Müller I. A history of thermodynamics. The doctrine of energy and entropy (1185104), страница 24

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The angle ij identifiesthe plane of the binary interaction, while ș is related to the angle ofdeflection of the path of an atom in the collision. ș ranges between 0 andʌ/2. ı is the cross section for a (ș,ij)-collision and g is the relative speed ofthe colliding atoms. The f ƍ s in the collision integral are the values of thedistribution function for the velocities cǯ, cƍ1 and c, c1 respectively as34…ever higher surges the chaos of formulae.J.C. Maxwell: (1867) loc.cit.36 S.G. Brush: (1976) loc.cit. p. 422 ff.3596 4 Entropie as S = k ln Windicated.

The form of the collision term represents the Stosszahlansatz37which was mentioned before; it is particularly simple for Maxwellianmolecules, because in their case ıg is a function of ș only, rather than afunction of ș and g. The combination f fc c1 ff 1 in the integrand reflectsthe difference of the probabilities for collisionscƍcƍ 1 ĺ cc 1 and cc 1ĺ cƍcƍ 1.This must have been easy for Boltzmann, since logically it is adapted fromthe argument which he had used before for the derivation of the Boltzmannfactor, see above.Generic equations of transfer follow from the Boltzmann equation bymultiplication by a function ȥ(x,c,t) and integration over c. We obtain˜ Ô ψ fdc ˜ Ô ψ fck dcÈ ˜ψ˜ψ ؘt˜xkf dc ÔÉ ck˜xk ÙÚÊ ˜t.1(ψ ψ 1 ψ ' ψ „1 )( f ' f „1 f f 1 )σ gsinθ dθ dϕ dc1 dc4ÔThis equation has the form of a balance law for the generic quantity Ȍ withdensityflux³\fdc ,³ c \fdc ,kÈ ˜ψ˜ψ Øfdc , and˜xk ÙÚintrinsic sourceÔ ÉÊ ˜tcollision source1(\ \ 1 \ ' \ c1 )( f ' f c1 f f 1 )V gsin T dT dM dc 1dc4³ ckȥ1, ȥƍ, and ȥƍ1 stand for ȥ(x,c1,t), ȥ(x,cƍ,t), and ȥ(x,cƍ1,t).Stress and heat flux in the kinetic theoryIn terms of the distribution function the densities of mass, momentum, and energycan obviously be written as37That cumbersome word – even for German ears – describes a formula for the number ofcollisions which lead to a particular scattering angle by the binary interaction of atoms.The expression is not due to Maxwell, of course, nor to Boltzmann.

As far as I can findout it was first used by P. and T. Ehrenfest in “Conceptual Foundations of the StatisticalApproach in Mechanics.” Reprinted: Cornell University Press, Ithaca (1959).The word seems to be untranslatable, and so it has been joined to the small lexicon ofGerman words in the English language like Kindergarten, Zeitgeist, Realpolitik and,indeed, Ansatz.Ludwig Eduard Boltzmann (1844–1906)Ô µfdc, ȡX iȡÔ µci fdc, ȡ u 12X 2Ô97µ 2c fdc.2u is the specific internal energy formed with Ci = ci – XiȡuWithW3 M2 P6µ2³ 2 C fdc .– appropriate for a monatomic ideal gas – we obtain 38µ32 kT³ 2C 2 fdc³ fdcso that T is the mean kinetic energy of the atoms. This may be considered as thekinetic definition of temperature, or the kinetic temperature.If \PP , PEK , 2 E 2 is introduced into the equations of transfer, one obtains theXconservation laws of mass, momentum and energywȡ wȡ i0wtwxiwȡ j w ( ȡ j i ³ µC j Ci fdc )0wxiwtµw §¨ ȡ( u 1 2 ) ³ µC C fdc ³ C 2 C fdc ·¸wȡ( u 12 2 )iijii22¹©wtwxiXXXXXXX0.Comparison with the corresponding macroscopic laws, cf.

Chap. 3, identifies stressand heat flux of a gas astij ³ µC j Ci fdcandqiµ³ 2 C 2 Ci fdc .Thus the stress is properly called a momentum flux.Insert. 4.4For special choices of ȥ, viz. ȥ = µ, ȥ = µci, ȥ = 1/2 µc2, one obtains theconservation laws of mass, momentum and energy from the genericequation, cf. Insert 4.4. In those cases both source terms vanish. For anyother choice of ȥ the collision term is not generally equal to zero.However, there is an important choice of ȥ for which a conclusion can bedrawn, although the source does not vanish. That is the case when theproduction has a sign. A sharp look at the source, – in the suggestive formin which I have written it – will perhaps allow the attentive reader toidentify that particular ȥ all by himself.

Certainly this was no difficulty for38The additive energy constant is routinely ignored in the kinetic theory.98 4 Entropie as S = k ln WBoltzmann. He chose ȥ = –k ln bf , where k and b are positive constants tobe determined. With that choice we havecollision source =kf ' f „1ln( f ' f „1 f f 1 )σ gsinθ dθ dϕ dc1 dc1Ô4f fand that is obviously non-negative, since lnf ' f „1f f1and ( f ' f c1 f f 1 )always have the same sign. In equilibrium, where f is given by theMaxwellian distribution, both expressions vanish so that there is no source.Both properties suggest thatM ³ f ln5fdcdxbis a candidate for being considered as the entropy of the kinetic theory ofgases.

If k is the Boltzmann constant, S is the entropy. Indeed, if we insertthe Maxwellian – the equilibrium distribution – we obtainS equ (T , p )È5 kTkp ØS equ (T R , p R ) m Éln lnÙ2TpµµÊRR Úwhich agrees with the entropy of a monatomic gas calculated by Clausius,see Chap. 3.Entropy fluxfThe interpretation of the quantity k Ô f ln b dc as entropy density is not completeunless we relate its rate of change, or its flux, to heat or heating, so as to recognizethe status of Clausius’s 2nd lawthat:dSdttQTwithin the kinetic theory.

Let us considerfIf indeed k Ô f ln b dcdx is the entropy, the non-convective entropy flux shouldbe given byΦif k Ô Ci f ln dc.bWe calculate that expression from the Grad 13-moment approximation3939All this is terribly anachronistic but it belongs here. Grad proposed the momentapproximation of the distribution function in 1949! H. Grad: “On the kinetic theory ofrarefied gases.” Communications of Pure and Applied Mathematics 2 (1949).Ludwig Eduard Boltzmann (1844–1906)fG99ÈØÉÙÈØÈØ1111 1 2 Ùf equ É1 tCCδqCC1ÉÙ ,ij Ù2 i iÉkkÉ 2 ȡ µk T ij Ê µk T i jÚ ȡ( µ T )Ê 5 µTÚÙÉ ÙÊÚijwhich is the most popular – and most rational – approximate near-equilibriumdistribution function available.

Insertion provides, if second order terms in ij areignoredȡst ij t ijȡsequ 24 ȡ kµ T5ȡqi qiand Φ i2k T3µThus s contains non-equilibrium terms and ĭiqi 2 t ij q j.T 5 ȡ kµ T 2qi– the Duhem expression for theTentropy flux, cf. Chap. 3 – holds only, if non-linear terms are neglected.Insert. 4.5Thus Boltzmann had given a kinetic interpretation for the entropy, aninterpretation in terms of the distribution function f and its logarithm. Thatinterpretation, however, is in no way intuitively appealing or suggestive,and as such it does not provide the insight into the strategy of nature whichI have promised; not yet anyway.In order to find a plausible interpretation, the integral for S has to bediscretized and extrapolated in the manner described in Insert 4.6.

It is thevery nature of extrapolations that there are elements of arbitrariness inthem; they are not just corollaries. In the present case – in the reformulationof the integral for S – I have emphasized the speculative nature of the extrapolating steps by introducing them with a bold-face if.The discretization stipulates that the element dxdc of the (x,c)-space hasa finite number Pdxdc (say) of occupiable points (x,c) – occupiable byatoms – and that Pdxdc is proportional to the volume dxdc of the element witha quantity Y as the factor of proportionality.

Thus 1/Y is the volume of thesmallest element, i.e. a cell, which contains only one point. In this mannerthe (x,c)-space is quantized and indeed, Boltzmann’s procedure in thiscontext foreshadows quantization, although at this stage it may beconsidered merely as a calculational tool rather than a physical argument.And it was so considered by Boltzmann when he says: … it seems needlessto emphasize that [for this calculation] we are not concerned with a realphysical problem. And further on: … this assumption is nothing more thanan auxiliary tool.4040L.

Boltzmann (1872) loc.cit.100 4 Entropie as S = k ln WIf the occupancy Nxc of all points, or cells, in dxdc is equal, Boltzmannobtained by a suitable choice of b viz. b = eY, cf. Insert 4.6S1k ln,P3 N xc !xcwhere P is the total number of cells – of occupiable points – in the (x,c)space.This is still not an easily interpretable expression, but it is close to one.Indeed, if we multiply the factor N! into the argument of the logarithm, wemay writeS = k ln W ,whereWN!P.3 N xc !xcAnd that expression is interpretable, because W – by the rules ofcombinatorics – is the number of realizations, often called microstates, ofthe distribution {Nxc} of N atoms.

[The combinatorial rule is relevant here,if the interchange of two atoms at different points (x,c) leads to differentrealizations.]Reformulation ofS k ³ f lnfd cd xbLet there be Pdxdc occupiable points in the element dxdc and let Pdxdc = Ydxdc.Let further each point in dxdc be occupied by the same number N xc ofatoms, cf. figure, so that we have Nxc Pdxdc = f dxdc.

Then the contributionof dxdc to S may be written as kflnfdc dxbkN xc Pdcdx ln2dcdxM ¦ 0 ZE lnZEN xcYb0 ZE;.DFig. 4.4 An element of (x,c)-spaceThe sum is really a sum over Pdxdc equal terms. b may be chosen arbitrarily and wechoose b = eY, where e is the Euler number so thatWe shall see later, cf. Chaps. 6 and 7 that it was S.N. Bose who took the cells seriously,and gave them a value and a physical interpretation.Ludwig Eduard Boltzmann (1844–1906) kf lnfdc dxb101Pdcdxk ¦ ( N xc lnN xc N xc )xck ln Pdcdx1.– N xc !xcThe last step makes use of the Stirling formula lna! = alna-a, which can be applied,if a – here Nxc – is much larger than 1.

Therefore the total entropy readsSk ln1,P–Nxc!xcwhere P is the total number of occupiable points in the (x,c)-space.Insert 4.6A first extrapolation of the formula for S is that we may now drop therequirement that the values Nxc within the element dxdc are all equal. Thismay be a constraint appropriate to the kinetic theory of gases, – where thereis only one value f(x,c,t) characterizing the gas in the element – but it has nostatus in the new statistical interpretation of S. In particular, it is nowconceivable that all atoms may be found in the same cell, so that they allhave the same position and the same velocity; in that case the entropy isobviously zero, since there is only one realization for that distribution.With S = k ln W we have a beautifully simple and convincing possibilityof interpreting the entropy, or rather of understanding why it grows: Theidea is that each realization of the gas of N atoms is a priori considered tooccur equally frequently, or to be equally probable.

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