Wednesday, March 2, 2011

Where does the equilibrium come from?

In my very first blogpost just over a year ago, I talked about how I really felt I understood classical physics even if I wasn't capable of every single calculation. The example I took was Maxwell's calculation of the equilibrium distribution of velocities in a gas. I said that even if I didn't know how to calculate the distribution, I could "see" how it more or less had to come about.

The irony in this example is that over the last two weeks, I've been agonizing over how to work out the exact same equilibrium, except as applied to the classical radiation field. And my problem was not merely one of how to do the calculation. It was the basic physics that had me baffled. Just why and how does an equilibrium come about in the first place? I know that an oscillating atom can either absorb or emit radiation. But I had no definite way of seeing why the tendency to absorb or radiate should depend on the magnitude of the ambient field. Rather than saying that there had to be an equilibrium point, I could have just as well argued, for example, that the amount of outward radiation would always excede the amount of absorption.

I went down a number of dead end roads before solving the problem. Mostly I was trying to analyze the ambient field as the sum of an in-phase and and out-of-phase component to the atomic vibration. To the extent that the fields are in quadrature (out-of-phase) there is no interaction, so the atomic oscillator is always emitting. Where does the absorption come into effect, to counterbalance the emission? As the fields go in and out of phase, they are either in a leading or lagging relationship. One is absorbing, the other is emitting. But if the phases are basically random, the quantities should apparently just cancel out. So there is no equilibrium between absorption and emission.

So then I looked for mechanisms whereby the ambient field would "drive" the atomic oscillator slightly off frequency, pulling the two fields into synchronism just like a motor when it is connected to a 60-Hz power line. I tried in vain to make this model work and simply gave up.

Then I had two essential inspirations that gave me the solution of the problem. The first was the drunkard's walk. I realized that even when the impulse is in completely random directions, there is still a net tendency for outward progress. This was the first part of the puzzle. Then I came up with a way of analyzing random fields whereby I could convert a power spectrum into a time-varying electric field. (See most "Harmonic Oscillator: The problem is solved" ). The amazing thing about this analysis is that no matter what I chose for an arbitrary frequency cutoff or discrete resolution, I ended up with waveforms, all of them different from each other, but all of which gave the same result when used in a "random walk" analysis of the harmonic oscillator.

At last I could say I understood the essential physics of the equilibrium process. Now, when you really understand something, you ought to be able to do some calculations with it; and that's what I'm going to do next. It's not the kind of thing where the numbers are going to come out exact, because I've made some approximations along the way; and to some extent, I'm allowing myself to be a little sloppy. The point of this kind of calculation is to see if you are at least in the ballpark, and I think I've been able to do that much.

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