Thursday, March 18, 2010

Ramanujan and the Casimir Effect

Damn, there's a lot of stuff I need to post. This is a calculation I did last year. The writeup it's pretty much barebones, but it's a very cool calculation.

EDIT: Right now (Jan 19 2014) this post is getting more hits than usual because of an upsurge of interest in the Ramanujan series. So I thought I'd revisit the whole thing and see if I couldn't explain it any better. You'll find my updated version here.)

Once in a long while I actually calculate something. I think I figured out how to do the Casimir effect. And I used that series 1 + 2 - 3 + 4....

First the series: it's one of those bizarre and counterintuitive results credited to the tragic Indian prodigy Ramanujan. For reasons that no one seems to know, he could intuitively recognize that certain alternating series which would appear to anyone else to be divergent actually had a finite sum. His conjectures were tested and almost always found to be correct.

I don't know how the mathematicians do it, but I can also make these series add up using my own little tricks. For example, the alternating arithmetic progression written above: you put it in an Excel spredsheet and then spread an envelope over it, a very gentle Gaussian. It quickly adds up to 0.25 which is the right answer. You can put the same Gaussian over the alternating squares, and it adds up to zero which is also correct. Of course you need a wider envelope to swallow up the much bigger terms.

The bizarre thing is how this weird mathematical series actually plays a role in modern physics. It can be used to calculate the Casimir Effect. Here's how the calculation goes.

We simplify things a little by putting everything in a one-dimensional box. So the energy modes are 1,2,3, etc. We can choose our units so that the pressure is numerically equal to the energy.

Now make the box twice small. The energy modes are 2,4,6...etc. But the PRESSURE is energy per unit volume (length, since it's one-dimensional) the pressures are (get this:) 4,8,12...
What's the difference in pressure between the big box and the small box?

1 + 2 + 3 .... - (4 + 8 + 12...)

which is...1 - 2 + 3 - 4 + 5 .... = 0.25!

If you take the pressure of the lowest mode in the smaller box to be 4, you see that the pressure difference between the small box and the big box is one sixteenth of the lowest-mode pressure. You can then take a chain of boxes, each one double the next one, to get the pressure relative to infinity: 1/16 + 1/64 + 1/256...= 1/12

And that's the calculatation. It's not really right for the three dimensional case, but I think if you apply it to a small cube and take the pressure defect to be 1/12 of the lowest mode pressure, you get something close to the ballpark for the parallel plate if you fill the gap with those small cubes. Or something like that. It's certainly true that the actual Casimir effect pressure is a definite fraction of the lowest mode pressure.


grebdlogj said...

regarding the alternating series, it is easier to demonstrate why putting a Gaussian envelope causes it to add up to 0.25.
consider the series 1/(1-x)^2=1+2x+3x^2+4*x^3+...
if you insert x=-1, you get the alternating series on the r.h.s., which is of course meaningless, but if you instead insert a value close to -1, eg x=-0.99, on the left hand side you'll get a value even closer to 0.25

Marty Green said...

Thanks, Grebdlogj. I had no idea why it worked...I had just figured it out experimentally by adding it up in Excel. I'm gratified that you approve (I think!) of my analysis.