Müller I. A history of thermodynamics. The doctrine of energy and entropy (1185104), страница 45
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And indeed, Einstein let himself be inspired by Bose’spaper. He followed it up with two papers of his own which he read in July1924 and January 1925 to the Preußische Akademie der Wissenschaften.38In these papers Einstein develops the novel theory of degenerate gases, i.e.ideal gases at low temperature and large density, which I proceed todescribe.Of course, S = klnW had to be retained, because of its inherentplausibility, and neither Bose nor Einstein touched that relation. But therealization of a distribution, and the distribution itself, were modified, andso was W. As before, cf. Insert 4.6, we concentrate on the infinitesimalelement dxdc at (x,c) in (x,c)-space where we havePdxdcN dxdcY dxdc¦ N xcf ( x , c ) dx d c No. of cells in dxdc No.
of atoms in dxdc.PdxdcThe new distribution in dxdc is given by the set{plxc} = {p0xc,p1xc,...pdxc}which represents the number of cells which are occupied by 0,1,…d atoms.Obviously the values plxc must satisfy the constraintsd¦ pldxcPdxdcandl 0¦ lpxclN dxdc .l 0A realization of this distribution is given by {Nxc}, the number of atomssitting in the individual cells (x,c) in dxdc. Thus by the rules of combinatorics the number of realizations of the distribution {plxc} is equal toWxcPdxdc !dpl 037xcl. Hence!S xc dxdck lnPdxdc !dpxcl!l 0S.N.
Bose: “Planck’s Gesetz und Lichtquantenhypothese.” [Planck’s law and thehypothesis of light quanta] Zeitschrift für Physik 26 (1924).38 A. Einstein: “Quantentheorie des einatomigen idealen Gases.” [Quantum theory of amonatomic ideal gas] Sitzungsberichte physikalisch mathematische Klasse, September1924 pp. 261–267.A. Einstein: “Quantentheorie des einatomigen idealen Gases II” [Quantum theory of amonatomic ideal gas. II] Sitzungsberichte physikalisch mathematische Klasse, February1925 pp. 3–14.1906 Third Law of Thermodynamicsis the entropy of the atoms in the element dxdc, andSk ln xcPdxdc !dpxcl!l 0is the total entropy of the gas, where·is the product over all elementsxcdxdc of the space (x,c).This new form of entropy lacks the inherent perspicuity of Boltzmann’sentropy, because the relation to Ndxdc, or to the distribution function f(x,c) isnot explicit. However, for fermions such an explicit relation does exist, andfor bosons it does exist in local equilibrium, where there is no knowledgeabout Nxc in dxdc except about the average value which isN xcN dxdc.PdxdcIn those cases the entropy may be written in the formSË k Ô Ì lnfÍ YY ÈfØ Èf ØÛÉÊ1 B ÙÚ ln ÉÊ1 B ÙÚ Ü fdcdxYY Ýffermionsbosons.This is the proper form of the entropy in a monatomic gas; the expressiongeneralizes Boltzmann’s relationfS k ³ ln fdcdx with b eY ,bfound – by accident or luck, as it were – in the kinetic theory of gases, cf.
Chap. 4.And it coincides with Boltzmann’s form, if the difference between fermions andbosons, i.e. the ± - alternative, becomes unimportant. This happens for f/Y<<1 orN dx dcN xc 1 i.e. for sparse occupancy of each element dxdc. 39PdxdcThis observation is eminently plausible because, if there is much less than one atomper element dxdc on average, it makes no difference whether the atom is a fermionor a boson, since even a double occupancy of a cell practically does not occur, letalone higher occupancies.Obviously S in terms of the distribution function f is a non-equilibriumentropy in general.
In a closed adiabatic gas, i.e. for a fixed number N ofatoms and for a fixed energy U, we expect S to tend to a maximum Sequ inequilibrium. The calculation provides39 Recallthat for Boltzmann it was a matter of course, that Nxc was greater than 1. In fact, ithad to be big enough that the Stirling formula could be applied.Classical LimitYexp[ kT f equPgPc22 kT191fermions,bosons]r1where g is the specific Gibbs free energy, and T the temperature, of course.This expression replaces the Maxwellian distribution function in a degenerate gas , i.e. a gas for which the quantum effects – evidenced by the± – alternative – make themselves felt. The thermal and caloric equationsof statep2U3Vp N ,TV andg N ,TVgare given implicitlyNV4SY2P3fx 2 dxU³o exp[ kTPg kTx2 ] r 1 , V4SY2P3fx 4 dx³0 exp[ kTPg kTx2 ] r 1 ,and the equilibrium entropy Sequ readsT Sequ = – Nµg + 5/3 U.Classical LimitThe Boltzmann limit occurs – just like in the non-equilibrium case – whenthe ±-alternative for the fermions and bosons does not matter, i.e.
forPgkT 1. In that case we havepNµ3kT and withVY3h :µgkT3ËN ÈØ Ûhln Ì ÉÙ Ü.Ì V Ê 2πµ kT Ú ÜÍÝIt follows that the classical limit is the one, in which an element of phasespace of the dimension of a typical thermal de Broglie wavelength, seeabove, contains practically no particle. In contrast, degeneration thereforeappears as the state, where the particles are so dense, or the temperature isso low, that the de Broglie wavelengths overlap.Note that for particles with a small mass the de Broglie wavelength is big. It is forthat reason that even at room temperature – and even for a few thousand K – theelectron gas in a metal is strongly degenerate, – also of course, because the electronNdensity /V is large.1926 Third Law of ThermodynamicsFor the non-degenerate state the equilibrium entropy has the formSequ3ÎËN ÈØ Û ÞÑhÑ5Nk Ï ln Ì ÉÙ Üß .2VÌ2kTπµÊÚ ÝÜ ÑÑÐÍàThat value is entirely explicit! Thanks to Bose’s choice Y = µ3/h3 there isno unknown constant.
The expression provides the absolute value of theentropy for a rarefied ideal gas. Hence, by integration over cp(T,p)/T – andsummation of latent heats divided by the temperatures of their occurrence –downward to lower temperatures, one may obtain the absolute value ofentropy of liquids and solids at absolute zero, or as close as we can getthere.If one proceeds with that integration – after having made all those caloricmeasurements – one obtains the value zero for entropy in most cases andthus confirms Planck’s extension of the third law of thermodynamics.Sometimes, however, the value zero is not obtained. That seems to happenonly when the solid phase is amorphous, – rather than crystalline – so thatthe third law must be qualified: the entropy at absolute zero for amorphoussolids is not zero. Handbooks record the value as the zero point entropy.Full Degeneration and Bose-Einstein CondensationThe opposite of the classical limit – the limit of full degeneracy – isdifferent for fermions and bosons.FermionsFor fermions the limit is characterized byf equY®¯0PgkTc2 g.1 c2 ! g21for!! 1 so that2At low temperature all atoms tend to assemble at zero kinetic energy, butthat desirable state cannot be achieved, since each velocity can only beassumed by just one atom.40 Therefore the atoms do the next best thing andfill all states with the lowest velocities.
N and U are given by40Actually, two atoms may assume the same velocity, if they have different spins.Full Degeneration and Bose-Einstein Condensation3NVUV2 1 3/ 24π Ygandµ 319332 1 5/ 24π Yg ,µ 5so that the energy is large, but the entropy vanishes.BosonsFor bosons – with the lower sign – we must realize that the biggest value ofg must be g = 0, lest negative values of the distribution function appear.Therefore g = 0 andf equYcexp[ 2µkT] 12characterize the Bose case of full degeneracy. The properties of thedistribution are much as expected, because it implies that there are lessparticles with larger speeds.
However, there is a problem, since fequ issingular for c = 0: To be sure, the values of N/V and p = 2/3 U/V are finite,namely41NVY232πµk35kÈ 3ØÈ 5ØT ζ É Ù and p Y 2π T ζ É Ù ,ÊÚÊ 2Ú2µµbut there is something strange. Indeed N/V and p are functions of T only, acircumstance that we have come to expect as an equilibrium condition forsaturated vapour coexisting with a boiling condensate.That observation may serve as a hint that the equation for the number Nof atoms is incorrect, because N cannot possible depend on T. And indeed,the equation holds only for the number of particles with c 0, while N0, thenumber of particles with c = 0, has somehow slipped through the(Riemann)-integration, although its density is singular.
Therefore the N/V –equation must be rewritten asN 0 YVNAnd, if YV2µ32µ32πk3È 3ØT ζÉ Ù .Ê 2Úµ32π µk T ζ ( 32 ) is the number of particles in the vapour, N0is the number of particles in the condensate. One says: The N0 particles with4135ζ ( /2) and ζ ( /2) are values of the Riemann zeta function which occurs in the integration ofthe distribution function for g = 0.1946 Third Law of Thermodynamicsc = 0 form the Bose-Einstein condensate.42 For T ĺ 0 there will be moreand more condensate, whose entropy is zero. The entropy of a Bose gas forfull degeneracy vanishes therefore for T ĺ 0.The observed decomposition of liquid helium into a normal fluid and asuper-fluid is often seen to be a reflection of the Bose-Einsteincondensation.
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