### 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.

### 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.

I then used the basketball as my metric. Wikipedia states that an NBA basketball is 29.5 inches in circumfe…

### Finding Blissful Clarity by Tuning Out

It's been a minute since I've posted here. My last post was back in April, so it has actually been something like 193,000 minutes, but I like how the kids say "it's been a minute," so I'll stick with that.
As I've said before, I use this space to work out the truths in my life. Writing is a valuable way of taking the non-linear jumble of thoughts in my head and linearizing them by putting them down on the page. In short, writing helps me figure things out. However, logical thinking is not the only way of knowing the world. Another way is to recognize, listen to, and trust one's emotions. Yes, emotions are important for figuring things out.
Back in April, when I last posted here, my emotions were largely characterized by fear, sadness, anger, frustration, confusion and despair. I say largely, because this is what I was feeling on large scales; the world outside of my immediate influence. On smaller scales, where my wife, children and friends reside, I…

### The Force is strong with this one...

Last night we were reviewing multiplication tables with Owen. The family fired off doublets of numbers and Owen confidently multiplied away. In the middle of the review Owen stopped and said, "I noticed something. 2 times 2 is 4. If you subtract 1 it's 3. That's equal to taking 2 and adding 1, and then taking 2 and subtracting 1, and multiplying. So 1 times 3 is 2 times 2 minus 1."

I have to admit, that I didn't quite get it at first. I asked him to repeat with another number and he did with six: "6 times 6 is 36. 36 minus 1 is 35. That's the same as 6-1 times 6+1, which is 35."

Ummmmm....wait. Huh? Lemme see...oh. OH! WOW! Owen figured out

x^2 - 1 = (x - 1) (x +1)

So $6 \times 8 = 7 \times 7 - 1 = (7-1) (7+1) = 48$. That's actually pretty handy!

You can see it in the image above. Look at the elements perpendicular to the diagonal. There's 48 bracketing 49, 35 bracketing 36, etc... After a bit more thought we…