“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.