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Diffstat (limited to 'source/know/concept/bose-einstein-distribution/index.md')
| -rw-r--r-- | source/know/concept/bose-einstein-distribution/index.md | 36 |
1 files changed, 18 insertions, 18 deletions
diff --git a/source/know/concept/bose-einstein-distribution/index.md b/source/know/concept/bose-einstein-distribution/index.md index 5640e69..ea5ca68 100644 --- a/source/know/concept/bose-einstein-distribution/index.md +++ b/source/know/concept/bose-einstein-distribution/index.md @@ -14,19 +14,19 @@ which do not obey the [Pauli exclusion principle](/know/concept/pauli-exclusion- distribute themselves across the available states in a system at equilibrium. -Consider a single-particle state $$s$$, +Consider a single-particle state $$\ket{i}$$, which can contain any number of bosons. -Since the occupation number $$N$$ is variable, +Since the occupation number $$n_i$$ is variable, we use the [grand canonical ensemble](/know/concept/grand-canonical-ensemble/), whose grand partition function $$\mathcal{Z}$$ is as shown below, -where $$\varepsilon$$ is the energy per particle, +where $$\varepsilon_i$$ is the energy per particle, and $$\mu$$ is the chemical potential. We evaluate the sum in $$\mathcal{Z}$$ as a geometric series: $$\begin{aligned} \mathcal{Z} - = \sum_{N = 0}^\infty \Big( e^{-\beta (\varepsilon - \mu)} \Big)^{N} - = \frac{1}{1 - e^{-\beta (\varepsilon - \mu)}} + = \sum_{m = 0}^\infty \Big( e^{-\beta (\varepsilon_i - \mu)} \Big)^{m} + = \frac{1}{1 - e^{-\beta (\varepsilon_i - \mu)}} \end{aligned}$$ The corresponding [thermodynamic potential](/know/concept/thermodynamic-potential/) @@ -35,42 +35,42 @@ is the Landau potential $$\Omega$$, given by: $$\begin{aligned} \Omega = - k T \ln{\mathcal{Z}} - = k T \ln\!\big( 1 - e^{-\beta (\varepsilon - \mu)} \big) + = k T \ln\!\big( 1 - e^{-\beta (\varepsilon_i - \mu)} \big) \end{aligned}$$ -The average number of particles $$\expval{N}$$ in $$s$$ +The average number of particles $$\expval{n_i}$$ in $$\ket{i}$$ is then found by taking a derivative of $$\Omega$$: $$\begin{aligned} - \expval{N} + \expval{n_i} = - \pdv{\Omega}{\mu} = k T \pdv{\ln{\mathcal{Z}}}{\mu} - = \frac{e^{-\beta (\varepsilon - \mu)}}{1 - e^{-\beta (\varepsilon - \mu)}} + = \frac{e^{-\beta (\varepsilon_i - \mu)}}{1 - e^{-\beta (\varepsilon_i - \mu)}} \end{aligned}$$ -By multiplying both the numerator and the denominator by $$e^{\beta(\varepsilon \!-\! \mu)}$$, +By multiplying both the numerator and the denominator by $$e^{\beta(\varepsilon_i - \mu)}$$, we arrive at the standard form of the **Bose-Einstein distribution** $$f_B$$: $$\begin{aligned} \boxed{ - \expval{N} - = f_B(\varepsilon) - = \frac{1}{e^{\beta (\varepsilon - \mu)} - 1} + \expval{n_i} + = f_B(\varepsilon_i) + = \frac{1}{e^{\beta (\varepsilon_i - \mu)} - 1} } \end{aligned}$$ -This gives the expected occupation number $$\expval{N}$$ -of state $$s$$ with energy $$\varepsilon$$, +This gives the expected occupation number $$\expval{n_i}$$ +of state $$\ket{i}$$ with energy $$\varepsilon_i$$, given a temperature $$T$$ and chemical potential $$\mu$$. {% comment %} -The corresponding variance $$\sigma^2 \equiv \expval{N^2} - \expval{N}^2$$ is found to be: +The corresponding variance $$\sigma^2 \equiv \expval{n_i^2} - \expval{n_i}^2$$ is found to be: $$\begin{aligned} \boxed{ \sigma^2 - = k T \pdv{\expval{N}}{\mu} - = \expval{N} \big(1 + \expval{N}\!\big) + = k T \pdv{\expval{n_i}}{\mu} + = \expval{n_i} \big(1 + \expval{n_i}\!\big) } \end{aligned}$$ {% endcomment %} |