Tuesday, July 22, 2014

Maxwell's Demon

In Theory of Heat, J C Maxwell - one of the greatest physicists of all time - attempted to illustrate the new theories of molecular flux and thermodynamics in a form as complete as the science allowed. He finished this section with a thought experiment that purported to show "Limitation of the Second Law of Thermodynamics". "Before I conclude, I wish to direct attention to an aspect of the molecular theory which deserves consideration". He introduced the idea we now call Maxwell's Demon, meant to illustrate the nature of Maxwell's thoughts on the Second Law. I'll let Maxwell illustrate:

"One of the best established facts in thermodynamics is that it is impossible in a system enclosed in an envelope which permits neither change of volume nor passage of heat, and in which both the temperature and the pressure are everywhere the same, to produce any inequality of temperature or of pressure without the expenditure of work. This is the second law of thermodynamics, and it is undoubtedly true as long as we can deal with bodies only in mass, and have no power of perceiving or handling the separate molecules of which they are made up. But if we conceive a being whose faculties are so sharpened that he can follow every molecule in its course, such a being, whose attributes are still as essentially finite as our own, would be able to do what is at present impossible to us. For we have seen that the molecules in a vessel full of air at uniform temperature are moving with velocities by no means uniform, though the mean velocity of any great number of them, arbitrarily selected, is almost exactly uniform. Now let us suppose that such a vessel is divided into two portions, A and B, by a division in which there is a small hole, and that a being, who can see the individual molecules, opens and closes this hole, so as to allow only the swifter molecules to pass from A to B, and only the slower ones to pass from B to A. He will thus, without expenditure of work, raise the temperature of B and lower that of A, in contradiction to the second law of thermodynamics."

The Wikipedia image is much better than the one I tried to make.

This brief thought experiment has given rise to a minor, but interesting, literature on whether Maxwell's reasoning is correct. Surprisingly, smart money says "No."! . The primary difficulty in Maxwell's thought experiment is his opinion that just because you can capture one fast particle that you can continue to capture more. In fact, this is a direct violation of the principle of detailed balance - if the door is open for any length of time, it is as likely to let a particle out as in (there are as many fast moving particles on one side of the door as the other after all!). Maxwell's reasoning is therefore circular, it assumes that if one could violate the second law, then he could. Another way of putting this is that he did not include the work done by the demon as a part of the system. If the demon is considered a rectifier or computing device, then the entropy of this device must be such that equilibrium will still be reached. This approach is demonstrated in a state of great excellence in this paper. Since the invariance of phase volume is a principle of mechanics, the circularity of reasoning described is revealed. This second approach - really the first approach in new clothing - was pioneered by Szilard and brought to a state of modernity by Landauer. In this paper, the Szilard-Landauer approach is given a simple model which is solved explicitly. They don't go into detail about the equivalence of these lines of thinking, in fact I don't know if anyone has bothered to do so. Incidentally, this literature has been tough for me to track down, even some of the most famous papers by Smoluchowski (as far as I know, Experimentally Verifiable Molecular Phenomena that Contradicts Ordinary Thermodynamics has never been translated!). Still, I can give examples of pieces of the literature. This literature is not pure theory, it also includes plentiful experimental and numerical examinations of these thought experiments. This excellent paper includes both a good summary of the issues and a formal model of the above trapdoor, showing precisely how it fails. This paper shows how well numerical experiments can clarify and elucidate, something near to my own heart.

Before I go, I should mention that I first became aware of this literature through the Feynman Lectures on Computation, which includes chapters on Quantum Computing, Reversible Computing and the physics of computation. His discussion of these issues probably influenced me a lot, but I don't want to dig it out of it's current location. Feynman also made a sizable contribution to this literature in his Lectures on Physics, where he introduced the Brownian Ratchet to illustrate the concepts above. This section is a very good example of how theory can be used to elucidate.
Much, much better than my attempts  

The two sides of the ratchet are in two boxes of gas, to the center a mass is tied. Randomly, the gas will push the blades in tank 1 (at temperature 1) left and the pawl in tank 2 (at temperature 2) stops the ratchet from moving right. This means that just like the Smoluchowski trap, the ratchet works as a rectifier. Feynman analyzes in detail why this fails - and it fails for the same reason that Maxwell's failed. There is a presumption that the pawl is not subject to the same random fluctuations, in other words that one need not worry about gas particles moving the other way. This whole chapter is worth reading (of course, the entire book is worth reading...) but I will only reprint his final words:

"If T2 were less than T1, [then] the ratchet would go forward, as anybody will believe. But what is hard to believe, at first sight, is the opposite. If T2 is greater than T1, the ratchet goes the opposite way! A dynamic ratchet with lots of heat in it runs itself backwards, because the ratchet pawl is always bouncing. If the pawl, for a moment, is on the incline somewhere, it pushes the inclined plane sideways. But it is [almost] always on an incline plane, because if it happens to lift up high enough to get past the point of a tooth, then the inclined plane slides by, and it comes down again on an inclined plane. So a hot ratchet in pawl is ideally built to go around in a direction exactly opposite to that for which it was originally designed!"

Exciting stuff!

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