/know/concept/thermodynamic-potential/) -is the Landau potential $\Omega$, given by: +is the Landau potential $$\Omega$$, given by: $$\begin{aligned} \Omega @@ -37,8 +37,8 @@ $$\begin{aligned} = k T \ln\!\Big( 1 - \exp(- \beta (\varepsilon - \mu)) \Big) \end{aligned}$$ -The average number of particles $\Expval{N}$ -is found by taking a derivative of $\Omega$: +The average number of particles $$\Expval{N}$$ +is found by taking a derivative of $$\Omega$$: $$\begin{aligned} \Expval{N} @@ -47,8 +47,8 @@ $$\begin{aligned} = \frac{\exp(- \beta (\varepsilon - \mu))}{1 - \exp(- \beta (\varepsilon - \mu))} \end{aligned}$$ -By multitplying both the numerator and the denominator by $\exp(\beta(\varepsilon \!-\! \mu))$, -we arrive at the standard form of the **Bose-Einstein distribution** $f_B$: +By multitplying both the numerator and the denominator by $$\exp(\beta(\varepsilon \!-\! \mu))$$, +we arrive at the standard form of the **Bose-Einstein distribution** $$f_B$$: $$\begin{aligned} \boxed{ @@ -58,9 +58,9 @@ $$\begin{aligned} } \end{aligned}$$ -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{ |