Müller I. A history of thermodynamics. The doctrine of energy and entropy (Müller I. A history of thermodynamics. The doctrine of energy and entropy.pdf), страница 66

The value of q is determined by theposition of the detector. The interferometer of type Fabry-Perot,23 cf.Fig. 9.6, superposes light that was scattered at different times in the past. Inthat way it registers the auto-correlation function E (0) E (τ ) of thescattered field or, in fact, the temporal Fourier transform of that function,i.e.

the spectral density I(q,Ȧ), whose essential part is the scatteringspectrum S(Ȧ) discussed above.Fig. 9.6. Schematic view of a Fabry-Perot interferometerIn practical physics and engineering the scattering of light has by nowbecome a powerful and elegant tool for the measurement of thermodynamic state functions and of transport coefficients. Let us consider this:23A lucid description of this remarkable instrument is given by G.

Simonsohn: “The role ofthe first order auto-correlation function in conventional grating spectroscopy.” OpticsCommunications 5 (1972). See also: I. Müller, T. Ruggeri: “Rational ExtendedThermodynamics.” loc.cit. pp. 233–236.More Information About Light Scattering287The Onsager hypothesis permits the calculation of the scatteringspectrum from the field equations of a gas (say). In particular, in a normallydense gas, where the Navier Stokes-Fourier theory is applicable we obtainthree well-developed peaks, cf.

Fig. 9.4. The heights and widths anddistance of the peaks permit the determination of the constitutive propertiesof the gas as listed in Table 9.1. Thus it is possible to read off specific heats,sound speed, and transport coefficients like thermal conductivity ț, andviscosity Ș from the properties of light scattered by a gas in equilibrium.Table 9.1. Constitutive data determine the shape of the scattering spectrumCentral PeakLateral Peaksω“c p È ˜p Øqcv ÉÊ ˜ρ ÙÚ TsiteȦ=0heightc p cv ρc p 1cpκ q2cv È c p cv κ4 ηØ 1c p ÉÊ cv ρcv 3 ρ ÙÚ q 2half widthκ 2qρc p1 È c p cv ț4 ηØ 2qÉ2 Ê cv ȡcv 3 ρ ÙÚ110 Relativistic ThermodynamicsThe theory of relativity must have implications in thermodynamics on twocounts.

Firstly, hot bodies are heavier than cold ones, because their atoms,or molecules have a bigger speed and therefore more mass. And secondly,since no particle can move faster than the speed of light, the velocity distribution of the particles must reflect the fact.To be sure, both effects are minuscule and it takes extraordinaryconditions – extraordinarily high temperatures – to make relativisticcorrections of classical formulae relevant numerically; the conditions insidethe sun are not sufficiently extreme, despite a temperature of millions of Kin the solar centre. In fact it seems that white dwarfs are the only bodies forwhich relativity matters, and where thermodynamic arguments may still beemployed without entering the realm of science fiction.

For white dwarfsthe relativistic effects are intermingled with quantum effects, because thedensity of the stars is so great that the de Broglie wave lengths of the freeelectrons overlap.In the beginning of this book I have given much space to theidiosyncrasies of the early authors in the field of thermodynamics. Onemust not think, however, that wild ideas, oversimplifications and shallowanswers are a privilege of physicists of the 19th century.

They do occur atall times and among the most distinguished people. As a case in point Idescribe – briefly – what has become known as the Ott-Planck imbroglio.Ferencz JüttnerAlthough Planck was slow to accept his own theory of quantization as true,he was quick to trust Einstein’s theory of relativity. It was therefore soonobvious to him that the Maxwell distribution had to be revised in order toaccommodate the upper bound on the speeds of atoms.

We recall fromChap. 2 that no mass can be accelerated beyond the speed of light c. Plancksuggested the problem to Ferencz Jüttner, who says in his paper:29010 Relativistic ThermodynamicsIt is a pleasant duty for me to express my warmest thanks to Hrn.Geheimrat1 Planck for the kind suggestion of this work and for hisbenevolent advise.Jüttner solved the problem in a satisfactory manner and published theresult in 1911.2 What he did was basically very simple. In effect he obtainsthe equilibrium distribution by maximizing the entropyS k lnW with WN!,·Nxp!x,punder the constraints of a fixed number N of atoms and fixed energy cP0and momentum P aN¦Nxpandx, pPA¦pAN xp ,x, pwhere Nxp is the number of atoms at place x with momentum p.Once again I apologize for a somewhat anachronistic presentation because, indeed,Jüttner did not employ the elegant four-dimensional notation of relativisticformulae which became standard later. Capital indices run from 0 to 3 such thatx0 = ct represents time, while xa are spatial coordinates.

The four-momentum of anq2atom of velocity qa and mass P P „ / 1 2 combines its energy cp0 = µc2 andcits momentum pa = µqa (a = 1,2,3) in one four-vector pA . In that notation we haveto distinguish between co- and contravariant components of a generic vectorVA and VA respectively. Both are related through VA = gABVB by the tensor gAB whichin Lorentz frames is given byg AB12ª1«0«0««¬0000º1 00 10»A, so that p A p»001»¼02 2Pc c .»Privy Councillor.

This is a honorific bestowed on eminent German – and Austrian –scientists in pre-WWI-times.F. Jüttner: “Das Maxwellsche Gesetz der Geschwindigkeitsverteilung in derRelativtheorie.” [Maxwell’s law of the velocity distribution in the theory of relativity]Annalen der Physik 84 (1911) pp. 856–882.F. Jüttner: “Die Dynamik eines bewegten Gases in der Relativtheorie.” [Dynamics of amoving gas in the theory of relativity] Annalen der Physik 35 (1911) pp. 145–161.The first paper deals with a gas at rest, while the second one deals with a moving gas.

Thesecond paper is much influenced by Planck’s erroneous opinion that temperature shouldbe transformed between two Lorentz frames, see below. In my account I present astreamlined modern version.Ferencz Jüttner291The maximization of entropy follows the usual steps known from thecorresponding non-relativistic arguments. The result is known as theMaxwell-Jüttner distributionequN xp§ U pA ·¸a ˜ exp¨ A¨kT ¸¹©withN§ U pA ·¸ ,a ¦ exp¨ AkT ¸xp ¨¹©where UA is the four-vector of velocity Xa of the gasUA§cXa¨,¨ 1 X2 2 1 X2 2cc©·¸,¸¹and T is its temperature, a scalar quantity with respect to Lorentz transformations. a is a Lagrange multiplier and it must be calculated as afunction of N and T from the constraint on N.

That calculation is best donein the rest frame of the gas, where UA = (c,0,0,0) holds. In the general casethe summation – or integration – leads to Hankel functions which makes theexpressions cumbersome although, of course, Hankel functions have beencalculated numerically and are tabulated. So, they are available, if needed.More instructive, however, than the full solutions in terms of Hankelfunctions are expansions in terms of what may be called the relativisticc 2coldness µkTc , which represents the ratio of the total energy µ´c2 of the restmass to the thermal energy kT. This is obviously a large number for normaltemperatures. The thermal and caloric equations of state may be given bysuch an expansion. Somewhat miraculously the thermal equation of state isunaffected by relativity; it still reads p = nkT, as it did for Mariotte andAvogadro.

But the caloric equation of state becomes more complex, namelyu23ÈØ3 È kT Ø 15 È kT Ø 15 È kT Ø.µ „c É1 2 É...ÙÙ2ÙÉ8 ÊÉ µ „c 2 ÚÙÊ µ „c 2 Ú 8 Ê µ „c ÚÊÚ2Thus the internal energy is still only a function of T , but its derivativewith respect to T – the specific heat cv – is no longer constant and universalas it is in the classical first order term. Rather it depends on T and on µ´, sothat the equipartition of energy is violated in a mixture of gases.Despite the successful completion of his task, Jüttner is ratherdespondent about observability and applicability of all this. He calculatesthe value of the relativistic coldness, and for helium he finds it equal toµ cc 24.32˜1013 . He comments that result by saying:kTT /KWe recognize that for all temperatures amenable to experiment theparameter has a very high value for all monatomic gases: Even when we29210 Relativistic Thermodynamicsconsider the temperatures of some stars, which have been calculated as20,000K, the parameter would not sink below1 billion.3 for any gas.Maybe Jüttner would have been less discouraged, had he known that thecentre of the sun has a temperature of 20 million K.

But even so, the relativistic coldness would still be roughly one million, so that no noticeablerelativistic effect can be expected in the sun.And yet, Jüttner’s work was to achieve some relevance in the end,although he had to wait for it.Seventeen years after the work of 1911, the phenomenon of quantum degeneration was brought to Jüttner’s attention. He studied the works ofEinstein,4 Fermi,5 and Dirac,6 in which Bose’s new method of countingrealizations of a state were employed – and in which the difference betweenFermions and Bosons was recognized.

Jüttner incorporated these modifications of classical, i.e. non-quantum physics into his relativistic formulaand obtained 7equN xp1UApA1expakT()#1with Nbosons.Afermionsxp 1 exp U A p#1akT¦1()The modification introduces more complex special functions even thanHankel functions into the equations of state, and the results are of littlesuggestive value to the non-expert. General results are listed in the literatureon relativistic thermodynamics, e.g.8, 9, 10. More instructive are the limitingexpressions of the equilibrium distribution function for either small or largerelativistic coldness, or small or large quantum mechanical degeneration.Some of these are exhibited in table 10.1.As for relevance under physically realistic circumstances Jüttner was stillpessimistic. He says:The significance of both generalized gas theories [relativistic only, andrelativistic plus quantum corrections] is, however, essentially theoretical.One has to consider that deviations of the relativistic from the Newtonianmechanics can only occur at such high temperatures that the speeds of the3I am using the American nomenclature here: What Wall Street calls a billion is a milliard,i.e.

109 , in the rest of the world.4 A. Einstein: Sitzungsberichte (1924) loc. cit.5E. Fermi: Zeitschrift für Physik (1926) loc.cit.6P.A.M. Dirac: Proceedings of the Royal Society (1927) loc.cit.7F. Jüttner: “Die relativistische Quantentheorie des idealen Gases.” [The relativisticquantum theory of the ideal gas] Zeitschrift für Physik 47 (1964), pp. 542–566.8 S.R. de Groot, W.A. van Leeuven, Ch.G. van Weert: “Relativistic Kinetic Theory.” NorthHolland Publishers Amsterdam (1980).9 I.

Müller, T. Ruggeri: “Rational Extended Thermodynamics.” loc.cit. (1998).10 C. Cercignani, G.M. Kremer: “The Relativistic Boltzmann Equation. Theory andApplications.” Birkhäuser Verlag, Basel (2002).White Dwarfs293particles become comparable with the speed of light. On the other hand,the quantization of the translational energy makes itself felt as gas degeneration only at small temperatures. Therefore the full theory could onlybe checked at intermediate temperatures by measurements conducted withextreme accuracy, and only, if the van der Waals corrections were takeninto account properly.In other words, Jüttner did not believe that his formulae had any actualrelevance anywhere. In that pessimistic evaluation he was wrong, however,as we shall see now.White DwarfsThe first white dwarf was detected by the eminent astronomer FriedrichWilhelm Bessel (1784–1846) in 1844, although the star was not actuallyseen by Bessel; it was only conjectured from the observation of the wavyline of the proper motion of the bright star Sirius.