Müller I. A history of thermodynamics. The doctrine of energy and entropy (1185104), страница 63
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At the present timeonly such problems have been resolved by extended thermodynamics – withmore than 13 moments – which do not need boundary and initial conditionsor which possess trivial ones. These include shock waves, which have beentreated with moderate success, see above, and light scattering, which hasbeen dealt with very satisfactorily indeed, cf. Chap 9.Minor intrinsic inconsistencies within extended thermodynamics havebeen removed by a cautious reformulation of the theory91,92.87I.
Müller, T. Ruggeri: “Stationary heat conduction in radially symmetric situations – anapplication of extended thermodynamics.” Journal of Non-Newtonian Fluid Mechanics119 (2004).88 E. Barbera, I. Müller: “Inherent frame dependence of thermodynamic fields in a gas.” ActaMechanica, 184 (2006) pp. 205-216.89 H. Struchtrup, W. Weiss: “Maximum of the local entropy production becomes minimal instationary processes.” Physical Review Letters 80 (1998).90E.
Barbera, I. Müller, D. Reitebuch, N.R. Zhao: “Determination of boundary conditions inextended thermodynamics.” Continuum Mechanics and Thermodynamics 16 (2004).91 I. Müller, D. Reitebuch, W. Weiss: “Extended thermodynamics – consistent in order ofmagnitude.” Continuum Mechanics and Thermodynamics 15 (2003).2708 Thermodynamics of Irreversible ProcessesHeat conduction between circular cylinders.Fourier theory and 13-moment theory 93For stationary heat conduction in a gas at rest between two concentric cylinders theBGK- version 94 of the 13-moment equations readsÈmomentum balance :Ø pδ ik t ikÊÚxk0, energy balance :q j Ø2 È qt ij balance : É i Ù5 Ê x j xi Úqi balance :qkxk0,1 t ij ,τÈØ 5 p µk T δ ik 7 µk Tt ikÊÚxk2 qi .τIn the physical cylindrical coordinates appropriate to the problem the solutioncan easily be foundp ~ homogeneous,qiª cr1 º« »«0» ,«0»¬ ¼t ijTª 45 W rc12«« 0« 0¬c2 0c 45 W r1200º»0» ,0»¼È 28 τØc1c1 r 2 Ù .ln Éτp Ê 25 pÚ5 µk92D.
Reitebuch: “Konsistent geordnete Erweiterte Thermodynamik.” [Consistently orderedextended thermodynamics] Dissertation TU Berlin (2004).93 I. Müller, T. Ruggeri: “Stationary heat conduction ...” loc. cit. (2004).94 P.L. Bhatnagar, E.P. Gross, M. Krook: “A model for collision processes in gases. I. Smallamplitude processes in charge and neutral one-component systems.” Physical Review 94(1954).The model approximates the collision term in the Boltzmann equation by W1 ( f equ f )with a constant relaxation time IJ of the order of a mean time of free flight. The BGKmodel is popular for a quick check and qualitative results.
In the present case it permits ananalytical solution, which cannot be obtained by a more realistic collision term.Extended Thermodynamics271Figure 8.9 shows the comparison of the temperature fields in this solution and of the NavierStokes-Fourier solution in a rarefied gas – with p = 1mbar – for a boundary value problem asindicated in the figureFig. 8.9.
Temperature field between coaxial cylindersAs expected, the difference becomes noticeable where the temperature gradientis big. Note that the Fourier solution becomes singular for r ĺ 0, but the Gradsolution remains finite.Insert 8.2Kinetic and thermodynamic temperatures 95,96We recall Insert 4.5 where the non-convective entropy flux ĭi was calculated. Itwas unequal to qi T . In fact it was given by)iqi 2 t ij q j,T 5 pTso that T is not continuous at a diathermic, non-entropy-producing – i.e.thermometric – wall, where the normal components of the heat flux and the entropyflux are continuous.In the case of heat conduction – treated in Insert 8.2 – there are only radialcomponents of ĭ and q and we haveΦ 11È2 t 11 Øq 1.1ÉTÊ5 p ÙÚ1Ĭ9596I.
Müller, T. Ruggeri: “Stationary heat conduction ...” loc. cit (2004).I. Müller, P. Strehlow: “Kinetic temperature and thermodynamic temperature.” In: DeanC. Ripple (ed.) “Temperature: Its Measurement and Control in Science and Industry.”Vol. 7 American Institute of Physics (2003).2728 Thermodynamics of Irreversible ProcessesThus Ĭ is the thermodynamic temperature, the temperature shown by a contactthermometer. Ĭ is not equal to T , the kinetic temperature, except in equilibrium, ofcourse.
Figure 8.10 shows the ratio of the two temperatures in a rarefied in thesituation investigated in Insert. 8.2 for the Grad 13-moment theory.Fig. 8.10. The ratio of thermodynamic to kinetic temperatureInsert 8.39 FluctuationsFluctuations are random and therefore unpredictable, except in the mean, oron average. They are due to the irregular thermal motion of the atoms. Aninstructive example – and the first one to be described analytically – is theBrownian motion of nearly macroscopic particles suspended in a solution.The velocity of such a particle fluctuates around zero in an apparently irregular manner.
Some regularity reveals itself, however, in the mean regression of the velocity fluctuations. In fact, in some approximation themean regression is akin to the non-fluctuating velocity of a macroscopicball thrown into the solution.That observation has been extrapolated to arbitrary fluctuating quantitiesby Lars Onsager. Applied to the fluctuating density field in a gas, or aliquid, Onsager’s mean-regression hypothesis furnishes the basis for theexploitation of light scattering experiments: The light scattered by a gascarries information about the transport coefficients of the gas, like thethermal conductivity and the viscosity, although the gas is macroscopicallyin equilibrium.In a rarefied gas, where extended thermodynamics is appropriate, theOnsager hypothesis – if accepted – permits the prediction of the shape ofthe scattering spectrum.
Experiments confirm that prediction.Brownian MotionBrownian motion is observed in suspensions of tiny particles which followirregular, erratic paths visible under the microscope. The phenomenon wasreported by Robert Brown (1773–1858) in 1828.1 He was not the firstperson to observe this, but he was first to recognize that he was not seeingsome kind of self-animated biological movement. He proved the point byobserving suspensions of organic and inorganic particles.
Among the lattercategory there were ground-up fragments of the Sphinx, surely a deadsubstance, if ever there was one. All samples showed the same behaviour1R. Brown: “A brief account of microscopic observations made in the months of June, Julyand August 1827 on the particles contained in the pollen of plants; and on the generalexistence of active molecules in organic and inorganic bodies.” Edinburgh NewPhilosophical Journal 5 (1828) p. 358.2749 Fluctuationsand no convincing explanation or description could be given for nearly 80years. According to Brush the phenomenon was mentioned in books on themicroscope which gave warnings about Brownian motion, lest observersshould mistake it for a manifestation of life and attempt to build fantastictheories on it.2After the kinetic theory of gases was proposed and slowly accepted, theimpression grew that the phenomenon provides a beautiful and directexperimental demonstration of the fundamental principles of themechanical theory of heat.3 That interpretation was supported by theobservation that at higher temperatures the motion becomes more rapid.However, none of the protagonists of the field of kinetic theory addressedthe problem, neither Clausius, nor Maxwell, nor Boltzmann.
It may be thatthey did not wish to become involved in liquids.A great difficulty was that the Brownian particles were about 108 timesmore massive than the molecules of the solvent so that it seemedinconceivable that they could be made to move appreciably by impactingmolecules.It was Poincaré – the mathematician who enriched the early history ofthermodynamics on several occasions with his perspicacious remarks – whoidentified the mechanism of Brownian motion when he said:4Bodies too large, those, for example, which are a tenth of a millimetre, arehit from all sides by moving atoms, but they do not budge, because theseshocks are very numerous and the law of chance makes them compensateeach other; but the smaller particles receive too few shocks for thiscompensation to take place with certainty and are incessantly knockedabout.Also Poincaré noted that the existence of Brownian motion was incontradiction to the second law of thermodynamics when he said:… but we see under our eyes now motion transformed into heat byfriction, now heat changed inversely into motion, and [all] that withoutloss, since the movement lasts forever.
This is the contrary of theprinciple of Carnot.5And indeed, the existence of Brownian motion demonstrates that thesecond law is a law of probabilities. It cannot be expected to be valid whenfew particles or few collisions are involved. If that is the case, there will besizable fluctuations around equilibrium.2345S.G.
Brush: “The kind of motion we call heat.” loc.cit. p. 661.G. Cantoni: Reale Istituto Lombardo di Scienze e Lettere. (Milano) Rendiconti (2) 1,(1868) p. 56.J.H. Poincaré: In: “Congress of Arts and Science. Universal Exhibition Saint Louis 1904.”Houghton, Miffin & Co. Boston and New York (1905).Ibidem.Brownian Motion as a Stochastic Process275Brownian Motion as a Stochastic ProcessAnd so we come to the third one of Einstein’s seminal papers of the annusmirabilis: “On the movement of small particles suspended in a stationaryliquid demanded by the molecular-kinetic theory of heat.”6 After Poincaré’sremarks the physical explanation of the Brownian motion was known, butwhat remained to be done was the mathematical description.Actually Einstein claimed to have provided both: The physicalexplanation and the mathematical formulation.
As a matter of fact, he evenclaimed to have foreseen the phenomenon on general grounds, withoutknowing of Brownian motion at all. Brush is sceptical. Says he:7… there is some doubt about the accuracy of these [claims]and he reminds the reader of Einstein’s own pronouncement quoted before,cf. Chap. 7:Every reminiscence is coloured by today’s being what it is, and thereforeby a deceptive point of view.8People do have a way of treading lightly around Einstein’s claims ofpriority, because there is a certain amount of hero-worship. The fact is,however, that in later life Einstein sometimes overreached himself; so whenhe claims to have developed statistical mechanics because he had no knowledge of Boltzmann and Gibbs’s work in 1905.9 In fact, however, he hadquoted Boltzmann’s book in an earlier paper published in 1902.10Be that as it may.
The fact remains that Einstein opened a new chapter ofthermodynamics when he treated Brownian motion.Obviously, after the insight provided by Poincaré, the Brownian motionhad to be considered as stochastic, i.e. random, or determined by chanceand probabilities. As far as I can tell, it was Einstein who invented a method6A. Einstein: “Die von der molekularkinetischen Theorie der Wärme geforderte Bewegungvon in ruhenden Flüssigkeiten suspendierten Teilchen.” Annalen der Physik (4) 17 (1905)pp.
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