Müller I. A history of thermodynamics. The doctrine of energy and entropy (1185104), страница 25
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That means that therealization where all atoms sit in the same place and have the same velocityis just as probable as the realization that has the first N1 atoms sitting in oneplace (x,c) and all the remaining N – N1 atoms sitting in another place, etc.N!In the former case W is equal to 1 and in the latter it equals N1! N N1 ! .
In thecourse of the irregular thermal motion the realization is perpetuallychanging, and it is then eminently reasonable that the gas – as time goeson – moves to a distribution with more possible realizations and eventuallyto the distribution with most realizations, i.e. with a maximum entropy.
Andthere it remains; we say that equilibrium is reached.So this is what I have called the strategy of nature, discovered andidentified by Boltzmann. To be sure, it is not much of a strategy, because itconsists of letting things happen and of permitting blind chance to take itscourse. However, S = klnW is easily the second most important formula ofphysics, next to E = mc2 – or at a par with it. It emphasizes the random102 4 Entropie as S = k ln Wcomponent inherent in thermodynamic processes and it implies – as weshall see later – entropic forces of considerable strength, when we attemptto thwart the random walk of the atoms that leads to more probabledistributions.However, the formula S = klnW is not only interpretable, it can also beextrapolated away from monatomic gases to any system of many identicalunits, like the links in a polymer chain, or solute molecules in a solution, ormoney in a population, or animals in a habitat.
Therefore S = klnW with theappropriate W has a universal significance which reaches far beyond itsorigin in the kinetic theory of gases.Actually S = klnW was nowhere quite written by Boltzmann in this form,certainly not in his paper of 187241. However, it is clear from an article of187742 that the relation between S and W was clear to him. In the firstvolume of Boltzmann’s book on the kinetic theory43 he revisits theargument of that report; it is there – on pp. 40 through 42 –, where he comesclosest to writing S = klnW. The formula is engraved on Boltzmann’stombstone, erected in the 1930’s after the full significance had beenrecognized, cf. Fig. 4.5.
From the quotation in the figure we see thatBoltzmann fully appreciated the nature of irreversibility as a trend to distributions of greater probability.Since a given system of bodies can neverby itself pass to an equally probable state,but only to a more probable one, … it isimpossibletoconstructa perpetuum mobilewhich periodically returns to the originalstate.44Fig. 4.5.
Boltzmann’s tombstone on Vienna’s central cemetery41L. Boltzmann: (1872) loc.cit.L. Boltzmann: „Über die Beziehung zwischen dem zweiten Hauptsatze der mechanischenWärmetheorie und der Wahrscheinlichkeitsrechnung respektive den Sätzen über dasWärmegleichgewicht“. [On the relation between the second law of the mechanical theoryof heat and probability calculus, or the theories on the equilibrium of heat.]Sitzungsberichte der Wiener Akademie, Band 76, 11. Oktober 1877.43 L.
Boltzmann: “Vorlesungen über Gastheorie I und II“. [Lectures on gas theory] VerlagMetzger und Wittig, Leipzig (1895) and (1898).44 L. Boltzmann: „Der zweite Hauptsatz der mechanischen Wärmetheorie“. [The second lawof the mechanical theory of heat] Lecture given at a ceremony of the KaiserlicheAkademie der Wissenschaften on May, 29th, 1886.
See also: E. Broda: “LudwigBoltzmann. Populäre Schriften”. Verlag Vieweg Braunschweig (1979) p. 26.42Reversibility and Recurrence103Boltzmann’s lecture on the second law45 closes with the words: Amongwhat I said maybe much is untrue but I am convinced of everything.
LuckyBoltzmann who could say that! As it was, all four bold-faced ifs on theforgoing pages – all seemingly essential to Boltzmann’s eventual interpretation of entropy – are rejected with an emphatic not so! by modernphysics:xxxxNeither is Nxc equal for all (x,c) in dxdc,nor is it true that all Nxc >> 1,nor does the interchange of identical atoms lead to a new realization,nor is the arbitrary addition of N! quite so innocuous as it might seem.And yet, S = klnW, or the statistical probabilistic interpretation standsmore firmly than ever.
The formula was so plausible that it had to be true,irrespective of its theoretical foundation and, indeed, the formula survived –albeit with a different W – although its foundation was later changed considerably, see Chap. 6.Reversibility and RecurrenceIf Clausius met with disbelief, criticism and rejection after the formulationof the second law, the extent of that adversity was as nothing comparedwith what Boltzmann had to endure after he had found a positive entropysource in the kinetic theory of gases. And it did not help that Boltzmannhimself at the beginning thought – and said – that his interpretation waspurely mechanical.
That attitude represented a challenge for themechanicians who brought forth two quite reasonable objectionsthe reversibility objectionandthe recurrence objection.The discussion of these objections turned out to be quite fruitful, although itwas carried out with some acrimony – particularly the discussion of therecurrence objection.
It was in those controversies that Boltzmann came tohammer out the statistical interpretation of entropy, i.e. the realization that Sequals k · lnW, which we have anticipated above. That interpretation isinfinitely more fundamental than the formal inequality for the entropy in thekinetic theory which gave rise to it.The reversibility objection was raised by Loschmidt: If a system of atomsran its course to more probable distributions and was then stopped and allits velocities were inverted, it should run backwards toward the less45L. Boltzmann: (1886) loc.
cit. p. 46.104 4 Entropie as S = k ln Wprobable distributions. This had to be so, because the equations ofmechanics are invariant under a replacement of time t by –t. ThereforeLoschmidt thought that a motion of the system with decreasing entropyshould occur just as often as one with increasing entropy. In his replyBoltzmann did not dispute, of course, the reversibility of the atomicmotions. He tried, however, to make the objection irrelevant in aprobabilistic sense by emphasizing the importance of initial conditions. Letus consider this:By the argument that we have used above, all realizations, or microstatesoccur equally frequently, and therefore we expect to see the distributionevolve in the direction in which it can be realized by more microstates, –irrespective of initial conditions; initial conditions are never mentioned inthe context.
This cannot be strictly so, however, because indeedLoschmidt’s inverted initial conditions are among the possible ones, andthey lead to less probable distributions, i.e. those with less possiblerealizations. So, Boltzmann46 argues that, among all conceivable initialconditions, there are only a few that lead to less probable distributionsamong many that lead to more probable ones. Therefore, when an initialcondition is picked at random, we nearly always pick one that leads toentropy growth and almost never one that lets the entropy decrease.Therefore the increase of entropy should occur more often than a decrease.Some of Boltzmann’s contemporaries were unconvinced; for them theargument about initial conditions was begging the question, and theythought that it merely rephrased the a priori assumption of equal probabilityof all microstates.
However, the reasoning seems to have convinced thosescientists who were prepared to be convinced. Gibbs was one of them. Hephrases the conclusion succinctly by saying that an entropy decrease seems(!) not to be impossible but merely improbable, cf. Fig. 4.6.Kelvin47 had expressed the reversibility objection even before Loschmidtand he tried to invalidate it himself. After all, it contradicted Kelvin’s ownconviction of the universal tendency of dissipation and energy degradation,which he had detected in nature. He thinks that the inversion of velocitiescan never be made exact and that therefore any prevention of degradation isshort-lived, – all the shorter, the more atoms are involved.46L.
Boltzmann: „Über die Beziehung eines allgemeinen mechanischen Satzes zum zweitenHauptsatz der Wärmetheorie“. [On the relation of a general mechanical theorem and thesecond law of thermodynamics] Sitzungsberichte der Akademie der Wissenschaften Wien(II) 75 (1877).47 W.
Thomson: „The kinetic theory of energy dissipation“ Proceedings of the Royal Societyof Edinburgh 8 (1874) pp. 325–334.Reversibility and Recurrence105… the impossibility of an uncompensateddecrease of entropy seems to be reduced to animprobability.48Fig. 4.6. Josiah Willard GibbsOne of the more distinguished person who remained unconvinced for along time was Planck.
He must have felt that he was too distinguished toenter the fray himself. Planck’s assistant, Ernst Friedrich FerdinandZermelo (1871–1953), however, was eagerly snapping at Boltzmann’sheels.49 Neither Boltzmann nor the majority of physicists since his timehave appreciated Zermelo’s role much; most present-day physics studentsthink that he was ambitious and brash, – and not too intelligent; they areusually taught to think that Zermelo’s objections are easily refuted.
And yet,Zermelo went on to become an eminent mathematician, one of the foundersof axiomatic set theory. Therefore we may rely on his capacity for logicalthought.50 And it ought to be recognized that his criticism moved Boltzmanntoward a clearer formulation of the probabilistic nature of entropy and,perhaps, even to a better understanding of his own theory.Zermelo had a new argument, because Jules Henri Poincaré (1854–1912)had proved51 that a mechanical system of atoms, which interact with forcesthat are functions of their positions, must return – or almost return – to its48J.W.
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