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Maxwell's demon

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Maxwell's Demon is the name of a creature thought up in 1867 by the likely heterosexualScottish physicist James Clerk Maxwell as part of a thought experiment meant to illustrate the Second law of Thermodynamics. This law forbids (among other things) that two bodies of equal temperature brought in contact with each other and isolated from the rest of the Universe, evolve to a state in which one of the two has a significantly higher temperature than the other. The second law is also expressed as the claim that entropy never decreases.

In Maxwell's thought experiment, two containers, A and B, filled with the same gas at equal temperatures are placed next to each other. A little 'demon' guards a trapdoor between the two containers, observing the molecules on both sides. Whenever a faster-than-average molecule from A flies towards the trapdoor, he opens it, and the molecule will move from A to B. Then he waits until a slower-than-average molecule from B comes flying towards the trapdoor, which he opens again, letting the molecule through to A. Thus, the average speed of the molecules in B increases and that in A decreases. But since average molecular speed corresponds to temperature, this means that the temperature in B increases and that in A decreases, which is a violation of the second law of thermodynamics.


A short history of Maxwell's Demon

In 1867, in a letter to his friend P.G. Tate, Maxwell introduced a 'finite being' with the power to exploit the differences in molecular speed which exist in any gas. By opening and closing a frictionless trapdoor between two containers of gas, depending on the speed of approaching molecules, this being could create a temperature difference and break the second law of thermodynamics. Maxwell concluded:

Or in short if the heat is the motion of finite portions of matter and if we can apply tools to such portions of matter so as to deal with them separately, then we can take advantage of the different motion of different proportions to restore a uniform hot system to unequal temperatures or to motions of large masses.

This finite being, which William Thomson (Baron Kelvin) later dubbed 'demon', was supposed to elucidate the second law of thermodynamics by showing it had only 'statistical certainty'.

The twentieth century turned out to be an eventful epoch for Maxwell’s Demon. Started as a helpful thought experiment, before long it was perceived as a threat to a beloved bulwark of physical truth. The second law had to be protected against the malicious attempts of the infernal rascal to violate it; the lofty foundations of thermodynamics had to be secured once and for all by providing a definite and unassailable proof of the demon’s non-existence. Thus began the exorcist tradition.

The history of exorcism can be divided into three main phases, in which thermal fluctuations, measurement and erasure of information were the consecutive notions thought to contain the key to any successful exorcism. The protagonist exorcists of every phase naturally saw the earlier attempts as well-meant but insufficient, and their own pet idea as the final piece of the puzzle, which at last made everything fit together and had the power to banish the demon for all eternity. Because the erasure-type exorcism is nowadays the predominant position among Maxwell’s Demon scholars, the tale which displays the history of exorcism as one of gradual improvement has become somewhat of an orthodoxy. But the debate rages on, and the past fifteen years have seen a veritable host of scholarly publications on the demon.

The original demon

(The rest of this extensive rewrite will follow very soon. For now, the old article is appended below. Because of vast differences in organisation between old and new text, I have deemed it unprofitable to keep up the appearance of unity. References named in the text will be added to the list of references in due time. Please bear with me for a couple of days.)

Two versions of the second law

[About the thermodynamical and the statistical-mechanical version of the second law.]

Maxwell's thought experiment

[About the demon, including quotes by Maxwell]

The purpose of the demon

[The demon points to the fact that the second law is only 'statistically certain' (quote Maxwell). What this does and does not mean.]

Exorcising the demon

The first stage: fluctuations

[Smoluchowksi's exorcism: the trapdoor fluctuates so fast that it doesn't work. Assumptions needed.]

Smoluchowski 1912, one of the early classics of the exorcist literature, contains a very short description of perhaps the simplest example of a mechanical demon: a one-way valve. Take two containers of gas, again called A and B, at equal temperatures with a little hole between them. Instead of a shutter, as in Maxwell's thought experiment, put a valve on it which opens only one way, say into B. The easiest way to visualise the valve is as a little trapdoor held closed by a very weak spring. Every time a molecule in A hits the trapdoor, it opens (we assume the spring is very weak) and lets the molecule pass. But whenever a molecule from B hits the valve, it remains shut, because it is pressed against the container wall. In this way, a pressure difference between A and B will build up without any change in the environment or the demon taking place, which qualifies the valve as a Maxwell’s Demon.

Smoluchowski, however, does not accept this prima facie correct reasoning. He points out that the valve will be at the same temperature as the gas. That means it will be subject to thermal fluctuations: its kinetic energy will randomly change by amount of – on average – 1/2kT. There are two possible scenario’s, depending on the strength of the spring which holds the trapdoor closed. Either the spring is so strong that the thermal fluctuations can hardly get the valve to move; but in that case, no molecules will be able to pass either. (After all, the kinetic energy of the molecules in any direction is also of the order kT.) Or the spring is so weak that molecules from A can push aside the trap door and enter B, but then the door will be a constant victim of thermal fluctuations which make it jump one way, then the other, and stop it from functioning as a pressure demon. Thus, the fact that the trapdoor has the same temperature as the gas, and therefore the same average kinetic energy fluctuations, defeats Smoluchowski’s design.

One may wonder whether the trapdoor would not still function as a demon, if somewhat less efficient than originally envisaged. It is open part of the time, in which case it does nothing; but it is closed another part of the time, in which case it does work. Therefore, it would create at least somewhat of a pressure gradient, although not as efficiently as we could have hoped. Bennett 1987 attempts to remove this objection. Indeed the trapdoor will stop some molecules in B from entering A (namely when it is closed). But this is compensated for by the fact that every time it bangs shut, any molecule in B which is in its path will actively be pushed into A, and molecules just passing from A to B may be bounced back. An analytical solution of this problem is quite involved, but computer simulations with a similar valve have confirmed Bennet’s result; see Skordos & Zurek 1992.

The second stage: measurements

[Szilard and Brillouin's exorcism: measuring the molecules costs energy. Assumptions needed.]

The third stage: information

[Landauer and Bennett's exorcism using information erasure. Controversy, assumptions needed.]

Is Maxwell's Demon still alive?

[Recent anti-exorcist voices; specifically Norton and Earman. NPOV conclusion: there is no consensus.]


Temporary repository of old article

The demon is an excellent demonstration of entropy, how it is related to

(a) the fraction of energy that is not available to do useful work, and
(b) the amount of information we lack about the detailed state of the system.

In Maxwell's thought experiment, the demon manages to decrease the entropy. In other words, it increases the amount of energy available by increasing its knowledge about the motion of all the molecules.

The Second law of Thermodynamics says this is impossible: you can only increase entropy (or rather, you can decrease it at one place as long as that is balanced by at least as big an increase somewhere else).

So, why wouldn't a setup like Maxwell's demon work? The question was first answered in 1929 by Leó Szilárd. Any real "demon" that does this would not be a disembodied spirit receiving its information telepathically; to acquire information about the world you must be in physical interaction with it. In determining what side of the gate a molecule must be on, the demon must store information about the state of the molecule. Eventually, the demon will run out of information storage space and must begin to erase the information that has been previously gathered. Erasing information is a thermodynamically irreversible process that increases the entropy of a system. Maxwell's demon therefore reveals a deep connection between thermodynamics and information theory.

Real-life versions of Maxwellian demons (with their entropy lowering effects of course duly balanced by increase of entropy elsewhere) actually occur in living systems, such as the ion channels and pumps that make our nervous systems work, including our brains. Molecular-sized mechanisms are no longer found only in biology; they are also the subject of the emerging field of nanotechnology.

Maxwell's demon plays a significant role in the book The Crying of Lot 49 by noted author Thomas Pynchon.


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