"Everybody listen up!" shouts the high school PE coach, "I'm splitting you up into N teams of equal size. I'm going to make the selection of players randomly, so with any luck the teams will be of equal skill. The players with high skills all the way down the continuum to the players with low skills are going to be partitioned among the ensemble of teams and I hope it will make all the games competitive. If it doesn't and some teams dominate while other teams collapse, I'm going to swap out players until we reach an even equilibrium. Them I'll keep scrambling things randomly just to keep it interesting." A student raises his hand. "You're a pretty unrealistic PE coach," the student says. The coach frowns.
"Shut up and give me 20 pushups!"
It's time for a short interlude before we get back to the math in our quest to understand the Bose-Einstein condensate state of matter. Just like the athletes are partitioned between the teams, atoms or other particles are partitioned among the various possible states of the system. There's a few different ways to describe the partition mathematically, such as the regally named Canonical Partition Function and the Grand Partition Function. The canonical partition function describes situations in which the constituent parts can exchange energy but not particles. The grand partition function describes systems which can exchange both energy and particles.
The canonical partition function is easier to deal with conceptually, but when dealing with certain problems (like, say, collections of bosons) you run squarely into a brick wall in the mathematics involving a horrific sum over states constrained in ways that make evaluation pretty much impossible. The grand partition function is not so constrained since the number of particles isn't fixed, which makes life a lot easier.
I'll be honest, they're hard to explain in a straightforward way because statistical mechanics is a hard subject. If there were an easy way to do it it there wouldn't be any need for graduate courses in the subject. So in the interests of not writing a deathly boring textbook and driving away some large fraction of my longsuffering readers, I'm going to present the following two equations as an accomplished fact.
This one is the grand partition function for an ideal Bose gas:
And this one is the number of particles in the gas. z is a quantity called the fugacity (no kidding) that measures how much energy is required to add particles to the system at a particular temperature. The term inside the sum is of course the number of particles in each energy state:
Monday (I think) we'll put all this together and actually describe how this leads to the Bose-Einstein condensation. It's a lot of effort, I know. But at the end of it whether you follow the math or not you'll have a qualitative understanding of the what and more importantly the why behind this particularly interesting piece of physics that even manages to percolate its way into the popular press every once in a while.
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the capital Pi indicates the product of the states, and not the sum over the states.
or am i hallucinating due to threat of the deathly boring textbook derivation?
i think i will play with my doll. when you pull her string she says "statisical mechanics is hard."
You're not hallucinating. In fact I screwed up an upload and had the wrong equation entirely! It's fixed now. The first equation is in fact supposed to have a product of states, the second has the sum.
Hi Matt.
It looks like you generated your mathematical formulas using TeX. How do you integrate TeX into HTML? Is it hard to do?
now i will be able to sleep at night, knowing i am not going crazy. at least, not because of Bose-Einstein statistics!
Jason: I'm not Matt, but I can see how he did it. The equations are actually graphics files. There are a number of ways you can convert the LaTeX to graphics--the one I use is called LaTeXiT, which runs on a Mac (I haven't had need to investigate Linux or Windows options). You just type the (La)TeX source code, choose the display style (equation array, displayed equation, inline, or text), and export the resulting image in one of several formats. Also comes in handy for presentations, since the equation editor that comes with PowerPoint sucks like a laboratory grade vacuum pump.
Right. I generate the images externally and just upload them like normal images. You don't even have to install a program, there's websites that will do it for you.
The webmasters around here say they're going to install an automatic renderer in our Movable Type so we can just put in the LaTeX code directly, but it's been months and it hasn't happened yet.
(Come on guys, hurry it up!)
Eric and Matt -
Thanks! I've been wanting to do more math posts but have been stymied by my ignorance of how to make things look pretty. Should have plenty of time to play over the summer break, though.
You might try OpenOffice Impress (ppt equiv) since it uses the same (superior to MS) equation editor as the word processor.
For the really adventurous, there are LaTeX styles for presentations. Seriously.
Finally, for Matt: SVG support is pretty widespread now, so you might be able to embed an SVG rendering of the equations instead (yes, OO.o exports to SVG -- as do several LaTeX renderers) and that way you don't get the crap jaggies from JPEG.
Just a couple of thoughts about this rather nice explanation, that I would think about if I were writing something similar for non-physicists:
1. If your aim is to avoid nasty statistical physics calculations, you could skip straight to quoting and explaining the Bose-Einstein distribution? This is a fast-track to showing why condensation happens, as the important part is evaluating the integral in Pt. 4.
2. You can introduce the fugacity or equivalently, the chemical potential - the latter is (in my experience) more widely-used and lends itself to an explanation.
3. You might want to note one of the enduring beauties of the grand canonical ensemble: the "systems" can be the single-particle energy levels of the box!
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Please don't take this as criticism though, the fact that anybody wants to write about this in a readable way - and succeeds - is exciting enough!