Abstract
The previous Lecture considered a system containing one atom of an ideal gas. The purpose of this Lecture is to generalize the discussion to treat a system of N such atoms. We could have gone directly from Lecture 3 to the calculation below, and skipped Lecture 4 and the Aside. There would have been a pedagogical penalty; our first serious calculation would have involved 6N-dimensional integrals for. Lecture 4 already involved quite enough new ideas.
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I use the convention that the limits → {ita+} and → {ita −} refer to the limit approaching {ita} from the positive and negative sides, respectively. These limits need not be equal, as seen trivially for functions which are discontinuous at {ita}.
The import of the first four K N was worked out in the social sciences a century ago. See K. Pearson, Phil. Trans. Roy. Soc. 185, 71 (1894).
E = ∞ is not a point (a single location) in the sense that “point” is used here.
There exist nonclassical experiments which are sensitive to these fluctuations.
Recall that laser fluorescence techniques sufficiently sensitive to identify and track the location of a single atom were demonstrated two decades ago.
G. D. J. Phillies, Elastic and Quasielastic Scattering of Light in P. J. Elving Treatise in Analytic Chemistry, Part I, Vol. 8, Wiley, New York (1986) lists many earlier treatises.
Strutt, J. W. (Lord Rayleigh), Phil. Mag. (IV) 41, 107 (1871), 224 (1871), 447 (1871); (V), 12, 81 (1881). Note that I refer to the luminosity of the sky (it glows, due to the scattering of sunlight or moonlight), not to the color of the sky. Simple Rayleigh scattering gives a bluish smoke-haze color, not the “big sky blue” seen on an especially clear day. The physiological color of the sky is a rather complicated issue.
The phrase “held fixed” carries with it the emendation “a record of momentum being kept, so that on releasing the atoms they are restored to the momenta they had before they were held fixed.”
If a = b, interchanging the two particles has no effect at all. There are a lot of states in which a ≠ b, but only a few states in which a = b. To be precise, the ratio of the former to the latter is the ratio of a line to a point. Relative to a line, a point is a set of measure zero, so the states with a = b make essentially zero contribution to the ensemble average. How one treats them does not matter. Point for thought:Does the Boltzmann argument that classical particles are distinguishable continue to apply if a = b— If it does, what does it mean— How can a state be distinguishable from itself— If the argument applies when a ≠ b, why does it break down when a = b (other than by giving a wrong answer, which is not why an argument breaks down)?
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© 2000 Springer-Verlag Berlin Heidelberg
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Phillies, G.D. (2000). N-Atom Ideal Gas. In: Elementary Lectures in Statistical Mechanics. Graduate Texts in Contemporary Physics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1264-5_6
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DOI: https://doi.org/10.1007/978-1-4612-1264-5_6
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