Poincaré recurrence theorem

In mechanics, Poincaré recurrence theorem states that an initial state or configuration of a mechanical system, subjected to conserved forces, will reoccur again in the course of the time evolution of the system. [1] The theorem was stated by French mathematician Henri Poincaré who in his 1890 article “On the Three-body Problem and the Equations of Dynamics”, building on the previous work of French mathematician Simeon Poisson, states that: [2]

“It is proved that there are infinitely many ways of choosing the initial conditions such that the system will return infinitely many times as close as one wishes to its initial position … there are also an infinite number of solutions that do not have this property, but it is shown that these unstable solutions can be regarded as ‘exceptional’ and may be said to have zero probability.”

The commonly used example to explain the theorem is that if one inserts a partition in a box, pumps out all the air molecules on one side, then opens the partition, the recurrence theorem states that if one waits long enough that all of the molecules will eventually recongregate in their original half of the box.

Second law
The theorem is often found mixed up with the second law of thermodynamics to the effect that some will loosely argue that there exists a very small probability that an isolated system will reconfigure to a more ordered state (thus effecting an entropy decrease). In the 1972 article "Thermodynamics of Evolution" Belgian thermodynamicist Ilya Prigogine uses the recurrence theorem, in stating that the probability that a macroscopic number of molecules will assemble to form higher ordered living structures is “vanishingly small” at ordinary temperatures, to justify his argument that classical thermodynamics cannot explain the formation of biological structures. [3]

It was German mathematician Ernst Zermelo, an assistant to Max Planck, who in 1896 first pointed out the apparent incompatibility between Poincaré’s recurrence theorem, Clausius’ second law, and Boltzmann’s H-theorem, in the sense that either the second law may be violated or connection to mechanical system cannot be made [4] Zermelo argued that the science of irreversible processes, thermodynamics, could not be reduced to mechanics, and reasoned that, for instance, German physicist Heinrich Hertz’s mechanical derivation of the second law must be impossible. [5]

Into the early to mid 20th century, following Max Planck's circa 1900-1910 push to define entropy as a measure of microstate disorder, people began to view humans as regions, eddies, of local entropy decrease. In 1946, Belgian-born English thermodynamicist Alfred Ubbelohde, for instance, utilized an unwritten version of the recurrence postulate:

“Considered from the standpoint of the trend of entropy, both the activities of living organisms, and their first appearance, may be termed ‘special happenings’ [extremely unlikely] not wholly amenable to the laws of molecular probability.”

This statement comes at the end pages on his chapter on thermodynamics and life, in which he concludes, in the end, rather ambivalently that "the contrast between inanimate matter and life is not to be explained starting solely from the laws that govern inanimate matter," but rather that certain experimental inquires will need to be made in the future, such as measuring the energy and entropy changes that accompany the activities of living organisms. [6]

See also
Loschmidt’s paradox

1. Dorfman, Jay R. (1999). An Introduction to Chaos in Nonequilibrium Statistical Mechanics (pg. 54). Cambridge University Press.
2. Poincaré, Henri. (1890). “On the Three-body Problem and the Equations of Dynamics” (“Sur le Probleme des trios corps ci les equations de dynamique”), Acta mathematica, 13: 1-270, in The Kinetic Theory of Gases (pgs. 368-81), 2003, by Stephen G. Brush and Nancy S. Hall. Imperial College Press.
3. (a) Prigogine, Ilya, Nicolis, Gregoire, and Babloyants, Agnes. (1972). "Thermodynamics of Evolution," (part I). Physics Today (pgs. 23-28), Vol. 25, November.
(b) Prigogine, Ilya, Nicolis, Gregoire, and Babloyants, Agnes. (1972). "Thermodynamics of Evolution," (part II). Physics Today (pgs. 38-44), Vol. 25, December.
4. (a) Zermelo, Ernst. (1896). “Ueber einen Satz der Dynamik und die Mechanische Warmetheorie (On a Theorem of Dynamics and the Mechanical Theory of Heat)”, Ann. 57: 485-94.
(b) Ebbinghaus, Heinz-Dieter, and Peckhaus, Volker. (2007). Ernst Zermelo (section: “The Boltzmann Controversy”, pgs 15-26). Springer.
(b) Petersen, Karl E. (1989). Ergodic Theory (pg. 34-35). Cambridge University Press.
5. Jungnickel, Christa and McCormmach, Russell. (1990). Intellectual Mastery of Nature: Theoretical Physics from Ohm to Einstein, Volume 2: the Now Mighty Theoretical Physics, 1870 to 1925 (pg. 214). University of Chicago Press.
6. Ubbelohde, Alfred R. (1947). Time and Thermodynamics (pgs. 104-105). Oxford University Press.

Further reading
● Brush, Stephen G. (1996). A History of Modern Planetary Physics: Nebulous Earth (pg. 129). Cambridge University Press.
● Zermelo, Ernst (1896). “On the Mechanical Explanation of Irreversible Processes” (“Ueber mechanische Erklarungen Irreversibler Vorgange”, Annalen der Physics, 59: 793-801, in The Kinetic Theory of Gases (pgs. 403-411), 2003, by Stephen G. Brush and Nancy S. Hall. Imperial College Press.

External links
Poincaré recurrence theorem – Wikipedia.

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