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10.5: Appendix - Jacobians

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Oct 27, 2024
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- 10.4: Average relative velocity and collision frequency
- 11: Boltzmann Statistics

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Appendix: Jacobians

A.1. 3D to 1D

A Jacobian is a mathematical entity used to switch partials inside an integral. For the case of velocity in three dimensions being converted into spherical coordinates, where $v$ is the net velocity and akin to a sphere’s radius:

\[\partial v_x\cdot \partial v_y\cdot \partial v_z=\left|\frac{\partial (v_x,v_y,v_z)}{\partial (v,\theta ,\phi )}\right|\cdot \partial v\cdot \partial \phi \cdot \partial \theta =v^2\cdot

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sin\ (phi )\cdot \partial v\cdot \partial \phi \cdot \partial \theta \nonumber \]

To evaluate the above, we should have some idea how to convert v ${}_{x}$ v ${}_{y}$ v ${}_{z}$ into spherical coordinates to begin with. It’s just a lesson in geometry:

\[v_x=v\cdot

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sin\ (phi )\cdot

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cos\ (theta )v_y=v\cdot

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sin\ (phi )\cdot

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sin\ (theta )v_z=v\cdot

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cos\ (phi ) \nonumber \]

where $v=\sqrt{v^2_x+v^2_y+v^2_z}$ . The Jacobian $\left|\frac{\partial (v_x,v_y,v_z)}{\partial (v,\theta ,\phi )}\right|$ is the absolute values of the determinant of the following matrix:

\[\left|

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detleft[\frac{∂v_x}{\partial v}frac{∂v_x}{\partial \theta }frac{∂v_x}{\partial \varphi }frac{∂v_y}{\partial v}frac{∂v_y}{\partial \theta }frac{∂v_y}{\partial \varphi }frac{∂v_z}{\partial v}frac{∂v_z}{\partial \theta }frac{∂v_z}{\partial \varphi }right]right| \nonumber \]

In our coordinate system the absolute value of the determinant of the matrix is:

\[\left|

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detleft[

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sin\ (phi )

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cos\ (theta )\ -v

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sin\ (phi )

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sin\ (theta )\ v

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cos\ (phi )

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cos\ (theta )\

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sin\ (phi )

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sin\ (theta )\ v

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sin\ (phi )

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cos\ (theta )\ v

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cos\ (phi )

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sin\ (theta )\

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cos\ (phi )\ 0\ -v

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sin\ (phi )right]right| \nonumber \]

\[=\left|-v^2(phi )(theta )-v^2

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sin\ (phi )(phi )(theta )+0-v^2(phi )(theta )-0-v^2

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sin\ (phi )(phi )(theta )\right|=\left|-v^2(phi )((theta )+(theta ))-v^2

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sin\ (phi )(phi )((theta )+(theta ))\right|=\left|-v^2(phi )-v^2

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sin\ (phi )(phi )\right|=\left|-v^2

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sin\ (phi )((phi )+(phi ))\right|=\left|-v^2

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sin\ (phi )\right| \nonumber \]

Since we take the absolute value of this result (the negative sign goes away), the final answer is: \(\partial v_x\cdot \partial v_y\cdot \partial v_z=v^2\cdot

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sin\ (phi )\cdot \partial v\cdot \partial \phi \cdot \partial \theta\)

When we apply the Jacobian to the Maxwell-Boltzmann formula we see that (bold emphasis added):

\[{\left(\frac{m}{2\cdot \pi \cdot k_B\cdot T}\right)}^{\frac{3}{2}}\cdot e^{\frac{-m\cdot v^2}{2\cdot k_B\cdot T}}\cdot \partial v_x\cdot \partial v_y\cdot \partial v_z={\left(\frac{m}{2\cdot \pi \cdot k_B\cdot T}\right)}^{\frac{3}{2}}\cdot e^{\frac{-m\cdot v^2}{2\cdot k_B\cdot T}}\cdot v^2\cdot

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sin\ (phi )\cdot \partial v\cdot \partial \phi \cdot \partial \theta \nonumber \]

Now we can integrate out the angles:

\[{\left(\frac{m}{2\cdot \pi \cdot k_B\cdot T}\right)}^{\frac{3}{2}}\cdot e^{\frac{-m\cdot v^2}{2\cdot k_B\cdot T}}\cdot v^2\cdot

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sin\ (phi )\cdot \partial v\cdot \int^{2\pi }_0{}

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sin\ (phi )\cdot \partial \phi \cdot \int^{\pi }_0{}\partial \theta \nonumber \]

which leaves us with just a constant:

\[\int^{\pi }_0{}

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sin\ (phi )\cdot \partial \phi \cdot \int^{2\pi }_0{}\partial \theta ={-

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cos\ (phi )]}^{\pi }_0\cdot 2\pi =4\pi \nonumber \]

leaving us with:

$4\pi \cdot v^2\cdot {\left(\frac{m}{2\cdot \pi \cdot k_B\cdot T}\right)}^{\frac{3}{2}}\cdot e^{\frac{-m\cdot v^2}{2\cdot k_B\cdot T}}\cdot \partial v \nonumber$

This is the velocity form of the Maxwell-Boltzmann equation.

A.2.

$\partial v_a\cdot \partial v_b\to \partial G\cdot \partial V_{rel}$ : The following Jacobian was used in the calculation of relative velocity:

$\partial v_a\cdot \partial v_b=\left|\frac{\partial (v_a,v_b)}{\partial (G,V_{rel})}\right|\cdot \partial G\cdot \partial V_{rel} \nonumber$

The determinant is:

$\left|\frac{\partial (v_a,v_b)}{\partial (G,V_{rel})}\right|=\left|det\left[\frac{\partial v_a}{\partial V_{rel}}frac{\partial v_a}{\partial G}frac{\partial v_b}{\partial V_{rel}}frac{\partial v_b}{\partial G}right]\right| \nonumber$

Given that $v_a=G+\frac{m_b}{m_a+m_b}V_{rel}$ and $v_b=G-\frac{m_a}{m_a+m_b}V_{rel}$ , we can fill out the matrix:

$\left|det\left[\frac{m_b}{m_a+m_b}\ 1\ -\frac{m_a}{m_a+m_b}\ 1right]\right|=\frac{m_b}{m_a+m_b}+\frac{m_a}{m_a+m_b}=\frac{m_a+m_b}{m_a+m_b}=1 \nonumber$

Done!

Appendix: Jacobians

A.1. 3D to 1D

A.2.

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