Abstract
One does not think of the nineteenth century as a century of scientific revolutions but as a period of growth and consolidation, and in the main this view is right.
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It is this seeming limitation that proves to be a source of strength in the early days of the quantum revolution. For while quantum theory radically changed our views on the structure of matter, the changes could never go far enough to violate either the first or the second law.
On the other hand, it follows from the laws of thermodynamics that the internal energy of a gas obeying the Boyle-Charles law depends only on temperature (that is, is independent of the volume), an important fact found experimentally by Joule.
In technical terms, these exceptional states form a set of measure zero in the set of all allowable states, the measure in question being determined by the laws of mechanics.
Paul and Tatiana Ehrenfest coauthored in 1911 a fundamental article for the Encyclopedie der Mathematischen Wissenschaften in which they explained with great clarity and precision Boltzmann’s views. The article is available in English translation (by M. J. Moravcsik) published by the Cornell University Press in 1959.
“Probability.” Scientific American, Vol. 211, No. 3, September 1964, pp. 92–108 (in particular p. 106).
Einstein’s first paper on the subject (published in Annalen der Physik in 1905) was entitled “On the Movement of Small Particles Suspended in a Stationary Liquid Demanded by the Molecular-Kinetic Theory of Heat” (I am quoting from an English translation of Einstein’s papers on Brownian motion by A.D. Cowper which first appeared in 1926 and has been published by Dover Publications Inc. in 1956). When Einstein wrote this paper, he was unaware that the movement he so brilliantly analyzed had actually been observed. His second paper “On the Theory of Brownian Movement,” published in 1906, begins with the words: “Soon after the appearance of my paper on the movements of particles suspended in liquids demanded by the molecular theory of heat, Siedentopf (of Jena) informed me that he and other physicists — in the first instance, Prof. Gouy (of Lyons)—had been convinced by direct observation that the so-called Brownian motion is caused by the irregular thermal movements of the molecules of the liquid.”
Even more importantly, the coefficient contains the so-called Boltzmann constant k = R/N, where R is the universal gas constant and N the “Avogadro number,” that is, the universal number (6.02 × 1023) of molecules on a mole of any substance. It was thus possible to determine N from Brownian motion experiments and compare the value thus obtained with determination based on different principles. The agreement was excellent.
This is a special case of what is technically known as the “weak law of large numbers.” A more familiar consequence of this law is that in a series of n independent tosses of a coin the excess of heads over tails (or vice versa) is, for large n, of the order √n
I am quoting from the English translation by Stephen G. Brush, University of California Press, 1964.
Martin J. Klein, “Maxwell, His Demon, and the Second Law of Thermodynamics.” American Scientist, Vol. 58, No. 1, 1970, pp. 84–97
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Kac, M., Rota, GC., Schwartz, J.T. (1986). Statistics and Its History. In: Discrete Thoughts. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4899-6667-4_5
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