Abstract
Statistical mechanics poses the problem of deducing macroscopic properties of matter from the atomic hypothesis. According to the hypothesis matter consists of atoms or molecules that move subject to the laws of classical mechanics or of quantum mechanics.
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N = 6.02 x 1023 particles per mole = “Avogadro’s number”: this implies, for instance, that 1 cm3 of hydrogen, or of any other (perfect) gas, at normal conditions (1 atm at 0°C) contains about 2.7 x 1019 molecules.
At least in the thermodynamic limit, see (1.5.6), in which the volume becomes infinite but the average density and energy per particle stay fixed.
Which it is worth stressing once more does not depend on the microscopic dynamics.
later, one finds k B = 1.38-16 erg K-1.
Mainly it simplifies the relation between T and (3 in the first of (1.6.8) below.
i. e. a thermodynamics that does not come into conflict with the basic principles, expressed by (1.6.5).
One should not think that it is difficult to devise ensembles which are orthodic and which may seem “not reasonable” (for a thermodynamic interpretation) : in fact Boltzmann’s paper, [Bo84], on the ensembles starts with such an example involving the motion of one of Saturn’s rings regarded as a massive line (in a parallel paper the example was the Moon, whose orbit was replaced by an ellipse of mass such that each arc contained an amount of mass proportional to the time spent on it by the Moon). This may have been one of the reasons this fundamental paper has been overlooked for so many years. Such “unphysical” examples come from Helmholtz, [He95a], [He95b], and played an important role for Boltzmann (who was considering them in a less systematic way even much earlier, [Bo66]). In fact if one can define the mechanical analogue of thermodynamics for any system, small or large, then it is natural to think that in large systems the average quantities will also satisfy the second law. And the idea (of Boltzmann) that the macroscopic observables have the same value on most of the energy surface makes the law easily observable in large systems, while this may not be the case in very small systems. In other words the one-degree-of-freedom examples are not at all unphysical; rather the contrary holds: see Appendix 1.A1 (to Chap. 1) for Helmholtz’s theory and Chap. 9 for a recent application of the same viewpoint.
Note that if one throws hard spheres randomly and independently in a box then some of them may overlap. It is convenient not to exclude such a possibility provided one disregards completely the interaction of the overlapping spheres as long as they overlap and starts considering it only after they separate because of the motion: this is clearly a trick that introduces some minor simplification of the discussion while not affecting the macroscopic properties of the (rarefied) gas.
Readers might be interested in the referee’s report to one of my papers, [Ga95a], as it shows, in my opinion, how blind to evidence an historian of science can be at times. The contents of the paper in question are reproduced here; the referee’s report and the corresponding unamended original version can be found in [Ga95b] (in English).
It is important, in this respect, to be aware that Boltzmann had studied the Greek language and, by his own account, quite well: see [Bo74], p. 133, to the point of having known at least small parts of Homer by heart. Hence there should be no doubt that he did distinguish the meanings of εεδoς, and ὁδóς which are among the most common words.
In checking my understanding of the original paper as partially discussed in [Ga81], I have profited from an English translation that Dr. J. Renn kindly provided me with later, (1984). He noticed this footnote in [Bo84] while performing his translation, (unfortunately still unpublished).
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© 1999 Springer-Verlag Berlin Heidelberg
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Gallavotti, G. (1999). Classical Statistical Mechanics. In: Statistical Mechanics. Texts and Monographs in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-03952-6_1
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