-
+
+{% 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" %}
-
-
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 @@ $$\begin{aligned}
}
\end{aligned}$$
-
-
-
-
-
+
+{% 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}$$
+{% include proof/end.html id="proof-moment1" %}
-
-
Finally, if we choose the kinetic energy $$Q = m |\vb{v}|^2 / 2$$,
we find that the **second moment** gives conservation of energy,
@@ -237,11 +229,8 @@ $$\begin{aligned}
}
\end{aligned}$$
-
-
-
-
-
+
+{% include proof/start.html id="proof-moment2" -%}
We insert $$Q = m |\vb{v}|^2 / 2$$ into our prototype and recognize $$\rho$$ wherever possible:
$$\begin{aligned}
@@ -349,9 +338,7 @@ $$\begin{aligned}
\end{bmatrix}
= \sum_{i=1}^{3} \sum_{j=1}^{3} \pdv{P_{ij}}{x_j} V_i
\end{aligned}$$
-
-
-
+{% include proof/end.html id="proof-moment2" %}
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