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

\left<D\right> = N^{\frac{1}{2}}l

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

t_{\rm tot} = \frac{N l}{c}  = \frac{(\Delta r)^2}{l^2}\frac{l}{c} = \frac{(\Delta r)^2}{l c}

It takes the photon this total time to traverse a distance $\Delta r$, so the diffusion speed is

v_{\rm diff} = \frac{\Delta r}{t_{\rm tot}}  = \frac{l}{\Delta r} c

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 $dT/dr$.

David Rodriguez said…
The only bad thing here is that it doesn't show up properly in the RSS feed.

An annual note to all the (NSF) haters

It's that time of year again: students have recently been notified about whether they received the prestigious NSF Graduate Student Research Fellowship. Known in the STEM community as "The NSF," the fellowship provides a student with three years of graduate school tuition and stipend, with the latter typically 5-10% above the standard institutional support for first- and second-year students. It's a sweet deal, and a real accellerant for young students to get their research career humming along smoothly because they don't need to restrict themselves to only advisors who have funding: the students fund themselves!
This is also the time of year that many a white dude executes what I call the "academic soccer flop." It looks kinda like this:

It typically sounds like this: "Congrats! Of course it's easier for you to win the NSF because you're, you know, the right demographic." Or worse: "She only won because she's Hispanic."…

On the Height of J.J. Barea

Dallas Mavericks point guard J.J. Barea standing between two very tall people (from: Picassa user photoasisphoto).

Congrats to the Dallas Mavericks, who beat the Miami Heat tonight in game six to win the NBA championship.

Okay, with that out of the way, just how tall is the busy-footed Maverick point guard J.J. Barea? He's listed as 6-foot on NBA.com, but no one, not even the sports casters, believes that he can possibly be that tall. He looks like a super-fast Hobbit out there. But could that just be relative scaling, with him standing next to a bunch of extremely tall people? People on Yahoo! Answers think so---I know because I've been Google searching "J.J. Barea Height" for the past 15 minutes.

So I decided to find a photo and settle the issue once and for all.