Muller A History of Thermodynamics The Doctrine of Energy and Entropy phần 10 pot

Table 10.1 Equilibrium distribution function in a gas at rest, i.e. with U A =(c,0,0,0) for a degenerate relativistic gas and limit values for weak and strong degeneration and for non-relativistic and ultra-relativistic case Non-relativistic 1 2 !! c kT c Relativistic Ultra-relativistic 1 2  c kT c non-degenerate lna<<1 )exp()exp( 2 22 kT p kT c a c c  Maxwell distribution )1exp( 2 22 )( c p kT c a c c  )exp( kT cp a  degenerate 1)exp()exp( 1 2 1 22 B kT p kT c a c c 1)1exp( 1 2 22 1 B c p kT c a c c  Maxwell-Jüttner distribution 1)exp( 1 1 B kT cp a 1ln 2 !! c  kT c a else0 )(ln20for1 2 kT c akT c d else0 1 ln 0for1 2 2  » ¼ º « ¬ ª dd c c kT c c p a else0 ln0for1 ap c kT dd 0ln 2 d c  kT c a 0 1)exp( 1 2 2 z  c p kT p 0 1)]1)1(exp[ 1 )( 2 22 z  c c p c p kT c 0 1exp 1 z  p kT cp Planck distribution for p = hȞ/c µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ strongly degenerate Bose strongly degenerate Fermi The only remaining source of energy for a white dwarf is gravitational contraction, Helmholtz fashion. That keeps the star hot in the centre, perhaps hot enough – a thousand times as hot as the sun – that it must be considered a relativistic gas. Note that the small electronic mass helps in this respect, because the relativistic coldness kT c 2 c is more than 10 3 times smaller for electrons than for nuclei, or atoms at the same temperature. Now, large speeds make for small de Broglie wave lengths, so that quantum effects should be small. However, the large gravitational pressure compresses the star to such a degree that even the small de Broglie wave lengths interfere and thus produce quantum degeneration. Therefore in a white dwarf the electron gas can perhaps be both: a relativistic gas and a quantum gas. Chandrasekhar adopted this assumption as the basis for his theory of white dwarfs. In this way he provided an application for Jüttner’s formulae. Thermal equation of state inside a white dwarf In relativistic thermodynamics the conservation of mass is replaced by the conservation of the number of particles, and momentum and energy conservation are combined in a vector equation. We have where0and0 , B , AB T A , A N N A is the particle flux vector and T AB is the energy-momentum tensor. The equilibrium quantities n, e, and p are related to N A and T AB as shown in the following table. number density energy density A A c NUn 2 1 AB BA c TUUe 2 1 2 11 3 () A B AB AB c pUUgT  In a gas in equilibrium N A and T AB are moments of Jüttner’s equilibrium distribution 1exp B ¸ ¸ ¹ · ¨ ¨ © §  kT A p A U a 1 Y F so that we have 13 13 dd d dd d and . oo 22 pp p pp p AA AB AB NpF TcppF pp ÔÔ White Dwarfs 295 µ pressure 296 10 Relativistic Thermodynamics 2 2 2 1with 3 d 2 d 1 d cµ p cµ o p o p ppp c  c is the scalar element of momentum space, and 1/Y – or h 3 – determines the cell of the phase space. For a strongly degenerate Fermi gas we thus have, cf. Table 10.1 z x o x z z Ycµʌc 3 1 pzzYcµʌn o d 2 1 4 4 )(4andd 23 )(4 ³³  c c , where 1 2 )ln(  akTx . It follows that p depends only on n, not on T ! An explicit form of the relation – the thermal equation of state – can be obtained, if the integrals are evaluated, so that x can be eliminated. If relativistic effects were ignored, the square root in the integrand for p would be absent. Insert. 10.1 Subramanyan Chandrasekhar (1910–1995) Chandrasekhar was an astrophysicist with a particular interest in white dwarfs. As Eddington did for normal stars, he argued that inside a white dwarf the atoms are broken down into nuclei and electrons, so that there is a lot of space for the particles to move in freely, even when the densities are as big as described above: If the total mass of the star is big enough, however, the free space between the particles can be squeezed out, as it were. The electrons are then pushed together and the resulting compact cluster of electrons resists the gravitational pull. That equilibrium can persist even when the white dwarf cools and becomes a red dwarf and eventually, a black one. But not all stars can follow that course as we shall now see. strongly degenerate relativistic Fermi gas. 11 In that case it was fairly easy to consider the limit of the ultimate white dwarf characterized by an infinite mass density at the centre and zero radius. Surely no other star could be denser and, presumably, have more mass. That ultimate white dwarf came 11 S. Chandrasekhar: “The maximum mass of ideal white dwarfs.” Astrophysical Journal 74 (1931) p. 81. S. Chandrasekhar: “The highly collapsed configurations of a stellar mass, I and II.” Monthly Notices of the Royal Astronomical Society 91 (1931) p. 456 and 95 (1935) p. 207. See also: S. Chandrasekhar: “An Introduction to the Study of Stellar Structure” University of Chicago Press (1939). This book is available in a Dover edition, first published in 1957. In part of his work Chandrasekhar assumed that the electron gas is a Subramanyan Chandrasekhar (1910–1995) 297 out to have a mass of approximately 1.4 solar masses, cf. Insert 10.2. This limiting mass for white dwarfs became known as the Chandrasekhar limit. It was confirmed by observation in the sense that no white dwarf was ever seen that has more than Chandrasekhar’s limit mass. The Chandrasekhar limit Since the mean value of the relative molecular mass is 2, by Insert 10.1 the mass density and the pressure are given by Y.cµc ʌ Bz x 0 z1 z Bp Ycµ ʌ o µAAxȡ 4 )( 3 4 withd and 3 )( 3 4 2with 3 2 4 c ³  c Therefore the momentum balance reads, see Chap. 7 rr r o rȡʌ r M r r M ȡG r p cc ³ c  d 2 )(4where 2 d d . Differentiation with respect to r and the use of the thermal equation of state, cf. Insert 10.1, provides 3/2 2 2/3 2 2/3 2 2 2 1( ) 1 d1() d 1 1 ȡ/A d ȡ/A 4ʌGA r rr dr B /L ÈØ ÉÙ  ÉÙ ÊÚ   ÈØ ÉÙ ÊÚ  . Non-dimensionalization with the unknown central value ȡ c of ȡ provides   3/2 2/3 )(1 1 2 2 )(1 2/3 )(1 2/3 )(1 2/3 )(1 d d d d 2 1 2/3 2 ¸ ¸ ¸ ¸ ¸ ¹ · ¨ ¨ ¨ ¨ ¨ © § ¸ ¸ ¸ ¸ ¸ ¹ · ¨ ¨ ¨ ¨ ¨ © §        /A c ȡ ĭ /A c ȡ ȡ/A Șĭ /A c ȡ ȡ/A Ș Ș Ș Ș  , where 2/3 (1 ( ) ) r L Șȡ/A c  is the dimensionless radius. We investigate the case that ȡ c is infinite. Presumably that assumption characterizes the ultimate white dwarf in the sense that no other one could be denser and have more mass. In that case it is easy to solve – numerically – the differential equation for the central values ĭ(0) = 1 and ĭ (0) = 0 and one obtains the graph shown in Fig. 10.1. On the surface of the star, at r = R, we must have ȡ = 0, hence ĭ = 0. According to the figure, that value occurs for Ș = 6.9, so that R is zero, but the mass is not. It can be calculated as follows: Ș c 298 10 Relativistic Thermodynamics Insert 10.2 Fig. 10.1. A kind of density distribution in the ultimate white dwarf The last step makes use of the differential equation in the form . d 2/3 )(1d 2 d d 2 1 2 ¸ ¸ ¹ · ¨ ¨ © § U  r /A r rr ALȡ Obviously, degeneration of the electron gas has played a decisive role in the forgoing analysis. It is less clear that the relativistic square root in the equation for p is essential for the result. However, it is! Without that relativistic contribution there is no mass limit. The usual interpretation of the Chandrasekhar limit is that the electron gas cannot withstand the gravitational pull of bigger masses than 1.4 M Ɓ . It is assumed that under great pressure the electrons are pushed into the protons of the iron nuclei to form neutrons. The star thus becomes a neutron star, with a truly enormous mass density: 10 15 times the already large density of a white dwarf. Neutron stars have their own mass limit – 3.2 M – according to a theory presented by J. Robert Oppenheimer (1904–1967) in 1939. If a star is bigger than that, – and does not get rid of the excess mass by nova- or supernova-explosions – it collapses into a black hole, at least according to current wisdom. There seems to be no conceivable mechanism to stop the collapse. It is tempting to pursue the matter further in this book. However, there is a touch of science fiction in the subject and I desist, – with regret. Chandrasekhar has left his mark in several fields of physics. In his autobiography he says that he was … motivated, principally, by a quest after perspectives…compatible with my taste, abilities and temperament. Stellar dynamics was the subject of only the first such quest. Others followed: x Brownian motion, x radiative transfer, x hydrodynamic stability, x relativistic astrophysics, x mathematical theory of black holes. Whenever Ɓ Maximum Characteristic Speed 299 The maximal mass of a white dwarf is not alone in having been named after Chandrasekhar. There is also the NASA X-ray observatory which is called Chandrasekhar observatory, and a minor planet,– one of about 15000 – which was named Chandra in 1958. Fig. 10.2. Subrahmanyan Chandrasekhar Maximum Characteristic Speed After Jüttner there was a period of stagnation in the development of rela- tivistic thermodynamics. To be sure, there was some interest, and in 1957 John Lighton Synge (1897–1995) streamlined Jüttner’s results in a neat small book 12 which, however, did not significantly add to previous results. Also Eckart provided a relativistic version of thermodynamics of irrever- sible processes, 13 in which he improved Fourier’s law of heat conduction by accounting for the inertia of energy, cf. Chap. 8. However, his differential equation for temperature was still parabolic so that the paradox of heat conduction persisted. Understandably that paradox has irritated relativists more than it did non-relativistic physicists. After all, if no atom, or molecule can move faster than the speed of light, heat conduction should 12 J.L Synge: “The Relativistic Gas.” North Holland, Amsterdam (1957). 13 C. Eckart: “The thermodynamic of irreversible processes III: Relativistic theory of the simple fluid.” loc. cit. he found that he understood the subject, he published one of his highly readable books, – in his words: a coherent account with order, form, and structure. Thus he has left behind an admirable library of monographs for students and teachers alike. His work on white dwarfs, but also his lifelong physics in 1983, fifty years after he discovered the Chandrasekhar limit. exemplary dedication to science, was rewarded with the Nobel prize in 300 10 Relativistic Thermodynamics not be infinitely fast. This problem was the original motive for Müller to develop extended thermodynamics, cf. Chap. 8, and its relativistic version. 14 Shortly afterwards, Israel 15 published a very similar theory and, eventually, it was shown by Boillat and Ruggeri 16 that extended thermodynamics of infinitely many moments predicts the speed of light for heat conduction. Thus the paradox was resolved; the field is conclusively explained by Müller in a recent review article. 17 14 I. Müller: “Zur Ausbreitungsgeschwindigkeit ” Dissertation (1966) loc. cit. A streamlined version of relativistic extended thermodynamics may be found in: I-Shih Liu, I. Müller, T. Ruggeri: “Relativistic thermodynamics of gases.” Annals of Physics 169 (1986). 15 W. Israel: “Nonstationary irreversible thermodynamics: A causal relativistic theory.” Annals of Physics 100 (1976). 16 G. Boillat, T. Ruggeri: “Moment equations in the kinetic theory of gases and wave velocities.” (1997) loc.cit. 17 I. Müller: “Speeds of propagation in classical and relativistic extended thermodynamics.” http:/www.livingreviews.org/Articles/Volume2/1999-1mueller. 18 N.A. Chernikov: “The relativistic gas in the gravitational field.” Acta Physica Polonica 23 (1964). N.A. Chernikov: “Equilibrium distribution of the relativistic gas.” Acto Physica Polonica 26 (1964). N.A. Chernikov: “Microscopic foundation of relativistic hydrodynamics.” Acta Physica Polonica 27 (1964). 19 H. Minkowski: “Raum und Zeit.” [Space and time] Address delivered at the 80th Assembly of German Natural Scientists and Physicists, at Cologne. September 21st, 1908. The address has been translated into English and is reprinted in “The Principle of Relativity. A collection of original memoirs on the special and general theory of relativity.” Dover Publications pp. 75–91 A decisive step forward in the general theory was done by N.A. Chernikov in 1964 18 when he formulated a relativistic Boltzmann equation. Let us consider this now. Boltzmann-Chernikov Equation I have already mentioned the elegant four-dimensional formulation which is now standard in relativity. It was introduced by Hermann Minkowski (1864–1909). Minkowski had taught Einstein in Zürich and later he became the most eager student of Einstein’s paper on special relativity. He sugge- sted that the theory of relativity makes it possible to take time into account as a kind of fourth dimension and he introduced the distance ds between two events at different places and different times 19 23222122 )()()( ddddddd xxxtcxxgs BA AB 2 c  c  c  c ccc . in a Lorentz frame with coordinates ct´,x´ a Boltzmann-Chernikov Equation 301 . CD B D A C AB g x x x x g c w c w w c w In particular, for a rotating frame – on a carousel (say) – with coordinates (ct,r,ș,z) given by tƍ = t, xƍ 1 = r cos(ș + Ȧt), xƍ 2 = r sin(ș + Ȧt), xƍ 3 = z the metric tensor reads ¸ ¸ ¸ ¸ ¸ ¸ ¹ · ¨ ¨ ¨ ¨ ¨ ¨ © §     1000 0 2 0 0010 00 2 22 1 r c Ȧr c Ȧr c rȦ AB g . The metric tensor has some significance, because it allows us to write the equation of motion of a free particle, whose orbit is parametrized by IJ, in the form . 2 where 1 d d d d 2 d 2 d ¸ ¸ ¹ · ¨ ¨ © § w w  w w  w w  D x AC g A x DC g C x DA g BD g B AC ī, IJ B x IJ A x B AC ī IJ B x Indeed, in a Lorentz frame, with B A C ī = 0, the solution of this equation is a motion in a straight line with constant velocity, which is the defining feature of an inertial frame. The parameter IJ is usually chosen as the proper time of the moving particle, i.e. the time read off from a clock in the momentarily co-moving Lorentz frame. With that, the equation of motion may be written in the form IJ x p,ppī µIJ p A ABAB AC B d d where 1 d d c  is the four-momentum of the particle as before. In this manner the tensor gƍ AB , whose invariance defines the Lorentz frames, may be interpreted as a metric tensor of space-time. Its components in a arbitrary frame x A = x A (xƍ B ) can be calculated from The equation of motion represents the equation of a geodesic in space-time. This is a nice feature, much beloved by theoretical physicists, because it supports their predilection for a specious geometrical interpretation of the theory of relativity. The notion was useful for Einstein, when he developed the theory of general relativity; but most often it is used to confuse laymen with talk about curved space, etc. 302 10 Relativistic Thermodynamics The relativistic – non-quantum – formulation of the Boltzmann equation was derived in a series of three remarkable papers by N.A. Chernikov. It is an integro-differential equation for the relativistic distribution function F(x A ,p a ) which reads Q.epq)h)F(qF(p)q)F(p(F( p F ppī x F p CCCC d BA d AB A A dd! cc w w  w w ³ Comparison with the classical Boltzmann equation, cf. Chap. 4, easily identifies the individual terms. I do not go into that, other than saying that x the term with ī represents the acceleration of a particle between two collisions, 20 and x the collision term on the right hand side vanishes for the Maxwell- Jüttner distribution because of conservation of the energy and mo- mentum vector p A in the collision. Chernikov uses the equation for the formulation of equations of transfer for moments of the distribution function and he concentrates on 13 moments, which is rather artificial for a relativistic theory; it is more appropriate to include the dynamic pressure and thus come up with a theory of 14 moments. 21 But we shall not pursue this question here, because so far – apart from the finite characteristic speeds – the multi-moment theory Seeing that the collision term vanishes for the Maxwell-Jüttner distribution, we must ask whether the Boltzmann-Chernikov equation is satisfied by that distribution, or what conditions on the fields a(x B ), T(x B ), and U A (x B ) are required by the equation. Insertion of the distribution leads to the requirements 0and ¸ ¹ · ¨ © §  ¸ ¹ · ¨ © § w w ;B A A; B A kT U kT U 0 x a , where the semi-colon denotes covariant derivatives. 20 The possibility of such a term was ignored in Chap. 4, because I wished to be brief. The term is only present in a non-inertial frame. 21 See: I. Müller, T. Ruggeri: “Rational Extended Thermodynamics.” loc. cit. Of course, nobody will try to solve the equation of the geodesic in its general form in order to calculate the orbit of a free particle. It is so much easier to solve it in a Lorentz frame and transform the straight line obtained there to an arbitrary frame. has not provided any suggestive results that go beyond Eckart’s reform- ulation of the Fourier law, see Chap. 8. Let us concentrate on equilibrium instead: Ott-Planck Imbroglio 303 Since a is a function of n and T, it follows that a temperature gradient must exist in equilibrium, if there is a density gradient. That conclusion may be made more concrete by exploiting the second condition for the special case of a gas at rest on a carousel. We obtain This result is eminently plausible, because it reflects the inertia of the thermal energy in the field of the centrifugal potential Ȧ 2 r 2 . Indeed, if energy has mass – and weight – it should be subject to sedimentation, as it were, by centrifugation. Einstein has postulated – in his general theory of relativity – that inertial forces and gravitational forces are equivalent. Accordingly non-homo- geneous temperature fields are also created by gravitational fields – not only by centrifugal fields – because they lead to stratification of mass density. I have already commented on that aspect in the context of Eckart’s relativistic paper. In view of the following argument, I should like to stress that the last relation does not imply a transformation formula for the temperature. It represents a property of the scalar temperature field as a solution of the energy balance equation in a centrifugal force field. Ott-Planck Imbroglio In 1907 the theory of relativity was new. A fundamental change had occurred in mechanics, and physics in the immediate aftermath was in a state of flux. The extension of the new concepts to thermodynamics was clearly desirable. Everything seemed possible and so Planck 22 came up with the idea to modify the Gibbs equation. Einstein 23 elaborated on that idea and introduced a working term –qdG into the heating of a body moving with the 22 M. Planck: “Zur Dynamik bewegter Systeme.” [On the dynamics of moving systems] Sitzungsberichte der königlichen preußischen Akademie der Wissenschaften (1907). 23 A Einstein: “Über das Relativitätsprinzip und die aus demselben gezogenen Folgerungen.” [On the principle of relativity and the conclusions drawn from it] Jahrbuch der Radioaktivität und Elektronik 4 (1907) pp. 411–462. Reprinted in: “Albert Einstein, die grundlegenden Arbeiten.” [Albert Einstein, the basic works] K.v. Meyenn (ed) Vieweg Verlag (1990). In the reprinting the modified Gibbs equation is misprinted: It says dQ instead of dG. Printed version: Annalen der Physik 26 (1908) p. 1. 22 2 homogeneous or, see above homogeneous. () : 00 TTr ~~ g Ȧ r 1 c  [...]... information about where and how to arrange seals and boreholes for lubrication, and how to operate the valves and where to install them What thermodynamics can do about engines is to give an account of the balance of in- and effluxes of mass, momentum, energy and entropy, and that is essentially what it can also do about life For the engine that task has been done satisfactorily; for animals and plants... (say) – than dictated by the stoichiometric formula The plant absorbs all that water in the roots, passes it upwards to the leaves and evaporates it there Thus a plant cools its leaves in the same manner as animals cool their skins: By evaporation of water.24 It is easy to calculate the value of x when we require that the temperature stay at 298K We obtain x 500 so that, for each gram of water that helps... breaking up of sugar, but also in the catabolism of fatty acids and of amino acids Fatty acids and amino acids are first broken down to acetic acid which can then enter the Krebs cycle just as the acetic acid originating from lactic acid does The catabolism of fatty acids is particularly productive of new ATP’s, which we shall now proceed to discuss Anabolism Obviously the energy – or enthalpy – of reactions... in the tissue does not all appear as heat, as it does in a flame Indeed, an animal and man are able to exert power, and they must do so, at least to the extent of the basal metabolic rate Also animals grow, and they are able – in their bodies – to produce fat even if they ingest primarily carbohydrates So they are building up complex molecules from the simpler ones that have entered their tissue The. .. cf Chap 4 – warned against an over-interpretation of entropy as a measure of disorder and I stress that caution again To be sure, an animal definitely seems more ordered than the sum of its atoms, loosely distributed, and it does probably have a lower entropy But then, what is the entropy of an animal? Or let us ask the easier question: What is the entropy of a molecule like hemoglobin, one of the simpler... this was understood, the distinction between organic and inorganic chemistry began to lose its original meaning Organic chemistry became the branch that deals with carbon compounds The chemical changes that take place in animals and humans are called metabolism; from Greek: to rearrange The metabolic rate may be measured in Watt – just like the power of a heat engine The maximal metabolic rate that a person... the stomach Thus Beaumont was able to study the changes which the food undergoes in the stomach, and he did so with so much enthusiasm that the patient eventually ran away from him That was a wise decision on the part of the patient, because away from his doctor he lived to the old age of 82 years ,10 always with the fistula Later, and in a different part of the world, the physiologist Claude Bernard (1813–1878)... hand breathe oxygen and use it to break down plant tissue In the process they release carbon dioxide and water The plants perform their task only in the light Jan Baptista van Helmont (1577–1644) was an alchemist on the verge of becoming a chemist or, perhaps, a biochemist On the one hand he claimed to have seen and used the philosopher’s stone – the hypothetical ultimate tool of alchemy – but on the. .. relation for a body at rest: The heating consists of the non-convective part of the energy flux and the internal energy is the non-convective part of the energy The power, or working of the force dG has no place in the Gibbs equation therefore, or it should not have Also, the heating of a body in the Gibbs equation is the integral of the heat flux over the surface And relativistically the heat flux forms... some way this is the worst possible case for a chemical reaction: The energy – or enthalpy – increases and the entropy decreases Since the process occurs at constant pressure pR = 1atm and at the normal temperature, roughly TR = 298K, the first law requires that we provide heat and the second law demands that we withdraw heat Indeed we have 0 by the first law, and 0 by the second law This is a clear contradiction . 299 The maximal mass of a white dwarf is not alone in having been named after Chandrasekhar. There is also the NASA X-ray observatory which is called Chandrasekhar observatory, and a minor. install them. What thermodynamics can do about engines is to give an account of the balance of in- and effluxes of mass, momentum, energy and entropy, and that is essentially what it can also. and thus produce quantum degeneration. Therefore in a white dwarf the electron gas can perhaps be both: a relativistic gas and a quantum gas. Chandrasekhar adopted this assumption as the basis