Müller I. A history of thermodynamics. The doctrine of energy and entropy (1185104), страница 23

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c here is the mean speed of the atoms – at least in order ofmagnitude – and l is the mean free path which we have previouslyintroduced by the equation N Vlπr2 1. Thus the viscosity is independent ofthe density of the gas and it grows with temperature, since the mean speedis proportional to T , see above. Maxwell says …this consequence of amathematical theory is quite surprising and he was doubtful, because … theJames Clerk Maxwell (1831–1879)91only experiment which I know does not seem to confirm the result.

In theevent, however, the theory was right and the experiment was wrong.Maxwell had been too timid and Boltzmann says when he reports theevent:25… observations revealed only the lack of confidence of Maxwell in thepower of his own weapons.It is true that the viscosity of liquids drops with growing temperature, butfor gases this relationship is reversed. When this was confirmed by newexperiments – by Maxwell himself – the fact provided a boost of confidencein the kinetic theory of gases. To be sure, the square root growth of Ș ontemperature is an artifact of the simple model. In 1867 Maxwell revisitedthe argument in a more systematic manner, when he derived equations oftransfer with collision terms, 26 see below. For that purpose he had to studythe dynamics of a binary collision between two atoms interacting at a shortdistance r with a repulsive force of the type 1/rs 27 .

It turned out that thetemperature dependence of the viscosity is then given by Tn with n =21s 1 2 . An infinite value of s corresponds to the billiard ball model, whiles=5 corresponds to the so-called Maxwellian molecules. Maxwellconcluded from his own experiments that Ș was proportional to T, whichmust have been wishful thinking, because forces of the type 1/r5 make for aparticularly simple form of the collision term in the equations of transfer,see below.28There is an element of probability in the kinetic theory, or the mechanicaltheory of heat, which was previously absent from mechanics: Indeed, whenwe say that Nij(ci)dci is the number of atoms with the i-component ci ofvelocity, we do not expect that statement to be strictly true.

Since the atomsare perpetually changing their velocities in frequent collisions, the numberof atoms with ci is fluctuating and Nij(ci)dci is merely the mean value orexpectation value of that number. Accordingly ij(ci)dci is the probability fora single atom to have the velocity component ci.. Assuming that the velocitycomponents of an atom are independent we have thatij(c1) ij(c2) ij(c3)dc1dc2dc3is the probability of an atom to have the velocity (c1,c2,c3). So physicists hadto learn the rules of probability calculus. For some of them there were25L. Boltzmann: “Der zweite Hauptsatz der mechanischen Wärmetheorie’’ [The second lawof the mechanical theory of heat]. Lecture given at the Kaiserliche Akademie derWissenschaften on May 29, 1886.26 J.C. Maxwell: loc.cit.

(1867).27 Actually in reality the force has repulsive and attractive branches, see below. However, fora rarefied gas the simple power potential is often good enough.28 Modern measurements give n a value of about 0.8; for argon n is equal to 0.816.92 4 Entropie as S = k ln Wpeculiar scruples. So also for Maxwell, a deeply religious man with thesomewhat bigoted ethics that often accompanies piety.

In a letter he wrote:… [probability calculus], of which we usually assume that it refers only togambling, dicing, and betting, and should therefore be wholly immoral, isthe only mathematics for practical people which we should be.The Boltzmann Factor. EquipartitionTrue to that recommendation Maxwell employed probabilistic argumentswhen he returned to the kinetic theory in 1867. Indeed, probabilisticreasoning led him to an alternative derivation of the equilibriumdistribution – different from the derivation indicated in Insert 4.2 above.The new argument concerns elastic collisions of two atoms with energiesP222E 2 , P2 E 1 which after the collision have the energies P2 E c2 , P2 E c1 .Boltzmann was not satisfied.

He acknowledges Maxwell’s arguments andcalls them difficult to understand because of excessive brevity. Therefore herepeats them in his own way, and extends them. Let us consider hisreasoning:29 Boltzmann concentrates on energy in general – rather than onlytranslational kinetic energy – by considering G(E)dE, the fraction of atomsbetween E and E+dE. The transition probability P that two atoms – with Eand E1 – collide and afterwards move off with Eƍ, Eƍ1 is obviously30proportional to G(E) G(E1).

Therefore we havec G( E)G( E1 ) .PE , E E „ , E „11The probability for the inverse transition is 31PE „ , E „  E , E11cG ( E „ ) G ( E „1 ) .In equilibrium both transition probabilities must be equal so that lnG(E)is a summational collision invariant. Indeed, in equilibrium we haveG(E)G(E1) G(E„)G(E„1) hence ln G(E) ln G(E1) ln G(E„) ln G(E„1) .29L. Boltzmann: “Studien über das Gleichgewicht der lebendigen Kraft zwischen bewegtenmateriellen Punkten.” [Studies on the equilibrium of kinetic energy between movingmaterial points] Wiener Berichte 58 (1868) pp. 517–560.30 Actually, what is obvious to one person is not always obvious to others.

And so there is anever-ending but fruitless discussion about the validity of this multiplicative ansatz.31 The most difficult thing to prove in the argument is that the factors of proportionality –here denoted by c – are equal in both formulae. We skip that.The Boltzmann Factor. Equipartition93Since E itself is also such an invariant – because of energy conservationduring the collision – it follows that lnGequ(E) must be a linear function of E,i.e.Gequ ( E )a exp( bE )1È EØexp É Ù .Ê kT ÚkTThe constants a and b follow from the requirement‡‡ÔGequ0( E )dE 1 andÔ EGequ( E )dEkT .0Boltzmann noticed – and could prove – that the argument is largelyindependent of the nature of the energy E.

Thus E may simply be equal toP 2– as it was for Maxwell – but then it may also contain the three2 cadditive contributions of the rotational energy of a molecule and thecontributions of the kinetic and elastic energy of a vibrating molecule.According to Boltzmann all these energies contribute the equal amount1/2kT – on average – to the energy U of a body. This became known as theequipartition theorem.The problem was only that the theory did not jibe with experiments. Tobe sure, the specific heat cv = ww76 of a monatomic gas was 3/2kT but for a twoatomic gas experiments showed it to be equal to 5/2kT when it should havebeen 3kT.

Boltzmann decided that the rotation about the connecting axis ofthe atoms should be unaffected by collisions, thus begging the question, asit were, since he did not know why that should be so. And vibration did notseem to contribute at all. The problem remained unsolved until quantummechanics solved it, cf. Chap. 7.If Boltzmann was not satisfied with Maxwell’s treatment, Maxwell wasnot entirely happy with Boltzmann’s improvement.

Here we have anexample for a fruitful competition between two eminent scientists.Maxwell acknowledges Boltzmann’s ingenious treatment [which] is, asfar as I can see, satisfactory:32 But he says: … a problem of such primaryimportance in molecular science should be scrutinized and examined onevery side…This is more especially necessary when the assumptions relateto the degree of irregularity to be expected in the motion of a system whosemotion is not completely known.

And indeed, Maxwell’s treatment doesoffer two interesting new aspects:32J.C. Maxwell: “On Boltzmann’s theorem on the average distribution of energy in a systemof material points.” Cambridge Philosophical Society’s Transactions XII (1879).94 4 Entropie as S = k ln WHe extends Boltzmann’s argument to particles in an external field,the force field of gravitation (say), and thus could come up with theequilibrium distribution of molecules of the earth’s atmosphere which readsf equÈ µc 2 µ gz Ø.exp É 3Ê 2kT kT ÙÚ2π µk T1The second exponential factor is also known as the barometric formula,it determines the fall of density with height in an isothermal atmosphere. Inthe same paper Maxwell provided a new aspect of a statistical treatment,which foreshadows Gibbs’s canonical ensemble, see below.So between them, Boltzmann and Maxwell derived what is now knownas the Boltzmann factor : exp kTE .It represents the ratio of probabilities for states that differ in energy byE – in equilibrium, of course.For practical purposes in physics, chemistry, and materials science theBoltzmann factor is perhaps Boltzmann’s most important contribution; it ismore readily applicable than his statistical interpretation of entropy,although the latter is infinitely more profound philosophically.

We proceedto consider this now.Ludwig Eduard Boltzmann (1844–1906)For those who had reservations about probability in physics, bad times werelooming, and they arrived with Boltzmann’s most important work.33Maxwell and Boltzmann worked on the kinetic theory of gases at aboutthe same time in a slightly different manner and they achieved largely thesame results, – all except one! That one result, which escaped Maxwell,concerned entropy and its statistical or probabilistic interpretation.

Itprovides a deep insight into the strategy of nature and explainsirreversibility. That interpretation of entropy is Boltzmann’s greatestachievement, and it places him among the foremost scientists of all times.33L. Boltzmann: “Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen”.[Further studies about the heat equilibrium among gas molecules] Sitzungsberichte derAkademie der Wissenschaften Wien (II) 66 (1872) pp.

275–370.Ludwig Eduard Boltzmann (1844–1906)95Boltzmann about Maxwell:immer höher wogt das Chaosder Formeln.34Maxwell about Boltzmann:... I am much inclined to put thewhole business in about six linesFig. 4.3. James Clerk MaxwellMaxwell had derived equations of transfer for moments of thedistribution function in 1867,35 and Boltzmann in 1872 formulated thetransport equation for the distribution function itself, which carries hisname. What emerged was the Maxwell-Boltzmann transport theory, socalled by Brush.36 Neither Maxwell’s nor Boltzmann’s memoirs are marvelsof clarity and systematic thought and presentation, and both privatelycriticized each other for that, cf. Fig. 4.3.

Therefore we proceed to presentthe equations and results in an modern form. The knowledge of hindsightpermits us to be brief, but still it is inevitable that we write lengthy formulaein the main text, which is otherwise avoided. Basic is the distributionfunction f(x,c,t) which denotes the number density of atoms at the point xand time t which have velocity c. The Boltzmann equation is an integrodifferential equation for that function˜f˜f ci˜t˜xi1Ô ( f „f „ ff 1 )σ g sin θ dθ d ϕ dc1 .The right hand side is due to collisions of atoms with velocities c and c1which, after the collision, have velocities cǯ and cƍ1.

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