Maxwell's demon

Schematic figure of Maxwell's demon

In the philosophy of thermal and statistical physics, Maxwell's demon is a thought experiment created by the physicist James Clerk Maxwell in which he suggested how the Second Law of Thermodynamics could hypothetically be violated.[1] In the thought experiment, a demon controls a small door between two chambers of gas. As individual gas molecules approach the door, the demon quickly opens and shuts the door so that fast molecules pass into the other chamber, while slow molecules remain in the first chamber. Because faster molecules are hotter, the demon's behavior causes one chamber to warm up as the other cools, thus decreasing entropy and violating the Second Law of Thermodynamics.

Origin and history of the idea

The thought experiment first appeared in a letter Maxwell wrote to Peter Guthrie Tait on 11 December 1867. It appeared again in a letter to John William Strutt in 1871, before it was presented to the public in Maxwell's 1872 book on thermodynamics titled Theory of Heat.[2]

In his letters and books, Maxwell described the agent opening the door between the chambers as a "finite being". William Thomson (Lord Kelvin) was the first to use the word "demon" for Maxwell's concept, in the journal Nature in 1874, and implied that he intended the mediating, rather than malevolent, connotation of the word.[3][4][5]

Original thought experiment

The second law of thermodynamics ensures (through statistical probability) that two bodies of different temperature, when brought into contact with each other and isolated from the rest of the Universe, will evolve to a thermodynamic equilibrium in which both bodies have approximately the same temperature.[6] The second law is also expressed as the assertion that in an isolated system, entropy never decreases.[6]

Maxwell conceived a thought experiment as a way of furthering the understanding of the second law. His description of the experiment is as follows:[6][7]

... if we conceive of a being whose faculties are so sharpened that he can follow every molecule in its course, such a being, whose attributes are as essentially finite as our own, would be able to do what is impossible to us. For we have seen that 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 molecules 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.

In other words, Maxwell imagines one container divided into two parts, A and B.[8][6] Both parts are filled with the same gas at equal temperatures and placed next to each other. Observing the molecules on both sides, an imaginary demon guards a trapdoor between the two parts. When a faster-than-average molecule from A flies towards the trapdoor, the demon opens it, and the molecule will fly from A to B. Likewise, when a slower-than-average molecule from B flies towards the trapdoor, the demon will let it pass from B to A. The average speed of the molecules in B will have increased while in A they will have slowed down on average. Since average molecular speed corresponds to temperature, the temperature decreases in A and increases in B, contrary to the second law of thermodynamics. A heat engine operating between the thermal reservoirs A and B could extract useful work from this temperature difference.

The demon must allow molecules to pass in both directions in order to produce only a temperature difference; one-way passage only of faster-than-average molecules from A to B will cause higher temperature and pressure to develop on the B side.

Criticism and development

Several physicists have presented calculations that show that the second law of thermodynamics will not actually be violated, if a more complete analysis is made of the whole system including the demon.[8][6][9] The essence of the physical argument is to show, by calculation, that any demon must "generate" more entropy segregating the molecules than it could ever eliminate by the method described. That is, it would take more thermodynamic work to gauge the speed of the molecules and selectively allow them to pass through the opening between A and B than the amount of energy gained by the difference of temperature caused by this.

One of the most famous responses to this question was suggested in 1929 by Leó Szilárd,[10] and later by Léon Brillouin.[8][6] Szilárd pointed out that a real-life Maxwell's demon would need to have some means of measuring molecular speed, and that the act of acquiring information would require an expenditure of energy. Since the demon and the gas are interacting, we must consider the total entropy of the gas and the demon combined. The expenditure of energy by the demon will cause an increase in the entropy of the demon, which will be larger than the lowering of the entropy of the gas.

In 1960, Rolf Landauer raised an exception to this argument.[11][8][6] He realized that some measuring processes need not increase thermodynamic entropy as long as they were thermodynamically reversible. He suggested these "reversible" measurements could be used to sort the molecules, violating the Second Law. However, due to the connection between thermodynamic entropy and information entropy, this also meant that the recorded measurement must not be erased. In other words, to determine whether to let a molecule through, the demon must acquire information about the state of the molecule and either discard it or store it. Discarding it leads to immediate increase in entropy but the demon cannot store it indefinitely: In 1982, Charles Bennett showed that, however well prepared, eventually the demon will run out of information storage space and must begin to erase the information it has previously gathered.[8][12] Erasing information is a thermodynamically irreversible process that increases the entropy of a system. Although Bennett had reached the same conclusion as Szilard’s 1929 paper, that a Maxwellian demon could not violate the second law because entropy would be created, he had reached it for different reasons. Regarding Landauer's principle, the minimum energy dissipated by deleting information was experimentally measured by Eric Lutz et al. in 2012. Furthermore, Lutz et al. confirmed that in order to approach the Landaeur limit, the system must asymptotically approach zero processing speed.[13]

John Earman and John D. Norton have argued that Szilárd and Landauer's explanations of Maxwell's demon begin by assuming that the second law of thermodynamics cannot be violated by the demon, and derive further properties of the demon from this assumption, including the necessity of consuming energy when erasing information, etc.[14][15] It would therefore be circular to invoke these derived properties to defend the second law from the demonic argument. Bennett later acknowledged the validity of Earman and Norton's argument, while maintaining that Landauer's principle explains the mechanism by which real systems do not violate the second law of thermodynamics.[16]

Recent progress

Although the argument by Landauer and Bennett only answers the consistency between the second law of thermodynamics and the whole cyclic process of the entire system of a Szilard engine (a composite system of the engine and the demon), a recent approach based on the non-equilibrium thermodynamics for small fluctuating systems has provided deeper insight on each information process with each subsystem. From this viewpoint, the measurement process is regarded as a process where the correlation (mutual information) between the engine and the demon increases, and the feedback process is regarded as a process where the correlation decreases. If the correlation changes, thermodynamic relations as the second law of thermodynamics and the fluctuation theorem for each subsystem should be modified, and for the case of external control a second-law like inequality[17] and a generalized fluctuation theorem[18] with mutual information are satisfied. These relations suggest that we need extra thermodynamic cost to increase correlation (measurement case), and in contrast we can apparently violate the second law up to the consumption of correlation (feedback case). For more general information processes including biological information processing, both inequality[19] and equality[20] with mutual information hold.

Applications

Real-life versions of Maxwellian demons occur, but all such "real demons" have their entropy-lowering effects duly balanced by increase of entropy elsewhere. Molecular-sized mechanisms are no longer found only in biology; they are also the subject of the emerging field of nanotechnology. Single-atom traps used by particle physicists allow an experimenter to control the state of individual quanta in a way similar to Maxwell's demon.

If hypothetical mirror matter exists, Zurab Silagadze proposes that demons can be envisaged, "which can act like perpetuum mobiles of the second kind: extract heat energy from only one reservoir, use it to do work and be isolated from the rest of ordinary world. Yet the Second Law is not violated because the demons pay their entropy cost in the hidden (mirror) sector of the world by emitting mirror photons."[21]

Experimental work

In the February 2007 issue of Nature, David Leigh, a professor at the University of Edinburgh, announced the creation of a nano-device based on the Feynman's Brownian ratchet. Leigh's device is able to drive a chemical system out of equilibrium, but it must be powered by an external source (light in this case) and therefore does not violate thermodynamics.

Previously, other researchers created a ring-shaped molecule which could be placed on an axle connecting two sites, A and B. Particles from either site would bump into the ring and move it from end to end. If a large collection of these devices were placed in a system, half of the devices had the ring at site A and half at B, at any given moment in time.

Leigh made a minor change to the axle so that if a light is shone on the device, the center of the axle will thicken, restricting the motion of the ring. It only keeps the ring from moving, however, if it is at A. Over time, therefore, the rings will be bumped from B to A and get stuck there, creating an imbalance in the system. In his experiments, Leigh was able to take a pot of "billions of these devices" from 50:50 equilibrium to a 70:30 imbalance within a few minutes.[22]

In 2009 Mark G. Raizen developed a laser atomic cooling technique which realizes the process Maxwell envisioned of sorting individual atoms in a gas into different containers based on their energy.[6][23][24] The new concept is a one-way wall for atoms or molecules that allows them to move in one direction, but not go back. The operation of the one-way wall relies on an irreversible atomic and molecular process of absorption of a photon at a specific wavelength, followed by spontaneous emission to a different internal state. The irreversible process is coupled to a conservative force created by magnetic fields and/or light. Raizen and collaborators proposed to use the one-way wall in order to reduce the entropy of an ensemble of atoms. In parallel, Gonzalo Muga and Andreas Ruschhaupt, independently developed a similar concept. Their "atom diode" was not proposed for cooling, but rather to regulate flow of atoms. The Raizen Group demonstrated significant cooling of atoms with the one-way wall in a series of experiments in 2008. Subsequently, the operation of a one-way wall for atoms was demonstrated by Daniel Steck and collaborators later in 2008. Their experiment was based on the 2005 scheme for the one-way wall, and was not used for cooling. The cooling method realized by the Raizen Group was called "Single-Photon Cooling," because only one photon on average is required in order to bring an atom to near-rest. This is in contrast to other laser cooling techniques which uses the momentum of the photon and requires a two-level cycling transition.

In 2006 Raizen, Muga, and Ruschhaupt showed in a theoretical paper that as each atom crosses the one-way wall, it scatters one photon, and information is provided about the turning point and hence the energy of that particle. The entropy increase of the radiation field scattered from a directional laser into a random direction is exactly balanced by the entropy reduction of the atoms as they are trapped with the one-way wall.

This technique is widely described as a "Maxwell's demon" because it realizes Maxwell's process of creating a temperature difference by sorting high and low energy atoms into different containers. However scientists have pointed out that it is not a true Maxwell's demon in the sense that it does not violate the second law of thermodynamics;[25][6] it does not result in a net decrease in entropy[25][6] and cannot be used to produce useful energy. This is because the process requires more energy from the laser beams than could be produced by the temperature difference generated. The atoms absorb low entropy photons from the laser beam and emit them in a random direction, thus increasing the entropy of the environment.[25][6]

As metaphor

Historian Henry Brooks Adams in his manuscript The Rule of Phase Applied to History attempted to use Maxwell's demon as a historical metaphor, though he misunderstood and misapplied the original principle.[26] Adams interpreted history as a process moving towards "equilibrium", but he saw militaristic nations (he felt Germany pre-eminent in this class) as tending to reverse this process, a Maxwell's demon of history. Adams made many attempts to respond to the criticism of his formulation from his scientific colleagues, but the work remained incomplete at Adams' death in 1918. It was only published posthumously.[27]

Sociologist Pierre Bourdieu incorporated Maxwell's demon into his work, "Raisons Pratiques" as a metaphor for the socioeconomic inequality among students, as maintained by the school system, the economy, and families.

The demon is mentioned several times in The Cyberiad, a series of short stories by the noted science fiction writer Stanisław Lem. In the book the demon appears both in its original form and in a modified form where it uses its knowledge of all particles in the box in order to surmise general (but unfocused and random) facts about the rest of the universe.

A machine powered by Maxwell's demon plays a role in Thomas Pynchon's novel The Crying of Lot 49.

Maxwell's Demon plays the role of spirit of perversity in Christopher Stasheffs books Her Majesty's Wizard and the Witch Doctor with the ability to concentrate or diffuse heat/energy

See also

Notes

  1. ↑ Cargill Gilston Knott (1911). "Quote from undated letter from Maxwell to Tait". Life and Scientific Work of Peter Guthrie Tait. Cambridge University Press. p. 215.
  2. ↑ Leff & Rex (2002), p. 370.
  3. ↑ William Thomson (1874). "Kinetic theory of the dissipation of energy". Nature 9 (232): 441–444. Bibcode:1874Natur...9..441T. doi:10.1038/009441c0.
  4. ↑ "The sorting demon Of Maxwell". Proceedings of the Royal Institution ix: 113. 1879.
  5. ↑ Alan S. Weber (2000). Nineteenth Century Science: a Selection of Original Texts. Broadview Press. p. 300.
  6. 1 2 3 4 5 6 7 8 9 10 11 Bennett, Charles H. (November 1987). "Demons, Engines, and the Second Law" (PDF). Scientific American (Scientific American Inc.) 257 (5): 108–116. doi:10.1038/scientificamerican1187-108. Retrieved November 13, 2014.
  7. ↑ Maxwell (1871), reprinted in Leff & Rex (1990) on p. 4.
  8. 1 2 3 4 5 Sagawa, Takahiro (2012). Thermodynamics of Information Processing in Small Systems. Springer Science and Business Media. pp. 9–14. ISBN 4431541675.
  9. ↑ Bennett, Charles H.; Schumacher, Benjamin (August 2011). "Maxwell's demons appear in the lab" (PDF). Nikkei Science (Scientific American Inc.): 3–6. Retrieved November 13, 2014.
  10. ↑ Szilard, Leo (1929). "Über die Entropieverminderung in einem thermodynamischen System bei Eingriffen intelligenter Wesen (On the reduction of entropy in a thermodynamic system by the intervention of intelligent beings)". Zeitschrift für Physik 53: 840–856. Bibcode:1929ZPhy...53..840S. doi:10.1007/bf01341281. cited in Bennett 1987. English translation available as NASA document TT F-16723 published 1976
  11. ↑ Landauer, R. (1961). "Irreversibility and heat generation in the computing process" (PDF). IBM Jour. of Research and Development (International Business Machines) 5 (3): 183–191. doi:10.1147/rd.53.0183. Retrieved November 13, 2014. reprinted in Vol. 44, No. 1, January 2000, p. 261
  12. ↑ Bennett, C. H. (1982). "The thermodynamics of computation—a review". International Journal of Theoretical Physics 21 (12): 905–940. doi:10.1007/BF02084158.
  13. ↑ jobs (2012-03-07). "The unavoidable cost of computation revealed : Nature News & Comment". Nature.com. Retrieved 2012-09-07.
  14. ↑ John Earman and John D. Norton (1998). "Exorcist XIV: The Wrath of Maxwell's Demon. Part I. From Maxwell to Szilard" (PDF). Studies in the History and Philosophy of Modern Physics 29: 435. doi:10.1016/s1355-2198(98)00023-9.
  15. ↑ John Earman and John D. Norton (1999). "Exorcist XIV: The Wrath of Maxwell’s Demon. Part II. From Szilard to Landauer and Beyond" (PDF). Studies in the History and Philosophy of Modern Physics 30: 1. doi:10.1016/s1355-2198(98)00026-4.
  16. ↑ Charles H. Bennett (2002–2003). "Notes on Landauer's principle, reversible computation, and Maxwell's demon" (PDF). arXiv:physics/0210005. Bibcode:2002physics..10005B.
  17. ↑ Hugo Touchette and Seth Lloyd (2000). "Information-Theoretic Limits of Control". Physical Review Letters 84: 1156–1159. Bibcode:2000PhRvL..84.1156T. doi:10.1103/PhysRevLett.84.1156.
  18. ↑ Takahiro Sagawa and Masahito Ueda (2010). "Generalized Jarzynski Equality under Nonequilibrium Feedback Control". Physical Review Letters 104: 090602. arXiv:0907.4914. Bibcode:2010PhRvL.104i0602S. doi:10.1103/PhysRevLett.104.090602.
  19. ↑ Armen E Allahverdyan, Dominik Janzing and Guenter Mahler (2009). "Thermodynamic efficiency of information and heat flow". Journal of Statistical Mechanics 2009: P09011. arXiv:0907.3320. Bibcode:2009JSMTE..09..011A. doi:10.1088/1742-5468/2009/09/P09011.
  20. ↑ Naoto Shiraishi and Takahiro Sagawa (2015). "Fluctuation theorem for partially masked nonequilibrium dynamics". Physical Review E 91: 012130. arXiv:1403.4018. Bibcode:2015PhRvE..91a2130S. doi:10.1103/PhysRevE.91.012130.
  21. ↑ "[physics/0608114] Maxwell's demon through the looking glass". Uk.arxiv.org. 2006-08-10. Retrieved 2013-02-18.
  22. ↑ Katharine Sanderson (31 January 2007). "A demon of a device". Nature. doi:10.1038/news070129-10.
  23. ↑ Raizen, Mark G. (June 12, 2009). "Comprehensive Control of Atomic Motion". Science (American Assoc. for the Advancement of Science) 324 (5933): 1403–1406. Bibcode:2009Sci...324.1403R. doi:10.1126/science.1171506. Retrieved November 14, 2014.
  24. ↑ Raizen, Mark G. (March 2011). "Demons, Entropy, and the Quest for Absolute Zero". Scientific American (Scientific American Inc.) 304 (3): 54–59. doi:10.1038/scientificamerican0311-54. Retrieved November 14, 2014.
  25. 1 2 3 Orzel, Chad (January 25, 2010). "Single-Photon Cooling: Making Maxwell’s Demon". Uncertain Principles. ScienceBlogs website. Retrieved November 14, 2014. External link in |publisher= (help)
  26. ↑ Cater (1947), pp. 640–647; see also Daub (1970), reprinted in Leff & Rex (1990), pp. 37–51.
  27. ↑ Adams (1919), p. 267.

References

  • Cater, H. D., ed. (1947). Henry Adams and his Friends. Boston. 
  • Daub, E. E. (1967). "Atomism and Thermodynamics". Isis 58 (3): 293–303. doi:10.1086/350264. 
  • Leff, Harvey S. & Andrew F. Rex, ed. (1990). Maxwell's Demon: Entropy, Information, Computing. Bristol: Adam-Hilger. ISBN 0-7503-0057-4. 
  • Leff, Harvey S. & Andrew F. Rex, ed. (2002). Maxwell's Demon 2: Entropy, Classical and Quantum Information, Computing. CRC Press. ISBN 0-7503-0759-5. 
  • Adams, H. (1919). The Degradation of the Democratic Dogma. New York: Kessinger. ISBN 1-4179-1598-6. 
  • Vladislav Cápek & Daniel P. Sheehan (2005). Challenges to the Second Law of Thermodynamics. The Netherlands: Springer. ISBN 1-4020-3016-9. 

External links

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