Müller I. A history of thermodynamics. The doctrine of energy and entropy (1185104), страница 58
Текст из файла (страница 58)
TIP became also known asthe thermodynamics of the Dutch school, because many Dutch thermodynamicists contributed to it. The major monograph on the subject was writtenby de Groot and Mazur.30 The book gives a fairly clear account of TIP; itputs some emphasis upon the so-called Curie principle by which thermodynamic forces and fluxes cannot be related linearly unless they have thesame tensorial rank.Clifford Ambrose Truesdell (1919–2000) recognized the Curie principlefor what it is: a corollary of the representation theorems of isotropicfunctions.
Truesdell was openly disdainful of TIP and in the 1950’s and1960’s he waged war on Onsagerism31,32 which, by reaction, made mostthermodynamicists rally behind Onsager.But Truesdell exempted Eckart to some degree from his criticism,because Eckart had been straightforward in his assumptions, not hidingthem behind perceived principles. In fact Truesdell gives Eckart some faintpraise when he says:… C. Eckart, … who attempted to split inequalities into parts withoutappeal to any non-existent theorem, … – and who did not obfuscate thescene by any circular or inapplicable rule of symmetry.33One must realize that Truesdell had his own axe to grind, because he feltcalled upon to advertise rational thermodynamics, see below, and in thatendeavour he proved himself to be a master of subjectivity.Before we leave Eckart, we must mention his third important paper34which appeared along with the two papers already cited.
In that paperEckart laid the foundation for relativistic irreversible thermodynamics offluids, and he discovered the alternative form of Fourier’s law which isappropriate for a relativistic gas. The thermodynamic force that drives heatconduction is no longer the temperature gradient alone, rather it is equal tow66 2 XK ,wZ K EwhereXK is the acceleration, possibly the gravitational acceleration. Consequently, in equilibrium a gas in a gravitational field exhibits a temperaturegradient. The reason is clear: higher temperature means higher energy, i.e.higher mass, i.e. higher weight and therefore the temperature field must be3031323334S.R.
de Groot, P. Mazur : “Non-Equilibrium Thermodynamics” North Holland,Amsterdam (1963).C. Truesdell: “Six Lectures on Modern Natural Philosophy” Springer 1966.C. Truesdell: “Rational thermodynamics.” McGraw-Hill series in modern appliedmathematics (1969) Chap. 7.Ibidem, p. 141.C. Eckart: “The thermodynamics of irreversible processes III: Relativistic theory of thesimple fluid.” Physical Review 58 (1940).2488 Thermodynamics of Irreversible Processesbarometrically stratified, just like the mass density.
Of course the 1/c2 in thedenominator indicates that the effect is relativistically small.Onsager RelationsOnsager relations in their proper form refer to some generic set of variablesuĮ (Į = 1,2…n) which all vanish in equilibrium and which satisfy linear ratelaws of the typedW DdV / DE W E .For obvious reasons we may call M a relaxation matrix.The entropy S depends on the u ’s in such a manner that it has amaximum in equilibrium. Thus in second order approximation – which issufficient for a linear theory – the entropy reads55 GSW 1 w 25W DW E2 wW D wW E125 GSW I DE W DW E ,where g is symmetric and positive definite. In this case, where fluxes areabsent, the entropy source is simply given by the rate of change of entropySduD wS,dt wuDwhich may be considered as a sum of products of thermodynamic fluxes andforces as shown in Table 8.3.Table 8.3.
Generic fluxes and forcesThermodynamic FluxesJĮdu DdtThermodynamic Forces:Dw5wW DI DE W ELinear relations between fluxes and forces, namelyJĮ = LĮȕ XȕwithLĮȕ – positive semi-definiteOnsager Relations249guarantee that the entropy source is non-negative.
And Onsager relations 35require thatLĮȕ = MĮȕ g-1Įȕbe symmetric.Onsager relations in this form – and for these forces and fluxes – can beproved on the basis of Onsager’s hypothesis about the mean regression offluctuations, cf. Chap. 9. A good presentation of the proof is contained inthe popular monograph by de Groot and Mazur. The authors are remarkablecandid when they call Onsager’s hypothesis not altogether unreasonable.36There are two qualifications of the Onsager relations, of which one is due toOnsager himself.37 It concerns the presence of a magnetic flux density B and itrefers to the well-known fact that the path of a charged particle in a magnetic fieldcannot be reversed by reversing the velocity, unless B is also reversed. The otherqualification is due to Casimir38 who distinguished even and odd variables amongthe uĮ’s with respect to time reversal.
I shall not go into that and merely mentionthat the Onsager relations with Casimir’s amendment are often cited under theacronym OCRR, for Onsager-Casimir-Reciprocity-Relations.Meixner39 has extrapolated the OCRR to transport phenomena inmixtures. To wit, he applied them to Eckart’s phenomenological equationsfor mixtures, see above, where, according to Meixner, they readN CDN DCND~N Dand.DE.ED~.E.E .A convincing proof in this more complicated case is not available.4035 L.Onsager: “Reciprocal relations in irreversible processes.” Physical Review (2) 37(1931) pp.
405-426 and 38 (1932) pp. 2265–2279.36 S.R. de Groot, P. Mazur : loc. cit. p. 102.It is often said that microscopic reversibility is the key assumption in the proof of Onsagerrelations. And it is true that the proof makes use of the fact that atomistic trajectories arereversed when the velocities change sign. But this is so evident from the laws ofmicroscopic physics that it barely needs to be mentioned.
Certainly microscopicreversibility is infinitely more certain than the mean regression hypothesis.37 L. Onsager: (1932) loc.cit.38 H.B.G. Casimir: “On Onsager’s principle of microscopic reversibility.” Review ofModern Physics 17 (1945) pp. 343–350.39 J. Meixner, H.G. Reik: “Die Thermodynamik der irreversiblen Prozesse in kontinuierlichen Medien mit inneren Umwandlungen.” [Thermodynamics of irreversible processesin continuous media with internal transformations] Handbuch der Physik III/2, SpringerHeidelberg (1959).40 Again de Groot and Mazur, loc.cit. pp. 69–74 go farthest in the attempt to prove Onsagerrelations for transport processes, i.e. when the basic equations are partial differentialequations rather than rate laws.
They try to show that the tensor of thermal conductivity issymmetric, – Onsager’s original problem. But they do not quite succeed: All they canshow is, that the divergence of the anti-symmetric part vanishes.2508 Thermodynamics of Irreversible ProcessesHowever, there are some entirely macroscopic arguments which sufficeto prove the symmetry of the matrix of diffusion coefficients LĮȕ on thebasis of momentum conservation, and of the plausible assumption of binarydrag, so that the interaction between two constituents is unaffected by thepresence of a third constituent.
This was shown by Truesdell,41 and Müller42extrapolated that argument to show that in a mixture of Euler fluids we have~.E .E . The instances of valid Onsager relations often cited from thekinetic theory of gases are all of the type envisaged by Truesdell andMüller, so that there is not really confirmation for general Onsager relationsto be found in the kinetic theory.Also Meixner43 has proved the symmetry of lab from the principle ofdetailed equilibrium of several chemical reactions, – again without reference to any hypothesis on the mean regression of fluctuations.Rational ThermodynamicsIf the truth were known and admitted, rational thermodynamics is not allthat different from TIP. Both theories employ the Clausius-Duhem inequality and the Gibbs equation.
It is true that the arguments are shuffledaround some: The Curie principle of TIP is replaced by the principle ofmaterial frame indifference, and the Gibbs equation of rational thermodynamics is a result, whereas in TIP it is the basic hypothesis. With theClausius-Duhem inequality it is the other way round. When applied tolinear viscous, heat-conducting fluids, both theories lead to the same results.This is a good thing for both, because the field equations for such fluidswere perfectly well known before either theory was formulated, and theywere known to be reliable.The difference between the theories lies in the claims of the protagonists:Whereas TIP was never intended to represent anything but a linear theory,and could not be extrapolated, there was no such a priori restriction inrational thermodynamics.
Therefore the authors expected – and hoped for –more general validity. However, in that expectation they were eventuallydisappointed; they had overreached themselves, and the non-linear part ofthe theory crumbled. Let us consider this:One new feature of the theory is the principle of material frameindifference.44 This had been invented by Hanswalter Giesekus45 in the41C. Truesdell: “Mechanical Basis of diffusion.” Journal of Chemical Physics 37 (1962).I.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
