Vain Efforts of Maxwell Demon

Milan Kunz

I already published this stuff. It appeared in a rather special scientific journal (1). Since the demon is of general interest, I publish it once more.

In euphoria about Shannon's theory of communication (2) which axiomatic form leading directly to practical proposals how to optimize transfer of messages, bewitched all mathematicians. Brillouin rejuvenated an old and odd idea of Szilard that Maxwell demon works with information to decrease entropy.

Brillouin gave a verbal proof that demon must see molecules and light them actively for obtaining information about their velocities and energies. To get information, demon must spend energy. The information transforms into decreasing entropy and thus information is a negative of entropy, negentropy (3, 4).

The information entropy anchored in physics due to an authoritative review paper of Jaynes (5), who postulated that the information entropy is independent from thermodynamics.

To understand this, it is necessary to know that the entropy equation, proposed in the 19. century by Boltzmann, was never fully accepted by fellow scientists. It was only tolerated as a hypothesis.

Moreover, the information and thermodynamical entropies H have an identical mathematical form (but the meaning is different).

Negentropy found its place in biology (6), where it replaced the vague term of vital force. And of course, it undermines the base of cybernetics (7). On the other hand, it is a suitable part of the theories of extrasensory perception.

According to the negentropy principle, information should be complementary to the thermodynamical entropy (I claimed that they could be, under some conditions, additive [8]).

I, personally, have with the Maxwell demon an outstanding account. I wrote a paper about entropic measures (it was only one variation from many vain attempts to show that authorities are not right). The reviewer rejected the paper calling to the Maxwell demon. But a was prepared.

Enlightened by the practical socialism, which tried to regulate spontaneously going processes, I observed, where all analyses of the Maxwell demon (9) work failed. They did not analyze the full thermodynamical cycle.

Maxwell imagined a minuscule demon posted near a microscopic swinging door in a wall separating two gases A and B of equal temperature. The demon is instructed to open and close the door so as to allow only the swifter molecules to pass from A to B and only the slower ones to pass from B to A. Clearly, the demon can in this way make the gas in B hotter than in A. That means that it can unbind bound energy, and hence defeat the entropy law of statistical thermodynamics.

We reformulate the Maxwell's conditions. Let us suppose that at start all hotter molecules are in the compartment A and all slower ones in the compartment B. The demon is instructed to open and close the door so as to allow only the swifter molecules to pass from A to B and only the slower ones to pass from B to A. Clearly, the demon can in this way make the gas in B hotter than in A, as before.

But at first, the demon equilibrates temperatures of both compartments and in this step the entropy is growing. Only then the process proposed by Brillouin starts.

The demon is works in both steps exactly as instructed. Both steps are under his bureaucratic supervision. It was necessary to analyze the full cycle. But it seems to me as a vain task. The full cycle means, at least, that the same efforts can decrease and increase the entropy and that the relation between gained information and entropy cannot be a complementary function.

From the point of view of thermodynamics, the starting and final states are equivalent, it is equal if the swifter molecules are in A or in B. We can only rotate the system and A changes in B and B in A. Both states are symmetrical.

It is possible to show, which physical law is broken by the demon. Imagine a thoroid (pneumatic tire) with a wall inside. A microscopic swinging door is in a wall as before. The demon is instructed to open and close the door so as to allow only the swifter molecules to pass from left (A) to right (B) and only the slower ones to pass from B to A. Clearly, the demon can not in this way make the gas in B hotter than in A since the hotter molecules return after some time in the tire to the wall from the left, and the swifter molecule return after some time in the tire to the wall from the right. It is a Sisyphean labors.

The gas begins to circulate in the tire; the demon changes the mobility of the system. There are laws of mechanics against it. The moment must be conserved. This law is stronger one than laws of thermodynamics.

If my remarks about Brillouin appeared to be mocking, I apologize. I am sorry for him, since the product of two polynomial coefficients, which explains the relation between Boltzmann and Shannon entropies is known as Polya-Brillouin statistics. To have the right solution at hand and to miss it is for a scientist a tragedy.

Literature

1. Kunz M.: MATCH, 23, 3, (1988).

2. Shannon C. E.: Bell System Technical Journal, 27, 379, 623 (1948).

3. Brillouin L.: Science and Information Theory. Academic Press, New York, 1956.

4. Brillouin L.: Amer. J. Phys. 29, 318 (1961).

5. Jaynes E.T.: Phys. Review 106, 620 (1957).

6. Lenk R., Crespi P., Greppin H.: Arch. Sci. 40, 351 (1987).

7. Wiener N.: Cybernetics, ?.

8. Kunz M.: Information Processing and Management, 20, 519 (1984).

9. Partington J. R. An Advanced Treatise on Physical Chemistry Vol. I. p. 163, 299. Longmans, London, 1949.

10. Kunz M. in the book: Problems in Quantum Physics II, Gdansk', (Mizerski J., Posiewnik A., Pykacz J., Zukowski M., Eds.) p. 377. World Scientific, Singapore, (1990).