Müller I. A history of thermodynamics. The doctrine of energy and entropy (1185104), страница 29

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Tobe sure, the partition function cannot be calculated either in terms of thethermodynamic variables, like the volume Vand the temperature T – exceptin trivial cases like the gas and the rubber – but it may sometimes beapproximated.Gibbs’s statistical thermodynamics represents a daring and ingeniousextrapolation of Boltzmann’s ideas.

Boltzmann and Maxwell had alwaysapplied probabilistic arguments to systems of identical elements: atoms in agas, or dipoles in paramagnetic fluids or – one might add – links in a rubberchain. Gibbs proposed a giant step away from this by suggesting thatfor some purposes, however, it is desirable to take a broader view … Wemay imagine a great number of systems of the same nature, but differingin the configurations and velocities which they have at a given instant, anddiffering not merely infinitesimally, but it may be so as to embrace everyconceivable combination of configurations and velocities.77The great number of systems was called an ensemble by Gibbs. Heintroduced different kinds of ensembles:x An ensemble of systems with the same energy, now calledmicrocanonical.x An ensemble of systems with the same volume and temperature which,on account of its unique importance in the theory of statisticalequilibrium, I have ventured to call canonical.78For gases the energy constraint is linear in Nxc, because ĭ (| x xc |) is absent, whichmakes things easy.77 J.W.

Gibbs: “Elementary principles in statistical mechanics – developed with especialreference to the rational foundation of thermodynamics.” Yale University Press (1902).This memoir is available as a Dover booklet, first published in 1960. My page numbersrefer to the Dover publication.78 J.W. Gibbs: Ibidem p. XI.76Gibbs’s Statistical Mechanics119x a grand ensemble ... composed of h petits ensembles79 appropriate formixtures of h constituents.How does the ensemble-idea help? In order to see that let us concentrate onthe canonical ensemble – of total energy İ and total entropy ı – of Ȟ liquids,each in a volume V, with particle number N, and all in thermal contact, so thatthey have the same temperature.

Among the Ȟ imagined liquids let there beQ Z1 ....E0 in the state x1…cN with energy U ( x1 ...c N ) such thatݦ U ( x1 ...c N ) v x ....cN1x1 ...c N.The summation extends over all Z K 8 and all velocities.In a big step of extrapolation away from Boltzmann’s entropy of a gas,Gibbs writes the entropy ı of the ensemble asık ln W with Wν!· ν x ...c !1Nx1 ...cNsuch that it represents the number of realizations of the distribution Q Z1 ....E0 .In order to find the equilibrium distribution Q GSW Z1 ....E0 he maximizes ı, cf.Insert 4.7. Thus he was able to calculate the mean energy U = İ/Ȟ and themean entropy S = ı/Ȟ of a single liquid as˜(T ln P )˜TÈ U ( x1...cN ) Øboth in terms of the partition function P with P Ç exp É ÙÚ .ÊkTx ...ckT 2U˜ ln Pand S˜Tk1NCanonical ensembleWhat interests us are the thermal and caloric equations of state of a single liquid ofN atoms in a volume V with energyU ( x1 ...cN )Ç N xcxcµ2c2 1Ç N xc N x „c „ĭ (| x x „ |) .2 xc , x „c „Instead we consider an ensemble of Ȟ such liquids with a fixed total energy İ.Among the Ȟ liquids let there be Q Z1 ....E 0 in the state x1…cN such that – ignoring theinteraction between the liquids – we have79J.W.

Gibbs: Ibidem p. 190.120 4 Entropie as S = k ln WÇ U ( x1 ...c N ) ν x ....c .1Nx1 ...c NİThe entropy ı of the ensemble is calculated ask ln W with Wıν!· ν x ...c !x1 ...cN 1 NSummation and product extend over all positions Z K.8 and all velocities. Theequilibrium distribution Q Z1 ....E 0 is the one that maximizes ı under the constraints ofconstant İ and Ȟ, namely the canonical distributionν x ....c1Nνexp( βU ( x1 ...cN ))Ç exp( βU ( x1 ...c N ))x1...cN.Hence follows for the energy and the equilibrium entropy of the ensembleݘ ln Pν˜βand ıÈ İØν k É β ln P Ù ,Ê νÚÇ exp( βU ( x1...c N )) is the partition function.

ȕ is the Lagrangex1 ...cNmultiplier that takes care of the energy constraint.where PBy equipartition the energy of each atom must be equal tous to identify ȕ as1M6M6and that helps, because we haveÇ νx1...cNİkin32µ 2Øequ È µ 2ÉÊ c1 ... cN ÙÚ22x ....c1Nν¹N3 12 β.Thus the mean energy and entropy of a single liquid isUSFİνıkT 2k˜ ln P˜˜Tν kT ln P.˜Twith PÈ U ( x1...cN ) ØÙÚkTÇ exp É Êx1 ...cN(T ln P ) hence the free energy FInsert 4.7U TS :Gibbs’s Statistical Mechanics121Thus Gibbs arrived at a final result, after a fashion, even though it isquite impossible for liquids – and most other non-trivial systems – toevaluate the sum in the partition function and obtain P(V,T) explicitly.However, the problem is reduced to the evaluation of a multiple sum.

Inthat form it represents a challenge for mathematicians, and one may think ofmaking intelligent approximations. In factx For liquids J.E. Mayer and M.G. Mayer developed a cumbersome buteffective cluster method to approximate the thermal equation of state ofa real gas80x Lars Onsager was able to evaluate the partition function exactly for theIsing model of a ferromagnet, although I believe that the success wasrestricted to a two-dimensional case,x Recently Oliver Kaster81 has approximated the partition function of ashape memory alloy and was able to simulate the austenitic ļmartensitic phase transition that is typical for such alloys.So Gibbs’s idea proved to be quite useful. Conceptually, however, there areproblems: We may very well conceive of ensembles, of course. But inactual fact we have a single liquid – never an ensemble.

So, how do weargue in order to get the ensemble out of our minds and concentrate on thesingle liquid? The conventional idea is that the ensemble does no more thanprovide the individual liquids with a temperature. From that thought it is asimple conceptual step to forget the ensemble entirely, and replace it by aheat bath for the one and only liquid in our laboratory.Gibbs did not address such lingering misgivings nor do most books onstatistical mechanics.82 A notable exception is Schrödinger in a writtenaccount of thoughtful seminars.83 Says he:….here the Ȟ identical systems are mental copies of the one system underconsideration – of the one macroscopic device that is actually erected onour laboratory table.

Now what on earth could it mean, physically, todistribute a given amount of energy İ over these Ȟ mental copies? The ideais in my view, that you can, of course, imagine that you really had Ȟ copiesof your system, that they really were in “weak interaction” with eachother, but isolated from the rest of the world. Fixing your attention on one80J.E. Mayer, M.G. Mayer: “Statistical Mechanics.” John Wiley & Sons, New York (1940)Chap. 13.81 O. Kastner: “Zweidimensionale molekular-dynamische Untersuchung des Austenit ļMartensit Phasenübergangs in Formgedächtnislegierungen.” [Two-dimensional moleculardynamics of the austeniteļmartensite phase transition in shape memory alloys]Dissertation TU Berlin, Shaker Verlag (2003).82 In modern books on the subject it is not uncommon to have the partition function appearon the first half-page, and the rest of the book is given to its evaluation in special cases.That is what is known as the deductive approach, or understanding by doing.83 E.

Schrödinger: “Statistical thermodynamics. A course of seminar lectures.” Cambridge atthe University Press (1948).122 4 Entropie as S = k ln Wof them, you find it in a peculiar kind of “heat bath” which consists of theȞ – 1 others.The only treatment of a proper and realistic ensemble, known to me, is dueto Maxwell in his paper “On Boltzmann’s theorem on the averagedistribution of energy in a system of material points” 84 Maxwell considers agas of v . N atoms and – in imagination – he splits it into v gases of N atoms.Then he proceeds to determine the distribution85νequx ....cN11VN12 πµkT3NÈ µc 2 ¹¹¹ µc 2 Ø1NÙ2 kTÊÚexp É ,which is the canonical distribution for the case. For N=1 – one single atom –Maxwell thus recovers the Maxwell distribution, which he had derivedoriginally in two different manners, see above.

Maxwell is acknowledgedby Gibbs – along with Clausius and Boltzmann – as one of the principalfounders of statistical mechanics.Another question which Gibbs took in his stride – without much ado –concerns the mean value over the ensemble: What is the significance of thatmean value for the single liquid under consideration? The answer is givenby the ergodic hypothesis. This implies that the number Q GSW Z1 ....E0 calculatedfor the ensemble of v liquids is also the frequency of the state x1....cN in asingle liquid, if that liquid is observed v times at sufficiently large intervals.The hypothesis is often expressed by sayingensemble average = expectation value for single liquid86so that the average over the imagined ensemble is immediately relevant forthe one and only system under consideration.

Obviously, the prescriptionfor the calculation of the time average – or expectation value – can only berelevant for equilibria.In the wording of arguments and in the formulae I have so farconcentrated on liquids. This was for definiteness and suggestiveness only.Statistical mechanics of other bodies follows the same lines.

One of themore amazing applications87 is a single hydrogen atom, a proton with oneelectron which may occupy 2n2 orbits (n = 1,2,…) with energies84J.C. Maxwell: (1879) loc.cit.This paper of 1879 thus contains Maxwell’s third derivation of the Maxwell distribution.We have reviewed the other two derivations above.Maxwell’s third derivation is now a popular exercise for physics students, because itprovides them with the opportunity to acquaint themselves with volumes and surface areasof spheres in many dimensions.86 A trivial illustration is this: Suppose that on an aerial photo of a city you identify thefraction of cars which drive with 50 km/h.

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