Müller I. A history of thermodynamics. The doctrine of energy and entropy (1185104), страница 46
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The idea is appealing, although, of course, the reflection – ifthat is what it is – must be distorted, since helium is not a gas when thedecomposition occurs at 2.19K. The whole argument about degeneracyignores the van der Waals forces which enforce liquefaction of helium atthe comparatively high temperature of 4.2K.Erwin Schrödinger (1887–1961), the pioneer of quantum mechanics, has publisheda thoughtful and well-written small book on statistical ermodynamics,43 in which hediscusses quantum effects in gases of fermions and bosons in some detail. He callsthe theory of degeneracy of gases satisfactory, disappointing and astonishing. Hefinds the theory satisfactory, because for high temperature and small density ittends to the classical theory of ideal gases.
At the same time the theory isdisappointing, because all its fascinating peculiarities occur at temperatures that areso low, that van der Waals forces have overwhelmed the gases – and made themliquid – long before the effects of degeneracy can be expected to appear. 44 Themost astonishing feature of the theory occurs, because in the classical limit we haveNxc <<1, while the classical theory itself has Nxc>>1, in fact, Nxc must be big enoughin the classical case that the Stirling formula can be applied.The fact that the entropies of gases of both bosons and fermions vanish in the stateof full degeneracy is often quoted as collateral support of the third law.
The supportis somewhat precarious, however, since no gas exists close to absolute zero.Satyendra Nath Bose (1893–1974)As a student Bose had been a member of a small and isolated, but dedicatedgroup of scholars in Calcutta, and then for long years he was an underpaidlecturer at a measly salary of 100 rupees.
In the opinion of Dutta,45 hisobsequious biographer, Bose was thus being punished for his outspokenness. Dutta gives no examples for this characteristic, but he does not forgetto praise the youthful Bose as a person who – in his college days – prepared42We have seen that, if velocity and momentum of a particle are zero, it cannot be localizedbecause of the uncertainty relation. That effect seems to be secondary in the presentcontext and we have ignored it in the preceding argument.43 E. Schrödinger: “Statistical thermodynamics.” Cambridge at the University Press (1948).44 This is not true for the electron gas in a metal as I have explained and, perhaps, liquidhelium shows vestiges of gas-degeneracy in the phenomenon of super-fluidity.45 M. Dutta: “Satyendra Nath Bose – life and work.” Journal of Physics Education.
2 (1975).Bosons and Fermions. Transition Probabilities195bombs. Presumably those were to be used for patriotic – terroristic (?) –deeds against the colonial power.Bose had treated a photon gas, then called a gas of light quanta.46 As Ihave mentioned before, Einstein translated his paper and it inspired him todevelop the statistical mechanics of degenerate gases, in which hediscovered the condensation-like phenomenon which is now called theBose-Einstein condensation, see above. Fritz Wolfgang London (1900–1954) and his brother Heinz London (1907–1970) were first to suggest – in1937 – that the super-fluidity of Helium II might be due to the BoseEinstein condensation.Soon after the Bose-Einstein statistics Enrico Fermi (1901–1954)formulated a statistics for particles which satisfy the Pauli exclusionprinciple.
In his honour we call those particles fermions. It seems thatFermi’s work was independent of Bose’s and Einstein’s; at least that is whatBelloni implies in a somewhat diffuse article.47 Paul Adrien Maurice Dirac(1902–1984) showed that quantum mechanics of many particles permitstwo types of statistics, i.e. ways of counting: Bose-Einstein for bosons andFermi-Dirac for fermions.48Still as a young man, but after the publication of his salient paper withthe help of Einstein, Bose spent two years in Europe; in France andGermany. Then he returned to India and became an influential physicsteacher and administrator. He finished his career as an honoured elderscientist; except when, after his retirement, he tried to continue his activity.According to Dutta this attempt violated the maxims laid down by the poetRabindranath Tagore (1861–1941), and there was some public debate andsevere criticism of Bose.Bosons and Fermions.
Transition ProbabilitiesThe equilibrium distributions fequ for fermions and bosons acquire a certaininterpretability by the following argument which concerns the transitionprobabilities in a collision between atoms with velocities c and c1 which,46An account of Bose’s arguments is given in Insert 7.4 below.L. Belloni: “On Fermi’s route to Fermi-Dirac statistics.” European Journal of Physics 15(1994).Belloni informs us that Fermi’s detailed and definitive theory for the quantization of theideal gas was published in German. He does not say when and where, and merely citessomeone else’s opinion about the paper. Thus he provides a good example for modernwriting in the history of science, where historians of science cite other historians ofscience rather than the original authors.48 Actually Fermi’s article appeared in: E.
Fermi: Zeitschrift für Physik 86 (1926)p. 902. Diracs contribution may be found in: P.A.M. Dirac: Proceedings of the RoyalSociety (A) 41 (1927) p. 24.471966 Third Law of Thermodynamicsafter the collision, have velocities cƍ and c1ƍ. We assume that the transitionprobability is of the form.fermionsPcc1 c c cN xc N xc1 (1 B N xc )(1 B N xc 1 )1bosonsso that it depends not only on the occupation numbers Nxc of the elementsdxdc before the collision, but also on those numbers after the collision. c isa factor of proportionality.
Thus the transition of fermions is less probable,if the target elements are well-occupied, – maximally with Nxc = 1 – whilethe transitions of bosons into such target elements become more probablewhen they are already well-occupied.For the reverse transition we assume an analogous expression for thetransition probabilities, viz.2E cE c oEEfermionsE 0 ZE c 0 ZE c (1 B 0 ZE )(1 B 0 ZE )1111bosons.In equilibrium, where both transition probabilities are equal, we concludethatln0 ZE1 B 0 ZEis a collisional invariant.Therefore this expression must be a linear combination of the collisionalinvariants mass and energy of the atoms and we may writelnN xc1 B N xcµα β c 2 hence N xc21exp(α βµ2.c2 ) 1fermionsbosonsThis agrees with the equilibrium distribution calculated before by amaximization of entropy. Thus the ansatz for the transition probabilitiesacquires some credibility.
Comparison of the whole argument withanalogous arguments by Maxwell and Boltzmann for the classical case, cf.Chap. 4, highlights the modification made necessary by quantum mechanics. Classically an effect of the target element on the transition probabilityis unthinkable. Of course the classical formula is recovered for the specialcase Nxc << 1.7Radiation ThermodynamicsAll energy available on earth – except nuclear and volcanic energy – comesfrom the sun through empty space by radiation, – or it came in previousgeological eras and was stored as coal, mineral oil, or natural gas.x Animals on the surface of the earth have evolved so as to see with theireyes those frequencies, – from red to violet – where the sunlight has itsmaximal intensity.x Plants utilize the red and yellow part of the visible spectrum for thethermodynamically precarious process of photosynthesis that hasevolved for the production of glucose and cellulose, the biomass ofplants.x And all creatures take advantage of the heating-part of the solarradiation which lies in the range of frequencies 3·1012 Hz < Ȟ <3·1014 Hzor in the range of wavelengths 10–6 m < Ȝ < 10–4 m.Despite the appearance of the numbers, these are small frequencies andlong wavelengths.
That is to say that the wavelengths (say) are longcompared to the dimensions of atoms and molecules. However, the solarradiation does contain shorter wavelengths which are of the dimension ofatoms and smaller. It stands to reason that the interaction of such highfrequency-radiation with matter is strongly influenced by the atomicstructure, which in turn is governed by the laws of quantum mechanics.Therefore the scientific research into radiation led to the discovery anddevelopment of quantum mechanics. This, of course, is no longer thermodynamics, but the pioneers of radiation physics, Stefan, Boltzmann, Planck,and Einstein were either thermodynamicists themselves or they were trainedto think thermodynamically. Therefore we follow their arguments in thischapter up to the point where they turn into quantum mechanics proper.Not only does radiation carry solar energy to the earth, the radiationpressure inside the sun serves to maintain the star in a stable mechanicalequilibrium.
Stellar physics is a paradigmatic application of the thermomechanical laws, and the consideration of radiation enriches the field in anon-trivial manner.198 7 Radiation ThermodynamicsBlack Bodies and Cavity RadiationThe history of the scientific study of light begins with Newton, of course,who concluded from his experiments with prisms that white light was amixture of colours, from red to violet.
Goethe, who occasionally dabbled inscience – and usually drew the wrong conclusions – ridiculed the idea ofwhite light as a mixture as clerical, because it reminded him of the Trinity,the hypostatic union of the Father, the Son, and the Holy Spirit in onegodhead. Newton carried the day, although his prisms were not goodenough to see more than just colours.Actually those colours were a nuisance for the users of microscopes,field-glasses and telescopes; they inevitably appeared at the rim of the fieldof vision and spoiled the view.
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