Maxwell's Demon: The Thought Experiment That Wouldn't Die

Maxwell's demon is a thought experiment, first described in an 1867 letter, in which a tiny being sorts fast and slow gas molecules, seemingly creating a temperature difference for free and breaking the second law of thermodynamics. The accepted resolution, completed by Charles Bennett in 1982 using Rolf Landauer's 1961 principle, is that the demon has to erase its memory to keep going, and erasure always costs at least kT ln 2 of heat per bit.

The setup and the paradox

The idea first appeared in a letter James Clerk Maxwell wrote to Peter Guthrie Tait on 11 December 1867. He returned to it in a letter to John William Strutt in 1871, and published it in his 1872 book Theory of Heat (Wikipedia).

His description: imagine "a being whose faculties are so sharpened that he can follow every molecule in its course." Take a vessel of air at uniform temperature, divided into two parts by a wall with a small hole. The molecules don't all move at the same speed; only their average is uniform. Let the being open and close the hole so that only the faster molecules pass one way and only the slower ones pass the other (quoted in Wikipedia).

After a while, one side holds the fast molecules and is hotter; the other holds the slow ones and is colder. A temperature difference has appeared, and you could run a heat engine on it. No work seems to have been done. The entropy of the gas has gone down, which the second law forbids for an isolated system.

Maxwell never called it a demon. He wrote of a "finite being", one "who can play a game of skill with the molecules". It was William Thomson, Lord Kelvin, who first used the word, in Nature in 1874, meaning the older Greek sense of a daemon working in the background rather than anything malevolent (Wikipedia).

The puzzle is simple to state. The demon does no work on the gas. It only uses information. So either the second law has a loophole for anyone who knows enough, or knowing has a cost nobody had counted.

Szilard, 1929: one molecule, one bit

Leo Szilard stripped the problem down. His engine, as it is usually described, holds a single molecule. Find out which side of the box it is on, and you can let it push a partition like a piston as it expands back to fill the box. Each cycle draws heat from surroundings at one temperature and turns it into work, and the only resource used is one binary fact: left or right. Toyabe and colleagues later summarized it as a feedback protocol in which a hypothetical intelligence "pumps heat from an isothermal environment and transforms it into work" (Toyabe et al., 2010).

Szilard's conclusion, in his 1929 paper in Zeitschrift für Physik, was that a system that measures and remembers could produce a lasting entropy decrease, breaking the second law, unless the measurements themselves necessarily produce entropy. He calculated the entropy that would exactly compensate, and showed that a lifeless measuring device could produce just that much (Szilard, 1929). This was the first time information was put into the thermodynamic books as a quantity with a price. It also linked the demon's gain to exactly one bit.

Brillouin, 1951: does looking cost?

Léon Brillouin pressed the same line. His 1951 paper was titled, bluntly, "Maxwell's Demon Cannot Operate: Information and Entropy" (Brillouin, 1951). On this view, which Szilard and Brillouin shared, a real demon needs some means of measuring molecular speeds, acquiring that information requires spending energy, and the resulting increase in the demon's entropy outweighs the decrease in the gas's (Wikipedia).

For decades that seemed to settle it. But it rested on an assumption that turned out to be wrong: that measurement must always be thermodynamically costly.

Landauer and Bennett: forgetting is the cost

In 1961 Rolf Landauer, at IBM, argued that the unavoidable cost in computing lies elsewhere. Logically irreversible operations, ones that throw information away, must dissipate heat, at minimum around kT per bit (Landauer, 1961). The more precise floor is kT ln 2. Landauer also realized that some measurements need not increase thermodynamic entropy at all, if they are carried out reversibly (Wikipedia).

Charles Bennett, also at IBM, put the pieces together in 1982. In his words, "the essential irreversible step, which prevents the demon from breaking the second law, is not the making of a measurement (which in principle can be done reversibly) but rather the logically irreversible act of erasing the record of one measurement to make room for the next" (Bennett, 1982).

Here is the logic. The demon can, in principle, look at molecules without paying. But it has to remember what it saw, at least long enough to decide whether to open the door. A real demon has finite memory. To keep sorting, it must eventually clear old records, and resetting a memory requires at least kT ln 2 of heat per bit to be passed to the environment (Wikipedia). That heat raises the entropy of the surroundings by at least as much as the sorting lowered the entropy of the gas.

Why does erasure cost? A bit that could be 0 or 1 has two possible states. Erasing it forces it into one known state. That is an entropy decrease for the memory, and the second law makes the environment pay for it. The number, the physics and the experiments behind it are the subject of the companion essay, Landauer's Limit: Why Deleting Data Costs Real Energy.

From Maxwell's letter in 1867 to Bennett's review in 1982 is 115 years. That is how long it took to find where the demon's bill is paid.

Has anyone built one?

Not one that beats the second law. Researchers have implemented forms of Maxwell's demon in experiments, but they all differ from the thought experiment in some way, and none has been shown to violate the second law (Wikipedia).

What experiments have done is make the demon's bookkeeping measurable.

In 2010, Shoichi Toyabe and colleagues used real-time feedback, based on measurements of a Brownian particle's position, to make it climb a spiral-staircase-like potential. The particle gained more free energy than the work done on it: a Szilard-type conversion of information into energy, which let them verify the generalized Jarzynski equality, a relation that builds the information gained into the energy accounting (Toyabe et al., 2010).

In 2012, Antoine Bérut and colleagues erased a one-bit memory made of a single colloidal particle in a double-well trap and measured the heat released. The dissipated heat approached the Landauer bound as the erasure was done more slowly, saturating at it in the limit of long erasure cycles (Bérut et al., 2012). Later tests used a particle in a feedback trap, nanomagnetic memory bits and a single trapped ion (Wikipedia).

(An earlier version of this page said the 2012 experiment measured "exactly kT ln 2". The result is subtler and more interesting: the heat comes down to the bound only when the erasure is slow.)

The demon in biology and computing

Cells are full of machinery that looks demon-like: enzymes that pick the right molecule out of many, and copying systems that correct their own errors. None of it is free. Bennett noted that the enzymes of DNA replication, transcription and translation are nature's closest approach to an idealized low-energy computer, and still dissipate 20 to 100 kT per step (Bennett, 1982). Living things pay for their precision in chemical energy.

Computers pay too, far more. Bennett observed that existing electronic computers dissipate energy "vastly in excess of the mean thermal energy kT" (1982). How far in excess, and whether anything can close the gap, is taken up in Landauer's Limit.

What the demon taught

Maxwell meant a small illustration: that the second law is statistical, and that a being with enough skill could, in principle, exploit the spread of molecular speeds. The demon turned into a century of argument, and the answer it forced was larger than the question. Information is physical. Remembering takes space, forgetting releases heat, and the second law keeps its books on both.

The laws of thermodynamics came out of the argument intact, and the synthesis essay ties them together.

The pebble: the demon could see for free; it was forgetting that sent the bill.

FAQ

Does Maxwell's demon violate the second law?

No. Once you include the demon itself, the second law holds. To keep sorting molecules, the demon must erase its memory, and erasure releases at least kT ln 2 of heat per bit, which raises the entropy of the surroundings at least as much as sorting lowers the gas's.

Who solved Maxwell's demon?

Several people contributed. Leo Szilard (1929) linked the demon to one bit of information, Léon Brillouin (1951) argued measurement has a cost, Rolf Landauer (1961) showed that erasing information must release heat, and Charles Bennett (1982) showed that erasure, not measurement, is the step that saves the second law.

Has Maxwell's demon been built?

Versions of it have, in the sense of devices that convert information about tiny particles into work or that measure the heat of erasing a bit. Toyabe and colleagues (2010) and Bérut and colleagues (2012) are well-known examples. None has violated the second law.

Further Reading

• Szilard, L. (1929). "Über die Entropieverminderung in einem thermodynamischen System bei Eingriffen intelligenter Wesen." Zeitschrift für Physik 53, 840-856. https://doi.org/10.1007/BF01341281
• Brillouin, L. (1951). "Maxwell's Demon Cannot Operate: Information and Entropy. I." Journal of Applied Physics 22, 334-337. https://doi.org/10.1063/1.1699951
• Landauer, R. (1961). "Irreversibility and Heat Generation in the Computing Process." IBM Journal of Research and Development 5(3), 183-191. https://doi.org/10.1147/rd.53.0183
• Bennett, C. H. (1982). "The thermodynamics of computation: a review." International Journal of Theoretical Physics 21, 905-940. https://doi.org/10.1007/BF02084158
• Toyabe, S. et al. (2010). "Experimental demonstration of information-to-energy conversion and validation of the generalized Jarzynski equality." Nature Physics 6, 988-992. https://doi.org/10.1038/nphys1821
• Bérut, A. et al. (2012). "Experimental verification of Landauer's principle linking information and thermodynamics." Nature 483, 187-189. https://doi.org/10.1038/nature10872
• "Maxwell's demon." Wikipedia: https://en.wikipedia.org/wiki/Maxwell%27s_demon

This is Part 7 of the Laws of Thermodynamics series. Next: Synthesis: Why the Laws of Thermodynamics Are Really One Law