The development of thermodynamics has proved to be a rewarding excursion for physicists in the past two centuries. The science of heat flow and equilibrium has spawned fundamental laws which strictly govern how the universe organises itself on a macroscopic level. Of these, the second law stands out as a bellwether for all thermodynamic processes. Among its many equivalent definitions of varying degrees of abstraction, it is based on the empirical observation that the universe tends from order to disorder. More specifically, it forbids heat flow from cold to hot. For centuries, physicists have tried to come up with creative ways of violating the second law, but it has stood the test of time. One of the more famous hypotheses was that of James Clerk Maxwell.

Though Maxwell was most famous for formulating the theory of electromagnetism, his work in other areas of physics shouldn’t be overlooked. His work in thermodynamics proved to be some of the most fruitful of the 19^{th} and 20^{th} centuries. Besides the kinetic theory of gases and the Maxwell-Boltzmann distribution, Maxwell left us with a thought experiment. Later named Maxwell’s Demon, the arrangement comprises a system of two gases at the same temperature, isolated from one another by a wall with a small door in it. A hypothetical being, the ‘demon’ controls the door at will, allowing individual gas particles he picks and chooses to transfer between the two gases. The idea is, the demon can select fast particles to move into one box and slow particles to move into the other, until two gases of different temperatures remain, violating the second law of thermodynamics and performing useful work with no energy input. Though the premise of this apparent paradox is fundamentally flawed, it raises some interesting points about information, entropy and work.

Figure 1: Illustration of the Maxwell’s Demon paradox. (Htkym: Creative Commons)

The issue lies in addressing how the demon can seemingly reduce the entropy of the system without increasing the entropy of the surroundings by an equivalent or greater amount. One may suggest that the act of opening and closing the door requires work, but since this is a theoretical scenario, this work can be reduced to an arbitrarily small amount. The resolution of the paradox arrives in how the demon obtains and stores information about the gas particles. The theory of information-entropy equivalence has become widely popular since John von Neumann proposed it in his 1932 book *Mathematical Foundations of Quantum Mechanics *[1]. It is argued that a logically irreversible process, say erasing a bit of information (0 or 1) by re-initialising the bit to one of its possible values, corresponds to a thermodynamically irreversible process, where entropy is produced. In his 1961 paper [2], IBM researcher and physicist Rolf Landauer discovered a theoretical minimum energy emission of for each bit of information erased (a tiny amount, several orders of magnitude smaller than what real computers dissipate for each erasure). This finding would have been of great relief to Maxwell, as it ensured that the entropy decrease of the gas system would always be compensated for by the demon erasing its memory and returning to its initial state after manipulating the system. It became known as Landauer’s principle and provided a physically insightful explanation for Maxwell’s paradox.

That hasn’t been the end of the story though. Landauer’s work has come under heavy scrutiny for alleged false assumptions about statistical mechanics, circular reasoning, and supposed inapplicability of his theorem to more general systems, in particular by Earman and Norton [3], and Shenker [4]. Furthermore, there is evidence that logically irreversible processes such as information erasure, can be performed in thermodynamically reversible ways [5]. However, Landauer’s Principle has never been experimentally disproven and in fact the numerical value has been reached experimentally [6]. The theoretical framework behind the principle has been generalised [7] to include entropy costs in other conserved quantities such as angular momentum.

A link between information and thermodynamics would be a momentous discovery, for it would introduce physical constraints to the idea of information. It would outlaw perfect computing efficiency and place limits on the abilities of quantum computers, which seek to avoid the phenomenon of decoherence.

So, Landauer’s principle, the supposed saviour of the second law of thermodynamics, remains a bone of contention. The paradox of Maxwell’s Demon has continued to challenge physicists with no universally accepted resolution in sight. Though Landauer proposed a clever solution, the debate behind its theoretical validity rages on, and will do so for years to come.

Eoin Whooley

References

[1] J. von Neumann, *Mathematical Foundations of Quantum Mechanics*. Princeton University Press, 1932.

[2] R. Landauer, “Irreversibility and Heat Generation in the Computing Process,” *IBM J Res Dev*, vol. 5, no. 3, pp. 183–191, 1961, doi: 10.1147/rd.53.0183.

[3] J. Earman and J. D. Norton, “Exorcist XIV: The Wrath of Maxwell’s Demon. Part I. From Maxwell to Szilard,” *Studies in History and Philosophy of Modern Physics*, vol. 29, no. 4, 1998.

[4] O. Shenker, “Logic and Entropy,” Jun. 2000.

[5] T. Sagawa, “Thermodynamic and logical reversibilities revisited,” *Journal of Statistical Mechanics: Theory and Experiment*, vol. 2014, no. 3, p. 3025, Mar. 2014, doi: 10.1088/1742-5468/2014/03/P03025.

[6] R. Gaudenzi, E. Burzurí, S. Maegawa, H. S. J. van der Zant, and F. Luis, “Quantum Landauer erasure with a molecular nanomagnet,” *Nat Phys*, vol. 14, no. 6, pp. 565–568, 2018, doi: 10.1038/s41567-018-0070-7.

[7] J. ~A. Vaccaro and S. ~M. Barnett, “Information erasure without an energy cost,” *Proceedings of the Royal Society of London Series A*, vol. 467, no. 2130, pp. 1770–1778, Jun. 2011, doi: 10.1098/rspa.2010.0577.

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