Foundations Postlude:Symmetry and the Conceptual Fou N Dations Ofthermostatistics

FOUNDATIONS POSTLUDE:SYMMETRY AND THE CONCEPTUAL FOU N DATIONS OFTHERMOSTATISTICS

2I.I STATISTICS

The overall structure of thermostatisticsnow has been established-of thermodynamicsin Part I and of statisticalmechanics in part II. Although these subjects can be elaboratedfurther, the logical basis is essentialy complete. It is an appropriate time to reconsiderand to reflect on the uncommon form of theseatypical subjects. unlike mechanics,thermostatistics is not a detailed theory of dynamic .responseto specifiedforces. And unlike electromagnetictheory (or the analogoustheories of the nuclear"strong" and "weak" forces),thermostatistics is not a theory of the forces themselves.Instead thermostatistics chancterizes the equilibrium state of microscopic systemswithout reference either to the specificforces or to the laws of mechanicalresponse. Instead thermostatisticscharacterizes the equilibrium state as the state that maximizes the disorder, a quantity associatedwith a conceptual framework ("information theory") outside of conventionalphysicai theory. The question arisesas to whetherthe postulatory basis of ihermosta- tistics thereby introduces new principles not contained in mechanics, electromagnetism,and the like or whetherit borrows principles in unrec- og:rizedform from that standardbody of physical theory. In either case, what are the implicit principles upon which thermostatisiicsrests? There are, in my view, two essentialbases underlying thermostatistical 4"ory. one is rooted in the statisticalproperties of large complexsystems. The second rests in the set of symmetriesof the fundamental laws of physics. The statisticalfeature ueilsthe incoherent,complexity of the atomic dynamics, therebyreuealing the coherentefects of the undirryingphysical svmmetries. 456 Postlude: Symmetry and the Conceptual Foundations of Thermostatistics

The relevanceof the statistical properties of large complex systemsis universally accepted and reasonablyevident. The essentialproperty is epitomized in the "central limit theorem"l which states(roughly) that the probability density of a variable assumesthe "Gaussian" form if the variable is itself the resultant of a large number of independentadditive subvariables.Although one might naively hope that measurementsof thermodynamicfluctuation amplitudescould yield detailedinformation as to the atomic structure of a system,the central limit theorem precludes such a possibility.It is this insensitivityto specificstructural or mechanical detail that underliesthe universalityand simplicity of thermostatistics. The central limit theoremis illustrated by the following example.

Example Considera systemcomposed of .& "elements,"each of whichcan take a valueof X in the range-, < X < 1. The valueof X for eachelement is a continuous randomvariable with a probabilitydensity that is uniform over the permitted region.The valueof X for the systemis the sumof the valuesfor eachof the elements.Calculate the probability density for thesystem for thecases fr : I,2,3. In eachcase find the standarddeviation o, definedbv

"r: If(x)x2dx where /(X) is the probability density of X (and where we have given the definition of o only for the relevantcase in which the meanof X is zero).Plot the probability density for f : \, 2, and 3, and in each caseplot the Gaussianor "normal" distribution with the samestandard deviation. Note that for even so small a number as fr : 3 the probability distribution /(X) rapidly approachesthe Gaussianform! It should be stressedthat in this examplethe uniform probability density of X is chosenfor easeof calculation; a similar approach to the Gaussianform would be observedfor any initial probability density. Solution - The probabilitydensity for.f :1is fr(X):1 for +

fo(X\ : (zr)-r/2 o-t*p-\zo- ( # ) I with o : o, is also plotted in Fig. 21.1a, for cpmparison.

'rJ. any standard reference on probability, such as L. G. Parratt, Probability and Experimental Errors in Sciezce (Wiley, New York, 1961) or E. Parzen, Modern Probability Theory and lts Applications (Wiley, New York, 1960). 457

1.5

0.5

0.0

1.0

0.5

_+ -1 _+ o + | + FIGURE 21 1 Convergenceof probability density to the Gaussianform. The probability density for systemscomposed of one, two and threeelements, each with the probability densityshown in FigureZL.La. ln eachcase the Gaussianwith the samestandard deviation is plotted.In accordancewith the central limit theorem the probability density becomesGaussian for hrge f.

To calculate the probability density fz(X), for fr :2, we note (problem 21.1-1)that f**r(x): [* f*(x - x')fr(x')dx' or, with fr(X)as given f **r(x) : ["' f *(x - x')dx' " -7/2 That is, f ,*r(X) is the average value of f ;,(X,') over a range of length unity centered at X. This geometric interpretation easily permits calculation of fr(X) as shown in Fig.2I.7b. From lr(X), in turn, we find (l-x' iflxl3 458 Postlude: Symmetry and the Conceptual Foundations of Thermostutistics

The valuesof o arecalculated to be o, : l/ ,h2, oz: l/ ,/6 andot : i. Thesevalues agree with a generaltheorem that for fr identicaland independent subsystems,o;1 : {Nay The Gaussiancurves of Fig. 2I.7 arecalculated with thesevalues of the standarddeviations. For evenso smalla valueof fr as 3 the probabilitydistribution is very closeto Gaussian,losing almost all traceof the initial shapeof the single-elementprobability distribution.

PROBLEMS

2l.l-1. The probabilityof throwinga "seven"on two dicecan be viewedas the sum of a) the probabilityof throwinga "one" on the first die multipliedby the probabilityof throwinga "six" on thesecond, plus b) theprobability of throwing a " two" on the first die multipliedby theprobability of throwinga "five" on the second,and so forth. Explainthe relationshipof this observationto the expressionfor f**r(X) in termsof. f;,(X - X') and fr(X') asgiven in the Example, and derivethe latter expression. 2l.l-2. Associatethe value+1 with oneside of a coin("head") and the value -1 with the otherside ("tail"). Plot theprobability of findinga given"value" whenthrowing one, two, three,four, and fivecoins. (Note that the probabilityis discrete-fortwo coins the plot consists of just threepoints, with probability : I for X : tl andprobability : ] for X : 0.)Calculate o for thecasen : 5, and roughlysketch the Gaussiandistribution for this valueof o.

2I-2 SYMMETRY2

As a basis of thermostatisticsthe role of symmetryis lessevident than the role of statistics.However, we first note that a basisin symmetrydoes rationalize the peculiar nonmetric character of thermodynamics.The results of thermodynamics characteristically relate apparently unlike quantities,yielding relationshipssuch as (07/0P)v: (0V/05)r, b:ut providing no numerical evaluation of either quantity. Such an emphasis on relationships,as contrastedwith quantitativeevaluations, is appropri- ately to be expectedof a subjectwith roots in symmetryrather than in explicitquantitative laws. Although symmetry considerationshave been seenas basic in science since the dawn of scientificthought, the developmentof quantum mechanics in 1925 elevatedsymmetry considerations to a more profound level of power, generality, and fundamentality than they had enjoyedin classical physics. Rather than merely restricting physical possibilities, symmetry was increasingly seenas playing the fundamental role in establishingthe

2H. Callen, Foundations ol Physics 4,423 (1974). Symmetry 459

The simplest and most evident form of symmetry is the geometric symmetry of a physicalobject. Thus a sphereis symmetricundeiarbitrary rotations around any axis passingthrough its center,under reflectionsin any plane containing the center, and under inversion through the center itself. A cube is symmetricunder fourfold rotationsaround ixes throueh the face centersand under various other rotations, reflections,and invJr- sion operations. Becausea sphereis symmetric under rotations through an angle that can take continuousvalues the rotational symmetry group of a sphereis said to be continuous.In contrast,the rotational symm-etrygroup of a cube is discrete.

The conceptof a geometricalsymmetry is easilygeneralized. A transformation of variables definesa symmetry operation. A function of those variables that is unchangedin form by the transformation is said to be symmetric with respect to the symmetry operation. Similarly a law of physicsis said to be symmetricunder the operationif the funciional form of the law is invariant under the transformation. Newton's law of dynamics,I : m(d2r/dt21 is symmetricunder time inversion (r --+r', t --+- t') for a systemin which the force is a function

3E. Wigner, "Symrnetry and Conservation Laws,,' physics Today, March 1964, p. 34. 460 Postlude: Symmetry and the Conceptual Foundations of Thermostatistics determinesthe forces.For quantum mechanicalproblems the dynamical equation is more abstract (Schrbdinger'sequation rather than Newton's law), but the principles of symmetryare identical.

2I-3 NOETHER'S THEOREM

A far reaching and profound physical consequenceof symmetry is formulated in "Noether's theorem4". The theorem asserts that e:uery continuoussymmetry of the dynamicalbehauior of a system (i.e., of the dynamical equation and the mechanicalpotential) implies a conseruation law for lhat system. The dynamical equation for the motion of the center of masspoint of any materialsystem is Newton'slaw. If the externalforce does not depend upon the coordinatex, then both the potentialand the dynamicalequa- tion are symmetricunder spatial translationparallel to the x-axis. The quantity that is conservedas a consequenceof this symmetry is the x-componentof the momentum.Similarly the symmetryunder translation along the y or z axesresults in the conservationof the y or z components of the momentum.Symmetry under rotation around the z axis implies cgnservationof the z-componentof the angularmomentum. Of enormoussignificance for thermostatisticsis the symmetryof dynamical laws under time translation.That is. the fundamental dvnamical laws of physics (such as Newton's law, Maxwell's equations, and Schrodinger'sequation) are unchangedby the transformationt --+t' * to (i.e.,by a shift of the origin of the scaleof time). If the externalpotentral is independentof time, Noether's theorempredicts the existenceof a conservedquantity. That conservedquantity is :alled the energy. Immediatelyevident is the relevancerf ume-translationsymmetry to what is often calledthe "first law of thermodynamics"-the existenceof the energyas a conservedstate function (recallSection 1.3 and Postulate r). It is instructive to reflect on the profundity of Noether's theorem by comparing the conclusion here with the tortuous historical evolution of the energy concept in mechanics (recall Section 1.4). Identification of the -t conserved energy began in 1693 when Leibniz observed that lmuz mgh is a conserved quantity for a mass particle in the earth's gravitational field. As successively more complex systems were studied it was found that additional terms had to be appended to maintain a conservation principle,

aSee E. Wigner, ibid. The physical content of Noether's theorem is implicit in Emmy Noether's purely mathematical studies. A beautiful appreciation of this brilliant mathematician's life and work in the face of implacable prejudice can be found in the introductory remarks to her collected works: Emmy Noether, Gesammelte Abhandlungen, (Collected Papers), Springer-Verlag, Berlin-New York, 1983. Energlt, Momentum, and Angular Momentum: the Generslized "First l-aw" of rhermostatistics 46I

but that in eachcase such an ad hoc addition was possible.The develop- ment of electromagnetictheory introduced the potential energy of the interaction of electric charges,subsequently to be augmentedby the electromagneticfield energy.In 1905Albert Einsteinwas inspired to alter the expressionfor the mechanicalkinetic energy, and even to associate energywith stationary mass,in order to maintain the principle of energy conservation.In the 1930sEnrico Fermi postulated the existenceof t[-e neutrino solely for the purposeof retaining the energyconservation law in nuclear reactions.And so the processcontinues, successively accreting additional terms to the abstractconcept of energy,which is definedby ii conseruationlaw. That conservationlaw was evolvedhistorically by a long seriesof successiverediscoveries. It is now basedon the assumptionoT time translationsymmetry. The evolution of the energy concept for macroscopicthermodynamic systemswas evenmore difficult.The pioneersof the subjectwere guided neither by a general a priori conservationtheorem nor by any specific analytic formula for the energy. Even empiricism was thwarted 6y the absenceof a methodof directmeasurement of heattransfer. only inspired insight guided by faith in the simplicityof nature somehowrevealed the interplay of the conceptsof energyand entropy,even in the absenceof a priori definitionsor of a meansof measuringeither!

2t-4 ENERGY, MOMENTUM, AND ANGULAR MOMENTUM: THE GENERALIZED "FIRST I,AW" OF THERMOSTATISTICS

In acceptingthe existenceof a conservedmacroscopic energy function 3s lhe first postulateof thermodynamics,we anchor that postulaiedirectly in Noether's theoremand in the time-translationsymmetry of physical laws. An astute reader will perhaps turn the symmetry argument around. There are seuen" first integralsof the motion" (as the conservedquantities are known in mechanics).These seven conserved quantities are the energy, the three componentsof linear momentum,and the three componentsof the angular momentum; and they follow in parallel fashion from the translationin "space-time" and from rotation. why, then, doesenergy appear to play a unique role in thermostatistics?Should not momenrum and angular momentumplay parallel roles with the energy? In fact, the energyis nol unique in thermostatistics.Tlie linear momentum and angular momentumplay preciselyparallel rores.The asymmetry in our accountof thermostatisticsis a purely conuentionalone that obscuris the true nature of the subject. we have followed the standard convention of restricting attention to systemsthat are macroscopicallystationary, in which casetlie momentum 462 Postlude: Symmetry and the Conceptual Founfuitions of Thermostatistics

and angular momentum arbitrarily are required to be zero and do not appear in the analysis. But astrophysicists, who apply thermostatistics to rotating galaxies, are quite familiar with a more complete form of thermostatistics. In that formulation the energy, linear momentum, and angular momentum play fully analogous roles. The fully generalized canonical formalism is a straightforward extension of the canonical formalism of Chapters 16 and 17. Consider a subsystem consisting of N moles of stellar atmosphere.The stellar atmospherehas a particular mean molar energy (U/N), a particular mean rnolar momentum (P/N), and a particular mean molar angular momentum (J/N). The fraction of time that the subsystem spends in a particular microstate i (with energy E,, momentum P,, and angular moment J,) is f,(E,,Pi,Ji,V, N). Then f is determinedby maximizing the disorder,or entropy, subject to the constraints that the averageenergy of the subsystem be the same as that of the stellar atmosphere, and similarly for momentum and angular momentum. As in Section \7.2, we quite evi- dently find - f,: +exp(-BE,-\o'4 x,.J,) (21.1)

The sevenconstants B, \rr, ^rr, \or, tr7", tr7u,.symmetricand trr, all ariseas Lagrangeparameters and they play bompletely roles in the theory (just as Bg,does in the grand canonicalformalism). The proper "first law of thermodynamics,"(or the first postulatein our formulation) is the symmetry of the laws of physics under space-time translation and rotation, and the consequentexistence of conseruedenergl, momentum,and angularmomentum functions.

2I.5 BROKEN SYMMETRY AND GOLDSTONE'S THEOREM

As we have seen,then, the entropy of a thermodynamicsystem is a function of various coordinates,among which the energyis a prominent member. The energy is, in fact, a surrogate for the seven quantities conservedby virtue of space-time translationsand rotations. But other independentvariables also exist-the volume,the magneticmoment, the mole numbers, and other similar variables.How do these arise in the theory? The operational criterion for the independentvariables of thermostatistics (recall Chapter 1) is that they be macroscopicallyobseruable. The low temporal and spatial resolving powers of macroscopicobservations re- f quire that thermodynamicvariables be essentiallytime independenton i the atomic scaleof time and spatially homogeneouson the atomic scaleof distance. The time independenceof the energy (and of the linear and 'll angular momentum) has been rationalized through Noether's theorem.

I Broken Symmetry and Coldstone's Theorem 463

The time independenceof other variables is based on the concept of broken symmetry and Goldstone'stheorem. These concepts are best introduced by a particular caseand we focus specificallyon the volume. For definiteness,consider a crystallinesolid. As we saw in section 16.7, the vibrational modes of the crystal are describedby a wave number k(: 2n/X, where tr is the wavelength)and by an angularfrequency o(k). For very long wavelengthsthe modesbecome simple sound waves,and in this region the frequencyis proportional to the wave numb er; a : ck (recall Fig. 16.1). The significantfeature is that o(k) vanishesfor k : 0 (i.e., for l, -- m). Thus, the very mode that is spatially homogeneoushas zero frequency.Furthermore, as we have seenin chapter 1 (refer also to Problem 21.5-7),the volume of a macroscopicsample is associatedwith the amplitude of the spatially homogeneousmode. consequently the volume is an acceptablytime independentthermodynamic cooidinaie.

first microscopic nucleation.In that nucleation processthe symmetry of the system suddenly and spontaneouslylowers, and it does so by a nonpredictable,random event. The symmetryof the systemis "broken.', Macroscopic sciences,such as solid state physics or thermodynamics, are qualitatively different from "microscopic" sciencesbecause of the effectsof broken symmetry,as was pointed out by P. W. Andersonsin an early but profound and easily readable essay which is highly recom- mended to the interestedreader. At sufficiently high temperaturesystems always exhibit the full symmetry of the "mechanical potential" (that is, of the Lagrangianor Hamilto- nian functions). There do exist permissible microstates with lower symmetry, but thesestates are grouped in sets which collectiuelyexhibit the full symmetry. Thus the microstatesof a gas do include stateswith crystal-like spacingof the molecules-in fact, among the microstatesall manner of different crystal-like spacingsare represented,so that collec-

5P. W. Anderson, pp. 175-182 in Conceptsin Solids(W. A. Benjamin Inc-, New york, 1964). 464 Postlude: Symmetry and the Conceptua! Foundations of Thermostatistics tively the statesof the gas retain no overall crystallinity whatever.How- ever, as the temperatureof the gas is lowered the moleculesselect that particular crystalline spacingof lowest energy,and the gascondenses into the corresponding crystal structure. This is a partial breaking of the symmetry. Even among the microstateswith this crystalline periodicity there are a continuum of possibilities available to the system, for the incipient crystal could crystallizewith any arbitrary position. Given one possible crystal position there exist infinitely many equally possibleposi- tions, slightly displacedby an arbitrary fraction of a "lattice constant". Among thesepossibilities, all of equal energy,the systemchooses one position (i.e.,a nucleationcenter for the condensingcrystallite) arbitrarily and "accidentally". An important generalconsequence of broken symmetryis formulatedin the Goldstone theorem6.It assertsthat any systemwith brokensymmetry (and with certain weak restrictionson the atomic interactions) has a spectrum of excitationsfor which the frequencyapproaches zero as the wauelengthbecomes infinitely large. For the crystal discussedhere the Goldstone theorem ensuresthat a phonon excitationspectrum exists, and that its frequencyvanishes in the long wavelengthlimit. The proof of the Goldstonetheorem is beyondthe scopeof this book, but its intuitive basiscan be understoodreadily in terms of the crystal condensationexample. The vibrationalmodes of the crystaloscillate with sinusoidaltime dependence,their frequenciesdetermined by the massesof the atoms and by the restoringforces which resist the crowding together or the separationof thoseatoms. But in a mode of very long wavelength the atomsmove very nearlyin phase;for the infinite wavelengthmode the atoms move in unison.Such a mode doesnot call into action any of the interatomicforces. The very fact that the original position of the crystal was arbitrary-that a slightlydisplaced position would havehad precisely the sameenergy-guarantees that no restoringforces are calledinto play by the infinite wavelengthmode. Thus the vanishingof the frequencyin the long wavelengthlimit is a direct consequenceof the broken symmetry. The theorem,so transparentin this case,is true in a far broadercontext, with far-reachingand profound consequences. In summary,then, the volume emergesas a thermodynamiccoordinate by virtue of a fundamentalsymmetry principle grounded in the conceptof broken symmetryand in Goldstone'stheorem.

PROBLEMS

21.5-1. Draw a longitudinal vibrational mode in a one-dimensional system, with a node at the center of the system and with a wavelength twice the nominal length

6P W Anderson. ibid Other Broken Symmetry Coordinates-Electric and Magnetic Moments 465

of the system.Show that the instantaneouslength of the systemis a linear function of the instantaneousamplitude of this mode.what is the order of magnitudeof the wavelengthif thesystem is macroscopicand if thewavelength is measuredin dimensionlessunits (i.e., relative to interatomiclensths)?

2I-6 OTHER BROKEN SYMMETRY COORDINATES_ELECTRIC AND MAGNETIC MOMENTS

- In the precedingtwo sectionswe havewitnessed the role of symmetryin determining several of the independent variables of thermostatistical

In addition to the energy and the volume, other common extensive parametersare the magneticand electric moments.These are also prop- erly time independent by virtue of broken symmetry and Goldston6's theorem. For definitenessconsider a crystal such as HCl. This material

The direction of the net dipole moment is the residue of a random accident associatedwith the processof cooling below the ordering temperature. Above that temperaturethe crystal had a higher symmetry; below the orderingtemperature it developsone uniqueaxis-the directionof the net dipole moment. Below the ordering temperaturethe dipoles are aligned generaly (but not precisely) along a common direction. Around this direction the dipoles undergosmall dynamicangular oscillations ("librations"), rather like a pendulum.The librational oscillationsare coupled,so that librational wavespropagate through the crystal.These librational wavesare the Goldstoneexcitations. The Goldstonetheorem implies that the librational modes of infinite wavelengthhave zero frequencyT.Thus the electric

?In the interests of clarity I have oversimplified slightly. The discussion here overlooks the fact that the crystal structure would have already destroyed the spherical symmetry even above the ordering temperature of the dipoles. That is, the discussion as given would apply to an amorphous (spherically symmetric) crystal but not to a cubic crystal. In a cubic crystal each electric dipole would be coupled by an "anisotropy energy" to the cubic crystal structure, and this coupling would (naively) appear to provide a restoring force even to infinite wavelength librational modes. However, under these circumstances librations and crystal vibrations would couple to form mixed modes, and these coupled "libration-vibration" modes would again satisfy the Goldstone theorem. 466 Postlude: Symmetry and the Conceptual Foundations of Thermostatistics

dipole moment of the crystal qualifies as a time independentthermodynamic coordinate. Similarly ferromagneticcrystals are characterizedby a net magnetic moment arising from the alignment of electron spins. These spins par- ticipate in collectivemodes known as "spin waves."If the spins are not coupled to lattice axes(i.e., in the absenceof "magnetocrystallineani- sotropy") the spin wavesare Goldstonemodes and the frequencyvanishes in the long wavelengthlimit. In the presenceof magnetocrystallineani- sotropy the Goldstonemodes are coupledphonon-spin-wave excitations. In either casethe total magneticmoment qualiflesas a time independent thermodynamiccoordinate.

2I.7 MOLE NUMBERS AND GAUGE SYMMETRY

We come to the last representativetype of thermodynamiccoordinate, of which the mole numbersare an example. Among the symmetryprinciples of physicsperhaps the most abstractis the set of "gauge symmetries."The representativeexample is the "gauge transformation" of Maxwell's equationsof electromagnetism.These equations can be written in terms of the observableelectric and magnetic fields, but a more convenientrepresentation introduces a " scalarpoten- tial" and a " vector potential." The electric and magnetic fields are derivable from these potentials by differentiation. However the electric and magneticpotentials are not unique. Either can be alteredin form providing the other is altered in a compensatoryfashion, the coupled alterations of the scalar and vector potentialsconstituting the "gauge transformation." The fact that the observableelectric and magneticflelds are invariant to the gauge transformation is the "gauge symmetry" of electromagnetictheory. The quantity that is conservedby virtue of this symmetry is the electriccharge8. Similar gauge symmetriesof fundamentalparticle theory lead to conservationof the numbersof leptons(electrons, mesons, and otherparticles of small rest mass)and of the numbersof baryons(protons, neutrons, and other particlesof largerest mass). In the thermodynamicsof a hot stellar interior, where nucleartransfor- mations occur sufficiently rapidly to achieve nuclear equilibrium, the numbers of leptons and the numbersof baryonswould be the appropriate "mole numbers" qualifying as thermodynamicextensive parameters. In common terrestrial experiencethe baryons form longJived associa- tions to constitute quasi-stableatomic nuclei. It is then a reasonable

E The result is a uniquely quantum mechanical result It depends upon the fact that the phase of the quantum mechanical wave function is arbitrary ("gauge symmetry of the second kind"), and it is the interplay of the two types of gauge symmetry that leads to charge conservation :1 1' I

I i Time Reuersal, the Equal Probabilities of Microstates, and the Entropy Principle 467 approximation to consideratomic (or even molecular) speciesas being in quasi-stableequilibrium, and to consider the atomic mole numbers as appropriate thermodynamiccoordinates.

2r-8 TIME REVERSAL, THF EQUAL PROBABILITIES OF MICROSTATES, AND THE ENTROPY PRINCIPLE

We come finally to the essenceof thermostatistics-to the principle that an isolated system spends equal fractions of the time in each of its permissible microstates.Given this principle it then follows rhat the number of occupiedmicrostates is maximum consistentwith the external constraints, that the logarithm of the number of microstates is also maximum (and that it is extensive),and that the entropy principle is validated by interpreting the entropy as proportional to ln O. . The permissible microstatesof a system can be representedin an abstract,many-dimensional state space (recall Section15.5). In the state spaceevery permissiblemicrostate is representedby a discretepoint. The system then follows a random, erratic trajectory in the space as it undergoesstochastic transitions among the permissiblestates. These transitions are guaranteedby the random externalperturbations which act on even a nominally "isolated" system(although other mechanismsmay dominatein particularcases-recall Section15.1). The evolution of the systemin state spaceis guided by a set of transitionprobabilities. If a systemhappens at a particularinstant to be in a microstate i then it may make a transition to the state 7, with probability. (per unit time) f,,. The transition probabilities { f,i} form a networkjoining pairs of statebthroughout the statespace. The formalism of quantum mechanicsestablishes that, at least in the absenceof externalmagnetic fieldse

f,i: fii (21.2)

That is, a systemin the state i will undergoa transition to the state/ with the sameprobability that a systemin state7 will undergoa transition to the state i. The "principle of detailed balance" (equation 27.2) follows from the symmetryof the releuantlaws of quantummechanics under time inuersion (i.e.,under the transformationt + -t').

eThe restriction that the external magnetic field must be zero can be dealt with most simply by including the source of the magnetic field as part of the system. In any case the presence of external magnetic fields complicates intermediate statements but does not alter final conclusions, and we shall here ignore such fields in the interests of simplicity and clarity. 468 Postlude: Symmetry and the Conceptual Foundations of Thermosratistics

Although we merely quote the principle of detailed balance as a quantum mechanical theorem, it is intuitively reasonable.Consider a system in the microstate i, and imagine a video tape of the dynamicsof the system(a hypotheticalform of video tape that recordsthe microstate of the system!).After a brief momentthe systemmakes a transitionto the microstate j. If the video tape were to be played backwardsthe system would start in the state 7 and make a transition to the state i. Thus the interchangeabilityof future and past, or the time reversibility of physical laws, associatesthe transitions i --+j and j -- i and leads to the equality (27.2)of the transitionprobabilities. The principle of equalprobabilities of statesin equilibrium(, : I/0) follows from the principleof detailedbalance (f , i : f ,,).To seethat this is so we first observethat l, is the conditionalpioba6ility that the system will undergo a transition tb statej if it is initially in state i. The number of such transitions per unit time is then the product of f ,, and the probability f, that the systemis initially in the state i. Hence the total numberof transitionsper unit time out of the statei isL,f if ,,.Similarly the numberof transitionsper unit time into the statei isL, fi\,. However in equilibrium the occupationprobability I of the ith'siate must be independentof time; or

df,-_ Lf,f,,+ Dfifi,:o (21.3) dt j+i j+i

some statesare "visited" frequently (i.e.,L,f11 is large), and others are visited only infrequently. Some states are tena-ciousof the system once it does arrive (i.e.,L 1fi i is small), whereasothers permit it to depart rapidly. Because of time ri:versal symmetry, however, those states that are visited only infrequently are tenaciousof the system.Those statesthat are visited frequently host the systemonly fleetingly. By virtue of thesecompensating attributes the system spendsthe same fraction of time in each state. The equal probabilities of permissible states for a closed system in equilibrium is a cons-equenceof time reuersal symmetry of the releuant quantum mechanicallawsro.

loIn fact weaker - :0, a condition, X;(I i lr,) which follows from a more abstract requirement of "causality," is also sufficient to ensure that , : 1/O in equilibrium. This fact does not invalidate the prevrous statement Symmetry and Completeness 469

2I.9 SYMMETRY AND COMPLETENESS

FIGURE 2I.2

suppose now that the permissibleregion in state spaceis divided into two subregions(denoted by A, and A,, in fig. 2t.Zy such that all transitionprobabilities f., vanishif the statet is in A, and i is in A,,, or vice versa.Such a set of transition probabilitiesis fully consistentwith time reversal symmetry (or detailed balance),but it doesnot lead to a probability uniform.o-verthe physicallypermissible region (A, + A,,).lf the systemwere initially in A' the probatility density would diffuse from the initial state to eventuallycover the region l, uniformly, but it would not crossthe internal boundary to the rcgjon A,,. The "accident" of such a zero transition boundary, separating the permissiblestates into nonconnectedsubsets, would leid to a failure of the assumptionof equalprobabilities throughout the permissibleregion of statespace. It is important to recognizehow incredibly stringentmust be the rule of vanishing of the f,i .between subregions if the principle of equal probabilities of statesis to be violated. It is not suffi-cientior transitlon probabilities betweensubregions to be very small-euery such transition probability must be absolutelyand rigorously zero. If even one or a few transition probabilitieswere merely very small acrossthe internal boundarv it would tu!-" u very long time for the probability density to fill both ,,4, and A" uniformly, but eventuallyit would The "accident" that we feared might vitiate the conclusion of equal probabilities appearsless and less likely-unless it is not an accideni at all, but the consequenceof someunderlying principle. Throughout quan- 470 Postlude: Symmetry and the Conceptual Foundations of Thermostatistics tum physicsthe occurrenceof outlandishaccidents is disbarred;physics is neither mystical nor mischievous.If a physical quantity has a particular value, say 4.5172... then a secondphysical quantity will not have preciselythat samevalue unless there is-a compellingreison that ensures equality. Degeneracyof energylevels is the most familiar example-when it occurs it always reflectsa symmetry origin. Similarly, transition probabilities do not accidentallyassume the precisevalue zero; when they do vanish they do so by virtue of an underlying symmetrybased reason. The vanishing of a transition probability as a consequenceof symmetry is called a "selectionrule." Selectionrules that divide the statespace into disjoint regionsdo exist. They alwaysreflect symmetryorigins and they imply conservationprinci- ples. An already familiar exampleis provided by a ferromagneticsystem. The statesof the systemcan be classifiedby the componentsof the total angular momentum. Stateswith different total angular momentum components have different symmetriesunder rotation, and the selectionrules of quantum mechanicsforbid transitionsamong such states. These selection rules give rise to the conservationof angular momentum. More generally,then, the state spacecan be subdividedinto disjoint regions,not connectedby transitionprobabilities. These regions are never accidental; they reflect an underlyingsymmetry origin. Each region can be labeled accordingto the symmetryof its states-such labelsare called the "charactersof the group representation."The symmetrythereby gives rise to a conservedquantity, the possiblevalues of which correspondto the distinguishinglabels for the disjoint regionsof statespace. In order that thermodynamicsbe valid it is necessarythat the set of extensiveparameters be complete.Any conservedquantity, such as that labelling a disjunctureof the statespace, must be includedin the set of thermodynamiccoordinates. Specifying the value of that conservedquantity then restrictsthe permissiblestate space to a singledisjoint sectoi(r' alone, or A" alone,in Fig. 21.2).The principle of equalprobabilities of states is restored only when all such symmetry based thermodynamic coordinatesare recognizedand included in the theory. Occasionallythe symmetrythat leads to a selectionrule is not evident, and the selection rule is not suspectedin advance.Then conventional thermodynamicsleads to conclusionsdiscrepant with experiment.Puzzle- ment and consternationmotivate exploration until the missing symmetry principle is recognized.Such an event occurredin the exploration of the properties of gaseoushydrogen at low temperatures.Hydrogen molecules can have their two nuclear spins parallel or antiparallel, the molecules then being designatedas "ortho-hydrogen"or "para-hydrogen,"respec- tively. The symmetriesof the two typesof moleculesare quite different. In one casethe moleculeis symmetricunder reflectionin a plane perpendicu- lar to the molecular axis, in the other casethere is symmetrywith respect to inversion through the center of the molecule.Consequently a selection Symmetry and Completeness 47I rule prevents the conversionof one form of molecule to the other. This unsuspectedselection rule led to spectacularlyincorrect predictionsof the thermodynamic propertiesof H, gas.But when the selectionrule was at last recognized,the resolutionof the difficulty was straightforward.Ortho- and para-hydrogenwere simply consideredto be two distinct gases,and the single mole number of "hydrogen" was replaced by separatemole numbers. With the theory thus extended to include an additional con- servedcoordinate, theory and experimentwere fully reconciled. Interestingly,a different"operational" solution of the ortho-H2,para-H, problem was discovered.If a minute concentrationof oxygengas or water vapor is added to the hydrogengas the propertiesare drasticallychanged. The oxygen atoms are paramagnetic,they interact strongly with the nuclear spins of the hydrogenmolecules, and they destroy the symmetry that generatesthe selectionrule. In the presenceof a very few atoms of oxygen the ortho- and parahydrogenbecome interconvertible, and only a single mole number need be introduced. The original "rtaive" form of thermodynamicsthen becomesvalid. To return to the generalformalism, we thus recognizethat all symmetries must be taken into accountin specifuingthe releuantstate spaceof a system. As additional symmetriesare discoveredin physicsthe scopeof thermostatistics will expand. Perhaps all the symmetries of an ideal gas at standard temperaturesand pressuresare known, but the caseof ortho- and para-hydrogencautions modesty even in familiar cases.Moreover thermodynamicshas relevanceto quasars,and black holes, and neutron stars and quark matter and gluon gases.For each of thesethere will be random perturbations, and symmetry principles, conservationlaws, and Goldstoneexcitations,- and thereforethermostatistics.