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Philosophy of Science

What Chains Does Liouville's Theorem Put on Maxwell's Demon?

Peter Mark Ainsworth

Published Jan 1, 2011• 2 citations
Publisher

Abstract

Recently Albert and Hemmo and Shenker have argued that, contrary to what is sometimes suggested, Liouville's theorem does not prohibit a Maxwellian demon from operating but merely places certain restrictions on its ability to operate. There are two main claims made in this article. First, that the restrictions Liouville's theorem places on Maxwell's demon's ability to operate depend on which notion of entropy one adopts. Second, that when one operates with the definition of entropy that is usual in this debate, the restrictions put on Maxwell's demon are not even as severe as Albert and Hemmo and Shenker argue.

DOI
10.1086/658113

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Recently Albert and Hemmo and Shenker have argued that, contrary to what is sometimes suggested, Liouville's theorem does not prohibit a Maxwellian demon from operating but merely places certain restrictions on its ability to operate. There are two main claims made in this article. First, that the restrictions Liouville's theorem places on Maxwell's demon's ability to operate depend on which notion of entropy one adopts. Second, that when one operates with the definition of entropy that is usual in this debate, the restrictions put on Maxwell's demon are not even as severe as Albert and Hemmo and Shenker argue.
"Maxwell's Demon", the famous thought experiment of James Clerk Maxwell, has been devised in 1867 as a counter example for the Second Law of thermodynamics. During the 140 years since the Demon was first suggested, numerous attempts have been made to counter Maxwell's argument. The attempts have been to show that Maxwell was wrong, since his Demon cannot work for one reason or another (see Leff and Rex 2003 for details and references). In this paper we show (following an argument by Albert 2000, Ch. 5.) that Maxwell was basically right, in the sense that his thought experiment is compatible with the laws of mechanics as well as with central principles of statistical mechanics. We then derive some (weak) restrictions on the Demon's efficiency. Finally, we prove that the Demon's cycle of operation can be completed (in particular, the Demon's memory can be erased) without increasing the total entropy of the universe. We draw some conclusions about the way to understand the meaning and role of probability in classical statistical mechanics
Demon, the famous thought experiment of James Clerk Maxwell, has been devised in 1867 as a counter example for the Second Law of thermodynamics. During the 140 years since the Demon was first suggested, numerous attempts have been made to counter Maxwell's argument. The attempts have been to show that Maxwell was wrong, since his Demon cannot work for one reason or another (see Leff and Rex 2003 for details and references). In this paper we show (following an argument by Albert 2000, Ch. 5.) that Maxwell was basically right, in the sense that his thought experiment is compatible with the laws of mechanics as well as with central principles of statistical mechanics. We then derive some (weak) restrictions on the Demon's efficiency. Finally, we prove that the Demon's cycle of operation can be completed (in particular, the Demon's memory can be erased) without increasing the total entropy of the universe. We draw some conclusions about the way to understand the meaning and role of probability in classical statistical mechanics
The information-theoretic arguments presented in a recent publication on "Quantum discord and Maxwell's demons" are discussed, and found not to address the problem specified by Maxwell. Two interrelated and definitive exorcisms of the demon, one purely thermodynamic, and the other quantum-thermodynamic are briefly discussed. For each of the two exorcisms, the demon is shown to be incapable to accomplish his assignment neither because of limitations arising from information-theoretic tools at his disposal, nor because of the value of his IQ. The limitations are due to the physics of the state of the system on which he is asked to perform his demonic acts.
Maxwell’s demon emanates from a thought experiment proposed by James Clerk Maxwell in 1867 to illustrate the statistical nature of the second law of thermodynamics. Subsequently researchers wondered whether such a demon could in fact violate the second law. Leon Brillouin argued that the entropy produced during the demon’s measurement precluded such a violation. Years later Oliver Penrose and Charles Bennett observed (independently) that a Maxwell’s demon gathers information and stores it in a memory. Penrose showed that erasure of such a memory sends sufficient entropy to the environment to preclude violation of the second law. Notably this is so even when measurement produces arbitrarily little entropy. Bennett obtained the same result using Rolf Landauer’s seminal research on the thermodynamics of computation. The stunning shift in focus from measurement to erasure provided the impetus to better understand the role of information in quantum mechanics and thermodynamics. Indeed the linkage of information with physics is the principal legacy of Maxwell’s demon. Szilard’s one‐particle classical “gas” model and its quantum mechanical extension, together with postulated connections between entropy and algorithmic information, have provided useful insights. We review the long history of Maxwell’s demon and assess the current status of the second law in the context of the demon’s operations.