Statistical Mechanics in the Entropy Representation

STATISTICALMECHANICS STATISTICALMECHANICS IN THE ENTROPYREPRESENTATION: THE MICROCANON ICAL FORMALISM

I$1 PHYSICAL SIGNIFICANCE OF THE ENTROPY FOR CLOSED SYSTEMS

Thermodynamicsconstitutes a powerful formalismof great generality, erectedon a basis of a very few, very simple hypotheses.The central conceptintroduced through those hypotheses is the entropy.It entersthe formulation abstractly as the variational function in a mathematical extremum principle determiningequilibrium states.In the resultantfor- malism, however,the entropy is one of a set of extensiveparameters, togetherwith the energy,volume, mole numbersand magneticmoment. As these latter quantities each have clear and fundamentalphysical interpretationsit would be strangeindeed if the entropyalone were to be exemptfrom physicalinterpretation. The subjectof statisticalmechanics provides the physicalinterpretation of the entropy, and it accordinglyprovides a heuristicjustification for the Gxtremumprinciple of thermodynamics.For some simple systems,for which we have tractable models, this interpretation also permits explicit calculation of the entropy and thenceof the fundamentalequation. We focus first on a closedsystem of given volume and given number of particles.For definitenesswe may think of a fluid, but this is in no way necessary.The parametersU, V, and N are the only constraintson the system.Quantum mechanicstells us that, if the systemis macroscopic, there may exist many discretequantum statesconsistent with the specified values of U, V, and N. The systemmay be in any of thesepermissible states. Naively we might expect that the system,flnding itself in a particular quantum state,would remainforever in that state.Such, in fact, is the lore

329 330 Srutistical Mechanics in Entropy Representation

of elementaryquantum mechanics;the "quantum numbers" that specifya particular quantum state are ostensibly"constants of the motion." This naive fiction, relatively harmlessto the understandingof atomic systems (to which quantum mechanicsis most commonly applied) is flagrantly misleadingwhen applied to macroscopicsystems. The apparent paradox is seatedin the assumptionof isoration of a physical system.No physicalsystem is, or euercan be, truly isorated.There exist weak, long-range,random gravitational, electromagneticand other forces that permeate all physical space.These forces not only couple spatially separatedmaterial systems,but the force flelds themseluescon- stitute physical systemsin direct interaction with the systemof interest. The very vacuum is now understoodto be a complexfluctuating entity in which occur continual elaborateprocesses of crealion and reabsorptionof electrons,positrons, neutrinos,and a myriad of other esotericsu6atomic entities.All of theseevents can couplewith the systemof interest. . For a simple systgmsuch as g hydrogen atom in spacethe vqJy weak interactions to which we have alluded seldominduce transitionslbetween quantum states.This is so becausethe quantum statesof the hydrogen atom are w_idelyspaced in energy,and the weak random fields in spice cannot easily transfer such large energydifferences to or from the aiom. Even so, such interegtionsoccassionally do occur. An excited atom may "spontaneously"emit a photon, decayingto a lower energystate. euan- tum field theory revealsthat such ostensibly "spontaneous" transiiions actually are inducedby the interactionsbetween the excitedatom and the modesof the vacuum.The quantumstates of atoms arenot infinitelylong lived, preciselybecause of their interaction with the random modesof thE vacuum. For a macroscopicsystem the energy differencesbetween successive quantum statesbecome minute. In a macroscopicassembly of atomseach energyeigenstate of_a singleatom "splits" into some1023 energy eigen- states of the assembly,so that the averageenergy difference-Letween successivestates is decreasedby a factor of - ro-23. The slightest random field, or the weakestcoupling to vacuum fluctuations, iJ then sufficient to buffet the systemchaotically from quantum state [o quanrum state. A realistic uiewof a macroscopicsystem is onein whichthe systemmakes enormouslyrapid random transitionsamong its quantumstates. A macroscopic measurementsenses only an auerageof the propertiesof myriads of quantumstates. All "statistical mechanicians"agree with the precedingparagraph, but n_ot.allwould agreeon the dominant mechanismfor inducing tiansitions. various mechanismscompete and others may well dominate in some or even in all systems.No matter-it is sufficientthat any mechanismexists, and it is only the conclusionof rapid,random transitions that is neededto validate statistical mechanicaltheory. Physical Significance of the Entropy lor Closed Systems 331

Becausethe transitions are induced by purely random processes,it is reasonableto supposethat a macroscopicsystem samples euery permissible quantumstate with equalprobability-a permissiblequantum state being one consistentwith the externalconstraints. The assumptionof equalprobability of all permissiblemicrostates is the fundamental postulate of statistical mechanics.Its justification will be exarninedmore deeplyin Part III, but for now we adopt it on two bases; its a priori reasonableness,and the successof the theory that flows from it. Supposenow that some external constraint is removed-such as the opening of a valve permitting the systemto expandinto a larger volume. From the microphysical point of view the removal of the constraint activates the possibility of many microstatesthat previously had been precluded.Transitions occur into thesenewly availablestates. After some time the systemwill have lost all distinction betweenthe original and the newly available states, and the system will thenceforth make random transitions that samplethe augmentedset of stateswith equalprobability. The number of microstatesamong which the systemundergoes ffansitions, and which therebyshare unifurm probability of occupation,inueases to the maximumpermitted by the imposedconstraints. This statement is strikingly reminiscent of the entropy postulate of thermodynamics,according to which the entropy increasesto the maximum permitted by the imposedconstraints. It suggeststhat the entropy can be identified with the number of microstatesconsistent with the imposed macroscopicconstraints. One difficulty arises: The entropy is additive (extensive),whereas the number of microstatesis multiplicative.The number of microstatesavailable to two systemsis the product of the numbers available to each (the number of "microstates"of two dice is 6 x 6: 36). To interpret the entropy, then, we require an additiuequantity that measuresthe number of microstatesavailable to a system.The (unique!) answeris to identifythe entropy with the logarithm of the number of auailable miuostates (the logarithm of a product being the sum of the logarithms).Thus

S: krlnO (1s.1) where Q is the number of microstatesconsistent with the macroscopic constraints.The constantprefactor merely determinesthe scaleof s; ii is chosento obtain agreementwith the Kelvin scaleof temperature,defined by f-t : dSl|U.We shallsee that this agreementis achievedby takingthe constantto be Boltzmann'sconstant kB: R/NA: 1.3807x 70-23J/K. wilh the definitionT5-l the basisof statisticalmechanics is established. Just as the thermodynamic postulates were elaborated through the formalism of Legendretransformations, so this singleadditional postulate will be renderedmore powerful by an analogousstructure of mathematical formalism. Neverthelessthis singlepostulate is dramatic in its brevity, 332 Statisticttl Mechanics in Entropy Representation simplicity, and completeness.The statistical mechanical formalism that d.erives directly from it is one in which we "simply" calculate the logarithm of the number of statesavailable to the system, thereby obtainin; s as a function of the constraints u, v, and N. That is, ii is statistical mechanics in the entropy representation, or, in the parlance of the field, it is statistical mechanics in the microcanonical formalism. - In the following seclions of this chapter we treat a number of systems by this microcanonical formalism as examplesof its logical compleieness. As in thermodynamics, the entropy representation is not ahvays the most convenient representation.For statistical mechanical calculations it is frequently so inconvenient that it is analytically intractable. The I.egendre transformed representationsare usually far preferable, and we shall turn to them in the next chapter. Neverthelessthe microcanonical formulation establishesthe clear and basic logical foundation of statistical mechanics.

PROBLEMS

15.1-1.A system is composedof two harmonic oscillatorseach of natural frequencyoro and eachhaving permissible energies (n + !)hao, wheren is any non-negativeinteger. The total energyof the systemis E, : n,hoss,where r,is a positiveinteger. How many microstatesare availableto the system?what is the entropy of the system? A secondsystem is alsocomposed of two harmonicoscillators, each of natural frequency2,0. The total energyof this systemis 8,,: n,,htis,where n,,is an even integer.How many microstatesare availableto this system?what is the entropy of this system? what is the entropyof the systemcomposed of the two precedingsubsystems (separatedand enclosedby a totally restrictivewall)? Expressthe entropy as a function of E' and E".

Answer: S,",:krln[ #\ \ zn-a6 I l5.l-2. A system is composed of two harmonic oscillators of natural frequencies -l cro and 2<,r0,respectively. If the systemhas total energy n : @ !)hao, where n is an odd integer, what is the entropy of the system? lf a composite system is composed of two non-interacting subsystemsof the type just described, having energies E, and Er, what is the entropy of the composite system? The Einstein Model of a Crvstalline Solid 333

I5.2 THE EINSTEIN MODEL OF A CRYSTALLINE SOLID

With a identification of the meaning of the entropy we proceed to calculate the fundamental equation of macroscopic systems.We first apply the method to Einstein'ssimplified model of a nonmetallic crystalline solid. It is well to pause immediately and to comment on so early an introduction of a specific model system.In the eleven chapters of this book devoted to thermodynamic theory there were few referencesto specific model systems,and those occasionalreferences were kept care- fully distinct from the logical flow of the generaltheory. In statisticalme- chanicswe almost immediatelyintroduce a model system,and this will be followed by a considerablenumber of others.The differenceis partially a matter of convention. To some extent it reflects the simplicity of the general formalism of statisticalmechanics, which merely adds the logical interpretation of the entropy to the formalism of thermodynamics;the interest thereforeshifts to applicationsof that formalism, which underlies the various material sciences(such as solid state physics, the theory of liquids, polymerphysics, and the like). But, mostimportant, it reflectsthe fact that counting the number of states available to physical systems requires computational skills and experiencethat can be developedonly by explicit application to concreteproblems. To account for the thermal properties of crystals,Albert Einstein, in 1907,introduced a highly idealizedmodel focusingonly on the vibrational modes of the crystal. Electronic excitations,nuclear modes,and various other types of excitations were ignored. Nevertheless,for temperatures that are neither very closeto absolutezero nor very high, the model is at least qualitatively successful. Einstein'smodel consistsof the assumptionthat eachof the .fr atoms in the crystal can be consideredto be bound to its equilibrium position by a harmonic force. Each atom is free to vibrate around its equilibrium position in any of the threecoordinate directions, with a natural frequency o)o. More realistically(recall Section1.2) the atomsof crystalsare harmoni- cally bound to their neighboring atoms rather than to fixed points. Accordingly the vibrational modesare strongly coupled,grving rise to 3N collective normal modes.The frequenciesare distributed from zero (for very long wave length modes)to somemaximum frequency(for the modes of minimum permissiblewave length, comparableto the interatomic distance).There are f.ar more high frequencymodes than low frequency modes,with the consequencethat the frequenciestend to clustermainly in a narrow range of frequencies,to which the Einstein frequency c^rois a rough approximation. 334 Skttistical Mechanics in Entropy Representation

In the Einstein model, then, a crystal of fr atoms is replacedby 3lV harmonic oscillators,all with the samenatural frequencyoo. For the presentpurposes it is convenientto choosethe zeroof energyso that the energyof a harmonic oscillator of natural frequencyoo can iake only the discjetevalues nh@s, with n : 0,1,2,3,... . Hereh : h/2n : 1.055x 10-34J-s., h beingPlanck's constant. In the languageof quantum mechanics,each oscillator can be o.occupied by an integral number of energyquanta," eachof energyhao. The number of possiblestates of the system,and hencethe entropy, can now be computed easily.If the energyof the systemis U it can be consideredas constitutingU/hao quanta.These quanta are to be distributed among 3fr vibrational'modes.The numberbf *ays of distributing the U/hao quanta among the 3fr modes is the nrrmb"r of states d available to the system. The problem is isomorphicto the calculationof the numberof waysof placing u/hoo identical (indistinguishable)marbles in 3,& numbered (distinguishable)boxes. 1l flfl fl fl |] o o o [loo l]ll oll o.....I ooo lJoo FIGURE 15.1 Illustrating the combinatorial problem of distributing U/ho4 indistinguishableobjects ("marbles") in 3fr distinguishable"boxes." The combinatorialproblem can be visualizedas follows. Supposewe have U/hao marblesand 3N - 1 match sticks.We lay theseout in a linear array, in any order. One such array is shown in Fig. 15.1. The interpretation of this array is that three quanta (marbles)are assignedto the flrst mode, two quantato the second,none to the third, and so forth, and two quanta are assignedto the last mode (the 3,&-th)._Thusthe numberof waysof distributingthe u/_haoquanta among the 3N modesis the number of permutationsof (3N - 1 + U/ha) objects,of which u/hao areidentical (marbles or quanta),and 3N - 1 areidentical (match sticks).That is

(zw- t + u/n")t. (zx _ --:------+ u/no:)t l,n : -_ (rs.z) (3iv- r)t(u7har)! (3tv)!(u/ha)t

This completesthe calculation,for the entropy is simply the logarithm of this quantity (multiplied by kr). To simplify the result we employ the Stirling approximation for the logarithm of the factorial of a large number

In\Ml)= MlnM - M + .'. (ifM>> 1) (15.3) The Eiwtein Model of a Crystalline Sotid 335

whencethe molar entropy is

s: 3Rrn(r+ - 3R-!rh(l. (15.4) t) +)

uo = 3N,thoso (15.5)

This is the fundamentalequation of the system. It will be left to the problems to show that the fundamentalequation implies reasonablethermal behavior. The molar heat capacity is zero at zero temperature, rises rapidly with increasing temperature, and ap- proachesa constant value (3R) at high temperature, in qualitative agreementwith experiment.The rate of increaseof the heat capacityis not quantitatively correct becauseof the naivet6 of the model of the vibrational modes.This will be improved subsequentlyin the "Debye model" (Section 16.7), in which the vibrational modes are treated more realistically. The heat capacity of the Einstein model is plotted in Fig. 15.2. The molar heat capacity c, is zero at T : 0, and it asymptotesto 3R at high temperature.The rise in cuoccurs in the region krT = ihor(in particular c,l3R:i andthe point of maximumslope both occurnear krTlhao= i). At low temperaturec, risesexponentially, whereas experimentally the heat capacityrises approximately as T3. The mechanicalimplications of the model-the pressure-volumerela- tionship and compressibility-are completelyunreasonable. The entropy, according to equation 15.5, is independentof the volume, whence the pressureTdSldV is identically zero! Such a nonphysicalresult is, of course, a reflection of the naive omission of volume dependenteffects from the model. Certain consequencesof the model give important general insights. Consider the thermal equation of state ?N \ L: g :\nh + (15.6) 1 du h,,o t fiNnh"o1

Now, noting that there are 3NN, oscillatorsin the system

U hro mean energyper oscillator : (15.7) 3NNA ,hoo/keT - 1

The quantity hao/k, is called the "Einstein temperature"of the crystal, and it generally is of the same order of magnitude as the melting 336 Statistical Mechanics in Entopy Representation

hpT

n@o t.o 1.s z 2.5 3 3.5 4.5

0.9

o.6 -/ f o.s 3n"1, oA , o.3 o.2 I

0.1

o I o o.2 n? OA o.5 ie-hnT FIGURE T5.2 Heat capacity of the Einstein model, or of a single harmonic oscillator.The upper curve refersto the upper scaleof ftsTlhas, and the lower curve to the lower (expanded)scale. The ordinate can be interpretedas the heat capacityof one harmonic oscillitor in units of /

PROBLEMS

15.2-1.calculate the molar heat capacity of the Einsteinmodel by equation15.7. Show that the molar heat capacity approaches3R at high temperatures.Show that the temperaturedependence of the molar heat capacity is exponentialnear zero temperature,and calculatethe leadingexponential term. 15.2-2. obtain an equation for the mean quantum number n of at Einstein oscillator as a function of the temperature. Calculate D for k"T/ho:o: 0,1,2,3,4,10,50,100 (ignorethe physicalreality of meltingof the crystal!). TheTwo-stateSystem 337

15.2-3. Assume that the Einstein frequencycoo for a particular crystal depends upon the molar volume: oo:oB - rt(i) a) Calculate the isothermal compressibility of this crystal. b') Calculate the heat transfer if a crystal (of one mole) is compressed at constant temperature from u, to ur.

I5.3 THE TWO-STATE SYSTEM

Anottrtr model that illustratesthe principles of statisticalmechanics in a simple and transparentfashion is the " two-statemodel." In this model each"atom" can be eitherin its "ground state"(with energyzero) or in its "excited state" (with energye). To avoid conflict with certain generaltheorems about energyspectra we assumethat eachatom has additional states,but all of suchhigh energyas to exceedthe total energyof the systemunder consideration.Such states are then inaccessibleto the systemand need not be consideredfurther in the calculation. If U is the energyof the systemthenU/e atomsare in the excitedstate and (N - U/e) atoms are in the ground state.The number of ways of choosing U/e atomsfrom the total number fr is

(15.8)

The entropy is therefore - - - s: karno:karn(ivr) r,rn(#') r"r"[(r #)'] (15.e) or, invoking Stirling'sapproximation (equation 15.3)

s: (#- - _lr"nfi (1s.10) ")r,rn(r# Again, becauseof the artificiality of the model, the fundamentalequation is independentof the volume.The thermal equation of stateis easily calculatedto be +:?"(f-') (1s.11) 338 Stutistical Mechanics in Entropy Represmtation

Recalling that the calculation is subject to the condition U < .&e, we observe that the temperatureis a properly positive number. Solving for the energy

(15.12)

The energy approachesIiIe/2 as the temperatureapproaches infinity in this model (although we must recall that additional statesof high energy would alter the high temperatureproperties). At infinite temperaturehalf the atoms are excitedand half are in their ground state. The rnolar heat capacityis t,

du e2 ,elkpT ' : N.tEP : *^#(eet2kar + e-enkBr)-2 aT: (l * e"rzk"r1z

(15.13)

A graph of this temperaturedependence is shown in Fig 15.3.The molar heat capacity is zero both at very low temperaturesand at very high

I o.4

o.3 I i I I c Ne, O.2 I

o.l I

oL I I o 01 0.2 03 0.4 0.5 0.6 07 0.8 0.9 1.O 1.2 h"T t+ FIGURE 15.3 Heat capacity of the two-state model; the "Schottky hump." i A Polymer Model-The Rubber Band Reuisited 339 temperatures,peaking in the regionof kBT : .42e.This behavioris known as a "Schottky hump." Such a maximum, when observedin empirical data, is taken as an indication of a pafuof low lying energystates, with all other energy states lying at considerably higher energies.This is an example of the way in which thermal properties can reveal information about the atomic structureof materials.

PROBLEMS t 15.3-1. In the two-statemodel systemof this section supposethe excited state energy e of an "atom" dependson its averagedistance from its neighboring atoms so that

a V e:- n: where a and y are positive constants.This assumption,applied to a somewhat more sophisticatedmodel of a solid, was introduced by Gruneisen,and y is the "Gruneisen parameter."Calculate the pressureP as a function of A and T.

Answer: p : *

I5.4 A POLYMER MODEL-THE RUBBER BAND REVISITED

There exists another model of appealingsimplicity that is euphemisti- cally referred to as a "polymer model." Its connectionwith a real polymer is tenuous, but that connection is perhaps close enough to serve the pedagogicalpurpose of providing some senseof physical reality while again illustrating the basic algorithm of statistical mechanics.And in particular the model provides an insight to the behavior of a "rubber band," as discussedon purely phenomenologicalgrounds in Section3.7. As we saw in that section the extensiveparameter of interest, which replacesthe volume, is the length; the correspondingintensive parameter, analogousto the pressure,is the tensiore.We are interestedin the equation of state relating tensionto length and temperature. The "rubber band" can be visualized as a bundle of loqg chain polymers. Each polymer chain is consideredto be composedof fr monomer units each of length a, and we focus our attention on one particular polymer chain in the bundle. One end of the polymer chain is fixed at a point that is taken as the origin of coordinates.The other end of the chain is subject to an externally applied tension V, parallel to the positive x-axis(Fig. 15.4). 340 Statistical Mechanics in Entropy Representation

wei*ht(=7) FrouRE15 4 '?olymer" model. The string should be much longer than shown, so that the end of th polymer is free to movein the y-direction, and the applied tension .Z is directedalong tl x-direction.

In the polymer model eachmonomer unit of the chain is permittedto lie either parallel or antiparallelto the x-axis,and zero energyis associ. ated with thesetwo orientations.Each monomerunit has the additiona

monomer. A somewhatmore reasonablemodel of the polymer might permit the perpendicularmonomers to lie along the xz directionsas well as alonl the ty directions,and, more importantly, would accountfor the inter- ferenceof a chain doubling back on itself. such modelscomplicate thr analysiswithout addingto the pedagogicclarity or qualitativecontent of the result. We calculatethe entropyS of one polymer chain as a function of the energyU = U'e, of the coordinatesL, and L, of the end of the polyme, chain, and of the numberN of monomerunit's in the chain. Let N,+ and N; be the numbersof monomersalong the * x and - x directions respectively,and similarly for Nu+and N,-. Then

NJ +N;*Nr*+N;:ii/

I Nt* _N; -"x - r, a^

L,, N; -N;:;=r;

Nu* +N;:I=U' (7s.r4) A Polymer Model-The Rubber Band Reuisited 341 from which we find Nl:+(iV-U'+Lt) N;:+(lv-u,-L1) N;:i(u,+L;)

N; :i(u,- L;) (ts.rs)

The number of configurationsof the polymer consistent with -given coordinates Z" and L, of its terminus,and with given energyU, is

(1s.16)

The entropy is, then, using the stirling approximation(equation 15.3)

S: ftrlnO: fr/calnU - W;*"lnN,* - N;kBlnN;

-Nr*k"lnN; - N, krlnN, (tS.tZ) or ^s: frkrhfrr_ +(jf_ rJ,+ L)kBtnl+(frr_ u, + a,)l - +(iV- (J,- L!,)karnli(fr- u, - L,,)l - i(u' + L;)kBrnIL(u'+ 4)l -i(u' - L;)kBtnlL(u,- z;)l (15.18)

L', 2::+:\^"rnu'- -o (15.1e) T 0L, 2a"^ U, + L; from which we conclude(as expected)that

Lv: L;: o (15.20) 342 Sutistical Mechanics in Entropy Representstion

Similarlv

_.%_as_kBhN-u'-L'*4-as:&,.N- (15.21) T AL, 2a"'frt- U'+ L'. and +:#: fi'4n*''. - u')+ finQr- r,,- u')- brnv (ls.22\ or - - (u u')'- L': ,2e/keT (ls.23) (J',2

This is the "thermal equationof state." The "mechanicalequation of state" (15.21)can be written in an analogousexponential form

N _ III _ I' o-2{,a/ksT:'= " ", (rs.24) N _ U'* ''

The two preceding equationsare the equations of state in the entropy representation,and accordingly they involve the energy U'. That is not generally convenient.We proceed,then, to eliminate U'between the two equations.With somealgebra we find (seeProblem 15.4-2)that

LI sinh(.%a/k"T) _ (1s.25) N cosh(.{,a/k"T) + "-c/kpr For small {a (relativeto krT) the equationcan be expandedto first order , t,:, 4fro' -r-" ' (ts.z6) W 1+Z-;iE;t

The modulus of elasticity of the rubber band (the analogue of the -[/V(AV/aP)r) compressibility is, for small {

+e-a*"r -'l (rs.27) #W),: #(L Counting Techniquesand their Circumoention; High Dimensionality 343

The fact that this elastic modulus deueases with increasing temperature (or that the "stiffness" increases)is in dramatic contrast to the behavior of a spring or of a stretched wire. The behavior of the polymer is sometimes compared to the behavior of a snake; if we grasp a snake by the head and tail and attempt to stretch it straight the resistance is attributable to the writhing activity of the snake. The snake, in its writhing, assumes all possible configurations, and more configurations are accessibleif the two ends are not greatly distant from each other. At low temperatures the rubber band is like a torpid snake. At high temperatures the number of configurations available, and the rate of transitions among them, is greater, resulting in a greater contractive tension. It is the entopy of the snake and of the rubber band that is responsible for the tendency of the endp to draw together! The behavior described is qualitatively similar to that of the simple phenomenological model of Section 3.7. But compared to a truly realistic model of a rubber band, both models are extremelv naive.

PROBLEMS

15.4-1.Is the signcorrect in equation15.19? Explain. 15.4-2. Eliminate U/e between equations 15.23 and 75.24 and show that the formal solution is equation 15.25 with a + sign before the secondterm in the denominator.Consider the qualitativedependence of L,/Fta on e, and show that physical reasoningrejects the negativesign in the denominator,thus validating equation15.25. 15.4-3. A rubber band consistingof n polymer chains is stretchedfrom zero length to its full extension(L : Na) at constanttemperature Z. Does the energy of the systemincrease or decrease?Calculate the work done on the systemand the heat transfer to the system. 15.4-4. Calculate the heat capacity at constant length for a "rubber band" consistingof n polymer chains.Express the answerin terms of T and,L,. 15.4-5. Calculate the "coefficient of longitudinal thermal expansion" defined by K7=r\.., - 7 [aLu\ n ). Express rc', as a function of Z and sketch the qualitative behavior. Compare this with the behavior of a metallic wire and discuss the result.

I5.5 COUNTING TECHNIQUES AND THEIR CIRCUMVENTION; HIGH DIMENSIONALITY

To repeat, the basic algorithm of statistical mechanics consists of counting the number of statesconsistent with the constraintsimposed; the 344 Statistical Mechanics in Entropy Representation

entropy is then the product of Boltzmann'sconstant and the logarithm of the permissiblenumber of states. unfortunately counting problems tend to require difficult and sophisti- cated techniquesof combinatorial mathematics(if they can be done at all!) In fact only a few highly artificial, idealized models permit explicit solution of the counting problem, even with the full armamentariumof combinatorial theory. If statistical mechanics is to be a useful and practical science it is necessarythat the difficulties of the counting problem somehowbe circumvented.one method of simplifying the counting problem is developedin this section. It is based on certain rather startlingproperties of systemsof "high dimensionality"-a conceptto be defined shortly. The method is admittedly more important for the insights it plovides to the behaviorof complexsystems than for the aid it provides in practical calculations. More general and powerful methods of circumventing the countingproblem are basedon a transfer from thermodynamics to statistical mechanicsof the techniqueof Legendretransfor- mations.That transferwill be developedin the followingchapters. For now we turn our attention to the simplifying effects of high dimensionality,a concept that can best be introduced in terms of an explicit model. we choosethe simplestmodel with which we are already familiar-the Einsteinmodel. Recall that the Einstein solid is a collection of f atoms.each of which is to be associatedwith three harmonic oscillators(corresponding to the oscillationsof the atom along the x, y, and z axes).A quantumstate of the systemis specifiedby the 3N quantumnumbers fl1,/t2tn3,...tn3fi; and the energyof the systemis zr't U(rr, fl2,..., nzn): \ nrhoso (1s.28) i-1

Each such state can be representedby a " point," with coordinates nylt2tfl3,...,ft3;1,itt a 3N-dimensional"state space." Only pointswith positive integral coordinatesare permissible,corresponding to the dis- cretenessor "quantization" of statesin quantum mechanics.It is to be stressedthat a single point representsthe quantum state of the entire crystal. The locus of stateswith a given energy U is a "diagonal" hyperplane with interceptsU/ho:o on eachof the 3N coordinateaxes (Fig. 15.5).All stateslying "inside" the plane(i.e., closer to the origin) haveenergies less than U, and all stateslying outsidethe plane, further from the origin, have energies greater than U. The first critical observationwhich is called to our attention by Fig. 15.5 is that an arbitrary "diagonalplane," correspondingto an arbitrary energy U, will generally pass through none of the discrete coordinate points in the space!That is, an arbitrarily selectednumber U generally Counting Techniquesand their Circumuention; High Dimensionality 345

ntx FIGURE 15.5 Quantum state spacefor the Einsteinsolid. The three-dimensionalstate space shown is for an Einstein solid composedof a single atom. Each additional atom would increasethe dimensionality of the spaceby three.The hyperplaneU has interceptsU/hoh on all axes. There is one state for eachunit of hypervolume,and (neglectingsurface corrections) the number of states with energy less than U is equal to the volume inside the diagonal hyperplane U. cannot be representedin the form of equation 15.28,such a decomposi- tion being possibleonly if U/hao is an integer. More generally,if we inquire as to the number of quantum statesof a system with an arbitrarily chosenand mathematicallyprecise energy, we almost always find zero. But such a question is unphysical.As we have stressedpreviously the random interactions of every system with its environment make the energy slightly imprecise.Furthermore we never know (and cannot measure) the energy of any system with absolute precision. The entropyis not the logarithmof the numberof quantumstates that lie on the diagonalhyperplane U of Fig. 15.5, but rather it is the logarithm of the number of quantumstates that lie in the closeuicinity of the diagonal hyperplane. This considerationleads us to study the number of statesbetween two hyperplanes:[/ and U - L. The energyseparation A is determinedby the imprecision of the energy of the macroscopicsystem. That imprecision may be thought of as a consequenceeither of environmentalinteractions or of imprecision in the preparation(measurement) of the system. The remarkableconsequence of high dimensionalityis that the uolume betweenthe twoplanes (U - A and U), and hencethe entropy,is essentially independentof the separationA, of theplanesl 346 Stufistical Mechanics in Entropy Reprcsentation

This result is (at first) so startlingly counter-intuitive,and so fundamental, that it warrants careful analysisand discussion.We shall first corrobo- rate the assertion on the basis of the geometricalrepresentation of the states of the Einstein solid. Then we shall reexamine the geometrical representationto obtain a heuristicunderstanding of the generalgeometrical basis of the effect. The number of states01U; wittr energiesless than (or equal to) a given value U is equal to the hypervolumelying "inside" the diagonal hyperplane U. This hypervolumeis (seeproblem 15.5-1)

0(U) : (numberof stateswith energiesless than U) , :oh'(#)'. (7s.2e) The fact that this result is proportional to (J3N,where 3lf is the dimensionality of the "state space," is the critical feature of this result. The precise form of the coefficientin equation 15.29will prove to be of only secondaryimportance. By subtraction we find the number of states with energiesbetween U-LandUtobe - - o(u)o(u- A): #(#)'" #(#)* or

o(u)- 0(u-a): otul[r-(r (1s.30)

But (1 -^!/U) is lessthan unity; raising this quantity to an exponent 3N = 10" resultsin a totally negligiblequantity (seeproblem 15.5-2),so that o(u):0(u)- 0(u-A) = 0(u) (rs.:r)

That is, the number A(U) of stateswith_energies between U - L and U is essentially equal to the total number 0(u) of states with energiesless than U-and this result is essentiallyindependent of A! Thus having corroboratedthe assertionfor our particular model, let us reexaminethe geometryto discern the more generalgeometrical roots of this strange,but enormouslyuseful, result. The physicalvolume in Fig. 15.5can'be looked at as one eighth of a regular octahedron(but only the portion of the octahedronin the physical Counting Techniquesand their Circumuention; High Dimensionality 347

octant of the spacehas physicalmeaning). With higher dimensionalitythe regular polyhedron would becomemore nearly "spherical." The dimen- sionlessenergy U/hao is analogousto the "radius" of the figure, being the distancefrom the origin to any of the cornersof the polyhedron.This viewpoint makes evident the fact (equation 15-29) that the volume is proportional,to the radius raisedto a power equal to the dimensionalityof the space(r2 in two dimensions,13 in three,etc.). The volumebetween two concentric polyhedra, with a difference in radii of dr, is dV: (0V/ 0r) dr. The ratio of the volumeof this "shell" to the total volumeis

dV 0V dr V: a,v (15.32) , or,if V: Anr'

dV dr (1s.33) V:N;

If we take n: l0z3 we find dV/V = 0.1 only if. dr/r - 10-24.For dr/r greater than - L0-'* the equation fails, telling us that the use of differentials is no longer valid. The failure of the differential analysisis evidencethat dV/V already-becomeson the order of unity for valuesof dr/r as small as dr/r = 10-23. In an imaginary world of high dimedsionality there would be an automatic and perpetual potato famine, for the skin of a potato would occupy essentiallyits entire volume! In the real world in which three-dimensionalstatistical mechanicians calculate entropies as volumes in many-dimensionalstate spaces,the propertiesof high dimensionalityare a blessing.We neednot calculatethe number of states"in the vicinity of the systemenergy (J"-it is quite as satisfactory,and frequently easier,to calculatethe number of stateswith energiesless than or equal to the energyof the physical system. Returning to the Einstein solid, we can calculate the fundamental equation using the result 15.29 for 01U;, the number of stateswith energiesless than U; the entropy is S: kalrl'0((l), and it is easily corroborated that this gives the sameresult as was obtained in equation t5.4. The two methodsthat we haueused to soluethe Einsteinmodel of a solid should be clearly distinguished.In Section I5.2 we assumedthat IJ/ho:o was an integer, and we countedthe number of waysof distributing quanta among the modes.This was a combinatorialproblem, albeit a simple and tractable one becauseof the extremesimplicity of the model. The second method, in this section,involved no combinatorialcalculation whatsoever. Instead we defineda volume in an abstract"state space"and the entropy was related to the total volume inside the bounding surfacedefined by the 348 Stutistical Mechanics in Enrropy Representation

energy u. The combinatorial approachis not easily transferableto more complicated systems-the method of hypervolumesis general and is usuallymore tractable.However the last methodis not applicableat very low temperaturewhere only a few statesare occupied,and where the occupied volumein statespace shrinks toward zero. PROBLEMS

15.5'1.To establishequation 75.29 let o, be the hypervolumesubtended by the diagonalhyperplane in n dimensions.Draw appropriatefigures for n : I,2, and. 3 and showthat if Z is theintercept on eachof the coordinateaxes

Qt: L o,: o,/'(r- t)*:# o,: or/'(r-;)'o*:# and by mathematicalinduction

o,_ o,_,/.(r-;)^-'a,:4

15.5-2. Recalling that

+ x)L/':, (= lyr(t 2.718...) show that

=s-NU (t-+) ro,$.. t

with this approximationdiscuss the accuracyof equation 15.31for a rangeof reasonablevalues of A,/U (rangingperhaps from 10-3 to l0-r0). with what precision a,/u would the energyhave to be known in order that correctio-nsto equation 15.31might becomesignificant? Assume a systemwith .& = 1023. 15.5-3. calculate the fraction of the hypervolumebetween the radii 0.9r and r for hyperspheresin L, 2, 3,4, and 5 dimensions.Similarly for 10, 30, and 50 dimensions.