Extended Thermodynamics 259 V is a characteristic speed and l Į and d Į are the left and right eigenvalues of the matrix 1 F u α β   in the one-dimensional field equations ), 2,1( 1 1 nȆ x F t u D w w  w w D DD . The solution of the Bernoulli equation reads )1()0(1 )0( )(   DV D C DV G# G# V# so that A(t) remains finite unless the initial amplitude A(0) is large. In general – for arbitrary solutions instead of merely acceleration waves – the condition for smooth solutions is not decisively known. There exists a sufficient condition for smoothness 65 which, however, is not necessary. Characteristic Speeds in Monatomic Gases We recall the generic equations of transfer in the kinetic theory of gases, cf. Chap. 4, and apply this to a polynomial in velocity components by setting N KKK EEE 21 P \ . In this manner we obtain equations of balance for moments cd 2121 fcccµu ll iiiiii ³ of the distribution function f which read ) 2,1,0( 21 2121 0N Z W V W N NN KKK C CKKKKKK w w  w w 3 . Since each index may assume the values 1,2,3, there are 1 6 equations. These equations fit into the formal framework of extended thermodynamics, see above, but they are simpler. Indeed, on the left hand side there is only one flux, namely CKKK N W 21 – the last one – which is not explicitly related to the fields N KKK W 21 (l = 1, N). 65 S. Kawashima: “Large-time behaviour of solutions to hyperbolic-parabolic systems of conservation laws and applications.” Proceedings of the Royal Society of Edinburgh A 106 (1987). n = /(N + 1)(N + 2)(N + 3) 260 8 Thermodynamics of Irreversible Processes Therefore the results of the previous sections may be carried over to the present case, in particular the exploitation of the entropy inequality. That inequality reads according to the kinetic theory of gases, cf. Chap. 4 ln d ln d 0 ee a a ff kf kcf tYx Y  ÈØÈ Ø   ÉÙÉ Ù ÊÚÊ Ú  ÔÔ cc. The exploitation makes use of the Lagrange multipliers N KKK 21 /   12 1 2 1 0 exp ll N ii i i i i k l fY µ cc cΛ  Ç so that the scalar and vector potentials may be written as     12 1 2 12 1 2 1 0 1 0 exp exp d . ll ll N ii i i i i k l N a aiiiiii k l hkY hkYc µccc Λ Λ       Ç Ô Ç Ô c Insertion into the characteristic equation for the calculation of wave speeds gives   11 det ( ) d 0 ln aa i i j j equ cn V c cc c f Ô c provided that the wave propagates into a region of equilibrium. f equ is the Maxwell distribution, cf. Chap. 4. Thus the calculation of characteristic speeds and, in particular, the maximal one, the pulse speed requires no more than simple quadratures and the solution of an nth order algebraic equation. It is true that the dimension of the determinant increases rapidly with N: For N = 10 we have 286 columns and rows, while for N = 43 we have 15180 of them. But then, the calculation of the elements of the determinant and the determination of V max may be programmed into the computer and Wolf Weiss (1956– ) has the values ready for any reasonable N at the touch of a button, see Fig. 8.6. We recognize that the pulse speed goes up with increasing N and it never (l = 1,2,…N ) and the moment character of the densities and fluxes implies that the distribution function has the form µc c c dc and Extended Thermodynamics 261 stops. 66 Indeed, Guy Boillat (1937– ) and Tommaso Ruggeri (1947– ) have provided a lower bound for V max which tends to infinity for N ĺ. 67 The fact that V max is unbounded represents something of an anticlimax for extended thermodynamics, because the theory started out originally as an effort to find a finite speed of heat conduction. Let us consider this: Fig. 8.6. Pulse speeds in relation to the normal speed of sound. Table and crosses: 68 )( 2 1 5 6  0 by Boillat and Ruggeri 69 Carlo Cattaneo (1911–1979) Fourier’s equation of heat conduction is the prototypical parabolic equation and it predicts an infinite speed of propagation of disturbances in tempe- ratures. This phenomenon became known as the paradox of heat conduction. Neither engineers nor physicists generally were much worried about the paradox. It is quantitatively unimportant in solids and liquids and even in gases under normal pressures and temperatures. And yet, the paradox represented an awkward feature of thermodynamics and in 1948 Carlo Cattaneo made an attempt to resolve it. Upon reflection it was clear to Cattaneo that Fourier’s law was to blame and he amended it. We refer to Fig. 8.7 and recall the mechanism of heat is a downward temperature gradient across a small volume element – of the dimensions of the mean free path – an atom moving upwards will, in the mean, carry more energy than an atom moving downwards. Therefore there 66 W. Weiss: “Zur Hierarchie der erweiterten Thermodynamik.” [On the hierarchy of extended thermodynamics] Dissertation TU Berlin. See also: I. Müller, T. Ruggeri: “Rational Extended Thermodynamics.” loc.cit. 67 G. Boillat, T. Ruggeri: “Moment equations in the kinetic theory of gases and wave velocities.” Continuum Mechanics and Thermodynamics 9 (1997). 68 W. Weiss: loc.cit. 69 G. Boillat, T. Ruggeri: “Moment equations …” loc.cit. Calculations by Weiss . Circles: Lower bound conduction in gases as described in the elementary kinetic theory. If there 262 8 Thermodynamics of Irreversible Processes is a net flux of energy upwards, i.e. opposite to the temperature gradient, associated with the passage of a pair of particles across the middle layer. That flux is obviously proportional to the temperature gradient, just as Fourier’s law requires for the heat flux. Fig. 8.7. Carlo Cattaneo. The Cattaneo equation Cattaneo 70 changed that argument slightly. He argued that there is a time- lag between the start of the particles at their points of departures and the time of passage through the middle layer. If the temperature changes in time, it is clear that the heat flux at a certain time depends on the tempe- rature gradient at a time IJ earlier, where IJ is of the order of magnitude of the mean time of free flight. Therefore it seems reasonable to write an non- stationary Fourier law in the form with 0 i ii TT q ț xtx ττ ÈØ    ! ÉÙ  ÊÚ . Now, this equation is badly flawed, because it predicts that for q i = 0 the temperature gradient tends exponentially toward infinity. Nor does this modified Fourier law lead to a finite speed, so that it does not resolve the paradox. Cattaneo must have known this – although he does not say so (!) – because he proceeded by converting his non-stationary Fourier law into something else in a sequence of three steps which deserve to be called mathematically creative. 70 C. Cattaneo: “Sulla conduzione del calore.” [On heat conduction] Atti del Seminario Matematico Fisico della Università di Modena, 3 (1948). Extended Thermodynamics 263 i ii TT q ț xtx τ ÈØ    À ÉÙ  ÊÚ i i t x T țq w w  W w w 1 1 1 i i T q ț tx τ  ÈØ À  ÉÙ ÊÚ  i i i x T ț t q q w w  w w W . The end result, now usually called the Cattaneo equation, is acceptable. It provides a stable state of zero heat flux for 0 w w K Z 6 and, if combined with the energy equation, it leads to a telegraph equation and predicts a finite speed of propagation of disturbances of temperature. So, however flawed Cattaneo’s reasoning may have been, he is the author of the first hyperbolic equation for heat conduction. Let us quote him how he defends the transition from the non-stationary Fourier law to the Cattaneo equation: Nel risultato ottenuto approfitteremo della piccolezza del parametro IJ per trascurare il termine che contiene a fattore il suo quadrato, conservando peraltro il termine in cui IJ compare a primo grado. Naturalmente, per delimitare la portata delle conseguenze che stiamo per trarre, converrà precisare un po’ meglio le condizioni in cui tale approssimazione è lecita. Allo scopo ammetteremo esplicitamente che il feno-meno di conduzione calorifica avvenga nell´intorno di uno stato stazionario o, in altri termini, che durante il suo svolgersi si mantengano abbastanza piccole le derivate temporali delle varie grandezze in giuoco. In the result we take advantage of the smallness of the parameter IJ so that terms with squares of IJ may be neglected. First order terms in IJ are kept, however. Of course, in order to appreciate the effect on the consequences, which we are about to derive, it would be proper to investigate the conditions when that approximation is valid. For that purpose we stress that the heat conduction should remain nearly stationary. Or, in other words, that the time derivatives of the various quantities at play remain sufficiently small, while the stationary state changes slowly. Well, if the truth were known, this is not a valid justification. How could it be, if it leads from an unstable equation to a stable one and from a parabolic to a hyperbolic equation. Let me say at this point that Cattaneo’s argument leading to the non-stationary Fourier law is the nut-shell-version of the first step in an iterative scheme that is often used in the kinetic theory of gases. In that field the objective is an improvement of the treatment of viscous, heat-conducting gases beyond what the 264 8 Thermodynamics of Irreversible Processes However, whatever the peculiarities of its derivation may have been, the Cattaneo equation on the paradox of heat conduction served as a stimulus. Müller 72 generalized Cattaneo’s treatment within the framework of TIP, taking care – at the same time – of a related paradox of shear motion. And then, after rational thermodynamics appeared, Müller and I-Shih Liu (1943– ) 73 formulated the first theory of rational extended thermo- dynamics, still restricted to 13 moments, but complete with a constitutive entropy flux – rather than the Clausius-Duhem expression – and with Lagrange multipliers. Thus the subject was prepared for being joined to the mathematical theory of hyperbolic systems. Mathematicians had studied quasi-linear first order systems for their own purposes, – without being motivated by the 74 Friedrichs and Lax, 75 and Boillat 76 discovered that such systems may be reduced to a symmetric hyperbolic form, if they are compatible with a convex extension, i.e. an additional relation of the type of the entropy inequality. Ruggeri and 71 The instabilities involved in the Chapman-Enskog iterative scheme have recently been reviewed by Henning Struchtrup (1956– ). H. Struchtrup: “Macroscopic Transport Equations for Rarefied Gases – Approximation Methods in Kinetic Theory” Springer, Heidelberg (2005). 72 I. Müller: “Zur Ausbreitungsgeschwindigkeit von Störungen in kontinuierlichen Medien.” [On the speed of propagation in continuous bodies.]. Dissertation TH Aachen (1966). See also: I. Müller: “Zum Paradox der Wärmeleitungstheorie.” [On the paradox of heat conduction]. Zeitschrift für Physik 198 (1967). 73 Archive for Rational Mechanics and Analysis 46 (1983). 74 Soviet Mathematics 2 (1961). 75 K.O. Friedrichs, P.D. Lax: “Systems of conservation equations with a convex extension.” Proceeding of the National Academy of Science USA 68 (1971). 76 Boillat: “Sur l´éxistence et la recherche d´équations de conservations supplémentaires pour les systèmes hyperbolique.” [On the existence and investigation of supplementary conservation laws for hyperbolic systems] Comptes Rendues Académie des Sciences Paris. Ser5. A 278 (1974). Navier-Stokes-Fourier theory can achieve. The iterative scheme is called the Chapman-Enskog method and its extensions are known as Burnett approximation and super Burnett. The scheme leads to inherently unstable equations and should be discarded. The reason why the fact was not recognized for decades is that the authors have all concentrated on stationary processes. 71 And the reason why it is still used is natural inertia and lack of imagination and initiative. The situation is quite similar mathematically and psychologically to the one mentioned in the context of rational thermodynamics of unstable equilibria of nth grade fluids with n > 1, see above. paradoxon of infinite wave speeds. Godunov, I-Shih Liu, I. Müller: “Extended thermodynamics of classical and degenerate gases.” S.K. Godunov: “An interesting class of quasi-linear systems.” Extended Thermodynamics 265 Strumia 77 recognized that the Lagrange multipliers – their main field – could be chosen as thermodynamic fields and, if they were, the field equations of of the theory was refined by Boillat and Ruggeri, 78 , 79 and eventually they although it is always finite for finitely many moments, see above. 80 outgrown its original motivation and had become a predictive theory for processes with large rates of change and steep gradients, as they might occur in shock waves. Let us consider this: Field Equations for Moments Once the distribution function is known in terms of the Lagrange multipliers, see above, it is possible – in principle – to change back from the Lagrange multipliers N KKK 21 / to the moments N KKK W 21 by inverting the relation   12 1 12 1 2 1 0 exp d lf ll N ii i i i ii i i i i k l uccY cccµ µ  Ç Ô c . Once this is done, we may determine the last flux  12 1 12 1 2 1 0 exp d NN ll N ii i a i i a ii i i i i k l ucccY cccµ µ  Ç Ô c ), 1.0( of termsin 21 0NW N KKK . Also in principle the productions may thus be calculated after we choose an appropriate model for the atomic interaction, e.g. the model of Maxwellian molecules, cf. Chap. 4. 77 T. Ruggeri, A. Strumia: “Main field and convex covariant density for quasi-linear hyperbolic systems. Relativistic fluid dynamics.” Annales Institut Henri Poincaré 34 A (1981). 78 T. Ruggeri: “Galilean invariance and entropy principle for systems of balance laws. The structure of extended thermodynamics.” Continuum Mechanics and Thermodynamics 1 (1989). 79 G. Boillat, T. Ruggeri: “Moment equations …” loc.cit. 80 Incidentally, in the relativistic version of extended thermodynamics the maximal pulse speed for infinitely many moments is c, the speed of light. extended thermodynamics were symmetric hyperbolic. The formal structure proved that for infinitely many moments the pulse speed tends to infinity, has its own appeal and anyway: Extended thermodynamics had by this time had originally set out to calculate finite speeds. However, the infinite limiting case As mentioned before this phenomenon is a kind of anti-climax for a theory that / / 266 8 Thermodynamics of Irreversible Processes In reality the calculations of the flux aiii N u 21 and of the productions ) 7,6( 21 0N N KKK 3 81 require somewhat precarious approximations, since integrals of the type occurring in the last equations cannot be solved analytically. However, when everything is said and done, one arrives at explicit field equations, e.g. those of Fig. 8.8, which are valid for N = 3 so that there are 20 individual equations. The equations written in the figure are linearized and the canonical notation has been introduced like ȡ for u, ȡ i for u i ,3ȡ k / µ T for the trace u ii , t <ij> for the deviatoric stress and q i for the heat flux. The moment u <ijk> has no conventional name, – other than trace- less third moment – because it does not enter equations of mass, momentum and energy. But it does have to satisfy an explicit fields equation, see figure. 81 Recall that the first five productions are zero which reflects the conservation of mass, momentum and energy. right: Navier-Stokes. Bottom left: Cattaneo. Bottom right: 13 moment Fig. 8.8. 4 times field equations of extended thermodynamics for N= 3 Top left: Euler. Top Extended Thermodynamics 267 Figure. 8.8 shows the same set of 20 equations four times so as to make it possible to point out special cases within the different frames: x On the upper left side we see the equations for the Euler fluid, which is entirely free of dissipation and thus without shear stresses and heat flux. x The upper right box contains the Navier-Stokes-Fourier equations with the stress proportional to the velocity gradient and the heat flux proportional to the temperature gradient. This set identifies the only unspecified coefficient IJ as being related to the shear viscosity Ș. We have 6 M P WUK 3 4 so that Ș grows linearly with T as is expected for Maxwellian molecules, cf. Chap. 4. x In the fifth equation of the third box I have highlighted the Cattaneo equation which has provided the stimulus for the formulation of extended thermodynamics, see above. The Cattaneo equation is essentially a Fourier equation, but it includes the rate of change of the heat flux as an additional term even though it ignores other terms. x The fourth box exhibits the 13-moment equations. These are the ones best known among all equations of extended thermodynamics, because they contain no unconventional terms, – only the 13 moments familiar from the ordinary thermodynamics, viz. ȡ, i, T, t <ij> , and q i . For interpretation we may focus on the upper right box in Fig. 8.8, the one that emphasizes the Navier-Stokes theory. In this way we see that some specific terms are left out of that theory, namely M K M KM K KL Z S Z V V S V V w w w w w w w w andandand . For rapid rates and steep gradients we may suspect that these terms do count and, indeed, they do, and we must go to the full set of 20 equations, or to equations with even more moments. Since rapid rates and steep gradients are measured in terms of mean times of free flight and mean free paths, we may suspect that extended thermodynamics becomes necessary for rarefied gases. Shock Waves Properly speaking shock waves do not exist, at least not as discontinuities in density, velocity, temperature, etc. What seems like shock waves turns out to be shock structures upon close experimental inspection, i.e. smooth but steep solutions of the field equations, which assume different equilibrium values at the two sides. Scientists and engineers are interested to calculate 268 8 Thermodynamics of Irreversible Processes the exact form of the shock structures; and they have realized that the Navier-Stokes-Fourier theory fails to predict the observed thickness. 82 Since this is a case of steep gradients or rapid rates, it is appropriate, perhaps, to apply extended thermodynamics. To be sure we cannot use the formulae of Fig. 8.8, because these are linearized. Their proper non-linear form is too complicated to be written here. Let it suffice therefore to say that, yes, extended thermodynamics does provide improved shock structures. But the work is hard, because even for rather weak shock – which move with a Mach number of 1.8 – the required number of moments goes into the hundreds as Wolf Weiss 83 and Jörg Au have shown. 84 An interesting feature of that research – first noticed, but apparently not understood by Grad 85 – is the observation that, when the Mach number reaches the pulse speed and exceeds it, a sharp shock occurs within the shock structure. Obviously those Mach numbers are truly supersonic and not just bigger than 1. That is to say that the upstream region has no way of being warned about the onrushing wave, if that wave comes along faster than the pulse speed. For the mathematician this is a clear sign that he has over-extrapolated the theory: He should take more moments into account and, if he does, the sharp shocks disappear, or rather they are pushed to a higher Mach number appropriate to the bigger pulse speed of the more extended theory. Boundary Conditions Extended thermodynamics up to 1998 is summarized by Müller and Ruggeri. 86 Since the publication of that book boundary value problems have been at the focus of the research in the field, and some problems of the 13- moment theory have been solved: x It has been shown for thermal non-equilibrium between two co-axial cylinders that the temperature measured by a contact thermometer is not 82 This was decisively shown by D. Gilbarg, D. Paolucci: “The structure of shock waves in the continuum theory of fluids.” Journal for Rational Mechanics and Analysis 2 (1953). 83 W. Weiss: “Die Berechnung von kontinuierlichen Stoßstrukturen in der kinetischen Gastheorie.” [Calculation of continuous shock structures in the kinetic theory of gases] Habilitation thesis TU Berlin (1997). See also: W. Weiss: Chapter 12 in: I. Müller, T. Ruggeri: “Rational Extended Thermodynamics” loc.cit. W. Weiss: “Continuous shock structure in extended Thermodynamics.” Physical Review E, Part A 52 (1995). 84 Au: “Lösung nichtlinearer Probleme in der Erweiterten Thermodynamik.” [Solution of non-linear problems in extended thermodynamics’’]. Dissertation TU Berlin, Shaker Verlag (2001). 85 H. Grad: “The profile of a steady plane shock wave.” Communications of Pure and Applied Mathematics 5 Wiley, New York (1952). 86 I. Müller, T. Ruggeri: “Rational Extended Thermodynamics.” loc.cit. [...]... molecular-kinetic theory of heat.”6 After Poincaré’s remarks the physical explanation of the Brownian motion was known, but what remained to be done was the mathematical description Actually Einstein claimed to have provided both: The physical explanation and the mathematical formulation As a matter of fact, he even claimed to have foreseen the phenomenon on general grounds, without knowing of Brownian motion at all... in other cases of fluctuating quantities; indeed, the decay may be a damped oscillation on other occasions On the other hand, when the particle has a small mass, the fluctuating force makes its velocity fluctuate as well about an average velocity zero as illustrated in the upper part of Fig 9. 1 The graph of this velocity fluctuation seems totally irregular, and certainly in no way related to the macroscopic... and has obtained the 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 experimental points well Nor can we adjust parameters to obtain a better fit, because there are no adjustable parameters in the theories of extended thermodynamics Or rather, one might say that the only parameter is the number of moments and moment equations... regressions are equal in their functional behaviour to the macroscopic law of decay, – according to the Onsager hypothesis – this is also true for their mean value, i.e the auto-correlation function The auto-correlation function is often easier to calculate and to measure than the mean regression of a particular size of fluctuation Therefore the Onsager hypothesis is most often pronounced by saying that the autocorrelation... a normally dense gas and in a moderately rarefied gas Dots: Measurements by Clarke for rarefied gas. 19 Lines: Calculation from a Navier-StokesFourier theory If the Onsager hypothesis is accepted, S( ) can also be calculated from the field equations of the gas, e.g the Navier-Stokes equations For dense gases the measured and calculated curves fit perfectly, and thus they support the hypothesis For the. .. We have already quoted the popular textbook by de Groot and Mazur,16 who give faint praise to Onsager by calling his hypothesis not altogether unreasonable.17 Light Scattering While Brownian particles and their erratic motion can be seen, albeit only under the microscope, fluctuations of mass density, and velocity and temperature in air cannot be seen And yet they are there, and they affect the transmission... pA A kT , of the gas a , 2 c2 1 , 2 c2 and T is its temperature, a scalar quantity with respect to Lorentz transformations a is a Lagrange multiplier and it must be calculated as a function of N and T from the constraint on N That calculation is best done in the rest frame of the gas, where UA = (c,0,0,0) holds In the general case the summation – or integration – leads to Hankel functions which makes... relation, cf Chap 7, Sirius A was twice as massive as the sun and, in order to be forced into the observed orbit by the companion, the companion had to have about the same mass as the sun This meant that the average density had to be a fantastic 140000 times that of the sun, or 200000 times that of water One cm3 has a mass of 200kg! Any scepticism about such numbers was quickly silenced when – at the. .. conclude that a gas itself adjusts the uncontrollable boundary values and the question is which criterion the gas employs It has been suggested 89 that the boundary values adjust themselves so as to minimize the entropy production in some norm Another suggestion is that the uncontrollable boundary values fluctuate with the thermal motion and that the gas reacts to their mean values .90 In all honesty,... theoretical chemistry at Yale University, where he taught Statistical Mechanics I and II to chemistry students Among the students his course was known as Norwegian I and II.18 Fig 9. 2 Lars Onsager receiving the Nobel prize for chemistry in 196 8 compressions and expansions of air – and gases generally – occur as a result of the random motion of molecules and atoms and they affect the dielectric constant, . framework of TIP, taking care – at the same time – of a related paradox of shear motion. And then, after rational thermodynamics appeared, Müller and I-Shih Liu ( 194 3– ) 73 formulated the. The Cattaneo equation Cattaneo 70 changed that argument slightly. He argued that there is a time- lag between the start of the particles at their points of departures and the time of passage. lack of imagination and initiative. The situation is quite similar mathematically and psychologically to the one mentioned in the context of rational thermodynamics of unstable equilibria of