diff options
Diffstat (limited to 'source/know/concept/boltzmann-equation')
-rw-r--r-- | source/know/concept/boltzmann-equation/index.md | 31 |
1 files changed, 9 insertions, 22 deletions
diff --git a/source/know/concept/boltzmann-equation/index.md b/source/know/concept/boltzmann-equation/index.md index 9ed2fd2..d2631b2 100644 --- a/source/know/concept/boltzmann-equation/index.md +++ b/source/know/concept/boltzmann-equation/index.md @@ -145,11 +145,8 @@ $$\begin{aligned} } \end{aligned}$$ -<div class="accordion"> -<input type="checkbox" id="proof-moment0"/> -<label for="proof-moment0">Proof</label> -<div class="hidden" markdown="1"> -<label for="proof-moment0">Proof.</label> + +{% include proof/start.html id="proof-moment0" -%} We insert $$Q = m$$ into our prototype, and since $$m$$ is constant, the rest is trivial: @@ -159,9 +156,8 @@ $$\begin{aligned} \\ &= \pdv{\rho}{t} + \nabla \cdot \big(\rho \Expval{\vb{v}}\big) - 0 \end{aligned}$$ +{% include proof/end.html id="proof-moment0" %} -</div> -</div> If we instead choose the momentum $$Q = m \vb{v}$$, we find that the **first moment** of the BTE describes conservation of momentum, @@ -174,11 +170,8 @@ $$\begin{aligned} } \end{aligned}$$ -<div class="accordion"> -<input type="checkbox" id="proof-moment1"/> -<label for="proof-moment1">Proof</label> -<div class="hidden" markdown="1"> -<label for="proof-moment1">Proof.</label> + +{% include proof/start.html id="proof-moment1" -%} We insert $$Q = m \vb{v}$$ into our prototype and recognize $$\rho$$ wherever possible: $$\begin{aligned} @@ -220,9 +213,8 @@ $$\begin{aligned} 0 &= \pdv{}{t}\big(\rho \vb{V}\big) + \nabla \cdot \big(\rho \vb{V} \vb{V} + \hat{P}\big) - n \vb{F} \end{aligned}$$ +{% include proof/end.html id="proof-moment1" %} -</div> -</div> Finally, if we choose the kinetic energy $$Q = m |\vb{v}|^2 / 2$$, we find that the **second moment** gives conservation of energy, @@ -237,11 +229,8 @@ $$\begin{aligned} } \end{aligned}$$ -<div class="accordion"> -<input type="checkbox" id="proof-moment2"/> -<label for="proof-moment2">Proof</label> -<div class="hidden" markdown="1"> -<label for="proof-moment2">Proof.</label> + +{% include proof/start.html id="proof-moment2" -%} We insert $$Q = m |\vb{v}|^2 / 2$$ into our prototype and recognize $$\rho$$ wherever possible: $$\begin{aligned} @@ -349,9 +338,7 @@ $$\begin{aligned} \end{bmatrix} = \sum_{i=1}^{3} \sum_{j=1}^{3} \pdv{P_{ij}}{x_j} V_i \end{aligned}$$ - -</div> -</div> +{% include proof/end.html id="proof-moment2" %} |