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Random walking our way to radiative diffusion

I just figured out how to insert LaTeX commands directly into Blogger! Check out these instructions for how to use MathJax.

In what follows is a piece of supplemental info for my Ay20 class. Since LaTeX stopped working on my office computer after my recent move across the street, this is the easiest way to get this info to my students...

Random Walks

A photon undergoing a random walk will travel a distance $l$ before suffering a "collision," which sends it off in a random direction. It seems counter intuitive, but this random walk process results in a net displacement. Read Astrophysics in a Nutshell, Chapter 3 pages 38-39 (starting with ``Thus, photons...'' and ending with equation 3.45. What you'll see is the expectation value for the mean displacement $\left<D\right>$ is
\begin{equation}
\left<D\right> = N^{\frac{1}{2}}l
\end{equation}
If we think of the distance $\Delta r$ as the net distance traveled, $\Delta r = \left<D\right>$, then each step in the random walk will take $t_{\rm step} = l/c$, and after N steps the total time to traverse $\Delta r$ will be
\begin{equation}
t_{\rm tot} = \frac{N l}{c}  = \frac{(\Delta r)^2}{l^2}\frac{l}{c} = \frac{(\Delta r)^2}{l c}
\end{equation}
It takes the photon this total time to traverse a distance $\Delta r$, so the diffusion speed is
\begin{equation}
v_{\rm diff} = \frac{\Delta r}{t_{\rm tot}}  = \frac{l}{\Delta r} c
\end{equation}
This is what I was trying to get you to derive using simple scaling arguments in problems 1 and 2 of this week's worksheet, but it occurred to me in class today that the students are unfamiliar with random walk processes. From here it should be pretty straight forward to derive the radiative diffusion equation for $dT/dr$.


Comments

David Rodriguez said…
Nice! I had to do this once and ended up uploading screenshots...
The only bad thing here is that it doesn't show up properly in the RSS feed.

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