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 65

We maydraw X (W ;X E ) as a graph of the type shown in the lower part of Fig. 9.1.According to Lars Onsager the mean regression is given by the samefunction as a macroscopic decay. This can be proved – after a fashion – forBrownian particles, see Insert 9.2.The mean regression of fluctuation for a Brownian particleThe formal solution of the Langevin equation reads with W 0XX(t )(t0 )et t 0W0 1µ etW0³tt0tcW0P6SK Te F (t c)dt cso that the mean regression of a fluctuation Xȕ comes out asAuto-correlation FunctionÇαeττ0ÈÉÉÊ(tÈ αβÉ (tα )e1ÊÉXXXX(τ ; β )1NNβ1 1µ NNÇαβτ ) tα βτ01µtα β τ etα βe1τ0Ôtα β τtα βτ0Ôtα β τtαβ281Øe F (t „)dt „ÙÚÙt„τ0t„Øe τ 0 F (t „)dt „ÙÙÚGiven time the force F(t) in the integrand is fluctuating between positive andnegative values so that the integral itself may have a positive or negative value, butit is definitely finite.

Therefore for large enough N the second term vanishes, so thatthe mean regression becomesXXX(τ ; β )βeττ0which is equal to the macroscopic decay. This may be considered proof of theOnsager hypothesis, at least for Brownian particles.[The fallacy of this proof for small values of IJ is obvious: Indeed, for smallvalues of IJ the force F(tƍ) in the interval tĮȕ < tƍ < tĮȕ + IJ is most likely close to theforce F(tĮȕ), because the force does not really jump, although it may changequickly.

Thus for small IJ the Onsager hypothesis fails. Another way to see this isas follows: For small values of IJ there are obviously equally many valuesX (VD E W )bigger and smaller thanX (VD E ) sothatX (W ;X E )must start outhorizontally, i.e. it cannot decay exponentially at the outset.]Insert 9.2Auto-correlation FunctionThe auto-correlation function – denoted by X (0)X (W ) for the velocity of aBrownian particle – is a mean value over mean fluctuation regressions of alloccurring initial values X  ; let their number be denoted by M. So as to avoidthe trivial result zero for the mean value, the mean regressions are premultiplied by X  before the mean value is taken.

Thus the auto-correlationfunction is defined asM(τ ., β )X1 βXÇβX1MXXXX(0) (τ ).N ,M1ββtt()(τ)ÇααMN α ,β 12829 FluctuationsBetween all N values tĮȕ and all M values with X  the summation covers acoherent large time interval T so that one may writeX (0)X (W )166³ X (V )X (V W )FV.0Since all mean fluctuation regressions are equal in their functionalbehaviour to the macroscopic law of decay, – according to the Onsagerhypothesis – this is also true for their mean value, i.e.

the auto-correlationfunction.The auto-correlation function is often easier to calculate and to measurethan the mean regression of a particular size of fluctuation. Therefore theOnsager hypothesis is most often pronounced by saying that the autocorrelation function is equal to the macroscopic decay function.Extrapolation of Onsager’s HypothesisBrownian particles provide the first fluctuating phenomenon that has beenstudied and they are simple enough to be amenable to intuitive argumentand to calculation.

Therefore they serve as prototypes for the treatment offluctuation and for the proof of Onsager’s hypothesis, see Insert 9.2.The hypothesis is not restricted to Brownian particles, however. It issupposed to hold for all fluctuating systems. And it is usually calledOnsager’s theorem. Physicists have a way to quickly become verydefensive of Onsager when challenged, probably because of the precariousness of the proof of the theorem, or because they do notunderstand it, or because Onsager has been canonized by the Nobel prize in1968, see Fig. 9.2.

There is some uneasiness, however. We have alreadyquoted the popular textbook by de Groot and Mazur,16 who give faint praiseto Onsager by calling his hypothesis not altogether unreasonable.17Light ScatteringWhile Brownian particles and their erratic motion can be seen, albeit onlyunder the microscope, fluctuations of mass density, and velocity and temperature in air cannot be seen. And yet they are there, and they affectthe transmission of light. Indeed, very tiny and very short-lived local1617S.R. de Groot, P.

Mazur: loc.cit.And we have seen above why the proof of the hypothesis is flawed for small times evenin Brownian motion. In the sequel I shall ignore that qualification. Physicists tell me that itis pedantic.Light Scattering283Onsager left his native Norway in 1928and came to the United States. Later heheld a chair of theoretical chemistry atYale University, where he taught StatisticalMechanics I and II to chemistry students.Among the students his course was knownas Norwegian I and II.18Fig. 9.2. Lars Onsager receiving the Nobel prize for chemistry in 1968compressions and expansions of air – and gases generally – occur as a resultof the random motion of molecules and atoms and they affect the dielectricconstant, because it depends on the mass density.Because of these fluctuations some light is scattered sideways, seeFig. 9.3.

Most of the scattered light has the frequency Ȧ(i) of the incidentmono-chromatic light, but neighbouring frequencies Ȧ are also present.Typically the spectrum S(Ȧ) of light – scattered in a gas and passed throughan interferometer to a photo-multiplier – exhibits three peaks, if the gas isnormally dense. In a moderately rarefied gas one sees a flatter curve withlateral shoulders, cf.

Fig. 9.4The blue frequencies in sunlight are 16times more efficiently scattered than thered frequencies. Therefore the cloudless skyappears blue. It was John Tyndall – theadmirer of Robert Mayer – who recognizedthis phenomenon after studying LordRayleigh’s work on electro-magnetic waves.Sir James Dewar – the low temperaturephysicist – had thought erroneously thatthe blue sky is due to the oxygen contentof the air; he knew that liquid oxygen hasa blue colour.Fig. 9.3.

Light scattering, schematic18According to J. Meixner: “Chemie Nobelpreis 1968 für Lars Onsager.” [Nobel prize 1968for chemistry for Lars Onsager] Physikalische Blätter 2 (1969).2849 FluctuationsFig. 9.4. Scattering spectrum S(Ȧ) in a normally dense gas and in a moderately rarefied gas.Dots: Measurements by Clarke for rarefied gas.19 Lines: Calculation from a Navier-StokesFourier theoryIf the Onsager hypothesis is accepted, S(Ȧ) can also be calculated fromthe field equations of the gas, e.g. the Navier-Stokes equations. For densegases the measured and calculated curves fit perfectly, and thus theysupport the hypothesis.

For the rarefied gas, however, the fit is not good, cf.Fig. 9.4, and we may conclude that the discrepancy is due to the NavierStokes equations which, indeed, according to Chap. 8 are expected to fail ina rarefied gas.So, this is a case where extended thermodynamics can prove itsusefulness and practicality. Wolf Weiss20 has applied the linearized fieldequations of 20, 35, 56, and 84 moments to the problem and has obtainedthe scattering spectra of Fig. 9.5 (top) for small pressures as in Fig. 9.4.They differ among themselves and none of them fits the experimentalpoints well. Nor can we adjust parameters to obtain a better fit, becausethere are no adjustable parameters in the theories of extendedthermodynamics. Or rather, one might say that the only parameter is thenumber of moments and moment equations.

So Weiss went ahead to 120through 286 moments and obtained convergence as well as a perfectagreement with experimental results, cf. Fig. 9.5 (bottom).Here we have another instance where a result of thermodynamics issatisfactory, amazing and disappointing at the same time.21192021N.A. Clarke: “Inelastic light scattering from density fluctuations in dilute gases.

Thekinetic-hydrodynamic transition in a monatomic gas.” Physical Review A 12 (1975).W. Weiss: “Zur Hierarchie der Erweiterten Thermodynamik.” loc. cit.See also: W. Weiss, I. Müller: “Light scattering and extended thermodynamics.”Continuum Mechanics and Thermodynamics 7 (1995).Recall Schrödinger’s comment on gas degeneracy in Chap. 6.Light Scattering285Fig.

9.5 . Light scattering spectra in a moderately rarefied gas. Top: ExtendedthermodynamicsN= 20, 35, 56, 84. Bottom: Extended thermodynamics N= 120, 165, 220,286. Dots: Experimental points measured by N.A. Clarke22x Satisfaction comes from the fact that extended thermodynamicscombined with the Onsager hypothesis is capable of representing lightscattering in rarefied gases.x The amazing feature is the convergence at some finite number ofmoments; that observation carries information about the range ofvalidity of the theory, see below.x Disappointment stems from the large number of moments needed toachieve convergence.

We might have hoped that 13 or, perhaps, 14 or20 moments could give good results. That would have given us amanageable system of equations. Instead, we need 120 of them, – atleast for the low pressure to which the curves of Fig. 9.5 refer.The convergence put in evidence by the plots of Fig. 9.5 permits us toconclude that extended thermodynamics determines its own range ofapplicability without any reference to experiments. This is something that isoften said a theory cannot do. But then, extended thermodynamics is not a22N.A. Clarke: loc.cit.2869 Fluctuationssingle theory, rather it is a theory of theories, one each for a given numberof moments. So if – after an increase of that number – we obtain the samefunction S(Ȧ) as before, in some norm, we have reached convergence andcan fully trust the theory and predict the light spectrum, without making asingle experiment.More Information About Light ScatteringIn the previous section we have used light scattering to advertise thevalidity and usefulness of the Onsager hypothesis in extended thermodynamics.

This is only one rather special aspect and application of lightscattering. There are others, more practical ones, and we give the briefestpossible description about the method and its practical application.The scattered electric field which arrives at the interferometer consists ofthe high frequency carrier wave – from a laser with Ȧ(i) = 4.7 1015 Hz (say) –modulated in its amplitude by a fluctuating spatial Fourier harmonic – ofwave number q – of the density field.