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Statistical Mechanics in the Entropy Representation

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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 macro- scopic 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 maxi- mum 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 microstatesavaila- ble 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 mathemati- cal 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 loga- rithm 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
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