arXiv:cond-mat/9708200v2 [cond-mat.soft] 28 Jan 1999THEPHYSICSANDMATHEMATICSOFTHESECONDLAWOFTHERMODYNAMICSElliottH.LiebDepartmentsofPhysicsandMathematics,PrincetonUniversityJadwinHall,P.O.Box708,Princeton,NJ08544, USAJakobYngvasonInstitutf urTheoretischePhysik,Universit atWien,Boltzmanngasse5,A1090Vienna,AustriaAbstract:The essential postulates of classical thermodynamics are formulated, from whichthesecondlawisdeducedastheprincipleofincreaseofentropyinirreversibleadiabaticprocessesthat take one equilibriumstate to another. The entropyconstructedhere is denedonlyforequilibriumstatesandnoattemptismadetodeneitotherwise. Statistical mechanicsdoesnotenter these considerations. One of the main concepts that makes everything work is the comparisonprinciple(which,inessence,statesthatgivenanytwostatesofthesamechemicalcompositionatleastoneisadiabaticallyaccessiblefromtheother)andweshowthatitcanbederivedfromsomeassumptionsaboutthepressureandthermal equilibrium. Temperatureisderivedfromentropy,butat thestartnoteventheconcept of hotness isassumed. Ourformulationoersacertainclarityandrigorthatgoesbeyondmosttextbookdiscussionsofthesecondlaw.1998PACS: 05.70.-aMathematicalSciencesClassication(MSC)1991and2000: 80A05,80A10ThispaperisscheduledtoappearinPhysicsReports310,1-96(1999)WorkpartiallysupportedbyU.S. National ScienceFoundationgrantPHY95-13072A01.WorkpartiallysupportedbytheAdalsteinnKristjanssonFoundation, Universityof Iceland.c _1997bytheauthors. Reproductionof thisarticle, byanymeans, ispermittedfornon-commercial purposes.1I.INTRODUCTIONA. ThebasicQuestions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3B. Otherapproaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6C. Outlineofthepaper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10D. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11II.ADIABATICACCESSIBILITYANDCONSTRUCTIONOFENTROPYA. Basicconcepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121. Systemsandtheirstatespaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132. Theorderrelation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16B. Theentropyprinciple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18C. Assumptionsabouttheorderrelation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20D. Theconstructionofentropyforasinglesystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23E. Constructionofauniversalentropyintheabsenceofmixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27F. Concavityofentropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .30G. IrreversibilityandCaratheodorysprinciple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32H. Somefurtherresultsonuniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33III. SIMPLESYSTEMSPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36A. Coordinatesforsimplesystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37B. Assumptionsaboutsimplesystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39C. Thegeometryofforwardsectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42IV.THERMALEQUILIBRIUMA. Assumptionsaboutthermalcontact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51B. Thecomparisonprincipleincompoundsystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551. Scaledproductsofasinglesimplesystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552. Productsofdierentsimplesystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56C. Theroleoftransversality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .59V.TEMPERATUREANDITSPROPERTIESA. Dierentiabilityofentropyandthedenitionoftemperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62B. Thegeometryofisothermsandadiabats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68C. Thermalequilibriumandtheuniquenessofentropy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69VI.MIXINGANDCHEMICALREACTIONSA. Thedicultyinxingentropyconstants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72B. Determinationofadditiveentropyconstants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73VII.SUMMARYANDCONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83LISTOFSYMBOLS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88INDEXOFTECHNICALTERMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89REFERENCES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 912I.INTRODUCTIONThe second law of thermodynamics is, without a doubt, one of the most perfect laws in physics.Anyreproducibleviolationofit,howeversmall,wouldbring thediscoverergreatrichesaswellasatriptoStockholm. Theworldsenergyproblemswouldbesolvedatonestroke. Itisnotpossibletondanyotherlaw(except, perhaps, forsuperselectionrulessuchaschargeconservation)forwhichaproposedviolationwouldbringmoreskepticismthanthisone. NotevenMaxwellslawsofelectricityorNewtonslawofgravitationaresosacrosanct,foreachhasmeasurablecorrectionscomingfromquantumeects or general relativity. Thelawhas caught the attentionof poetsandphilosophers andhasbeencalledthegreatestscienticachievementofthenineteenthcentury.Engels dislikedit, for it supportedoppositionto dialectical materialism, while Pope Pius XIIregardeditasprovingtheexistenceofahigherbeing(Bazarow,1964,Sect. 20).A.ThebasicquestionsIn this paper we shall attempt to formulate the essential elements of classical thermodynamicsof equilibrium states and deduce from them the second law as the principle of the increase of entropy.Classical meansthatthereisnomentionof statistical mechanicshereandequilibrium meansthat we deal only with states of systems in equilibrium and do not attempt to dene quantities suchas entropy and temperature for systems not in equilibrium. This is not to say that we are concernedonly with thermostatics because, as will be explained more fully later, arbitrarily violent processesareallowedtooccurinthepassagefromoneequilibriumstatetoanother.Moststudentsof physicsregardthesubjectasessentiallyperfectlyunderstoodandnished,andconcentrateinsteadonthestatisticalmechanicsfromwhichitostensiblycanbederived. Butmanywill admit, if pressed, thatthermodynamicsissomethingthattheyaresurethatsomeoneelseunderstands and theywillconfess tosomemisgivingabout thelogicofthe steps intraditionalpresentationsthatleadtotheformulationofanentropyfunction. Ifclassical thermodynamicsisthemostperfectphysical theoryitsurelydeservesasolid, unambiguousfoundationfreeof littlepicturesinvolvingunrealCarnotcyclesandthelike. [Forexamplesofun-ordinaryCarnotcyclessee(TruesdellandBharatha1977,p.48).]Therearetwoaims toour presentation. Oneis franklypedagogical, i.e., toformulatethefoundations ofthetheoryinaclearandunambiguous way. The secondistoformulateequilibriumthermodynamics as anideal physical theory, whichis tosayatheoryinwhichtherearewelldenedmathematical constructsandwell denedrulesfortranslatingphysical realityintotheseconstructs;havingdone sothe mathematicsthen grinds out whateveranswers itcanandthesearethentranslatedbackintophysical statements. Thepointhereisthatwhilephysical intuition isausefulguideforformulatingthemathematicalstructureandmayevenbeasourceofinspirationfor constructingmathematical proofs, it shouldnot be necessarytorelyonit once the initialtranslation intomathematical languagehasbeengiven. Thesegoalsarenotnew, of course; seee.g.,(Duistermaat,1968),(Giles,1964,Sect. 1.1)and(Serrin,1986,Sect. 1.1).Indeed, itseemstousthatmanyformulationsof thermodynamics, includingmosttextbookpresentations, suer frommixingthephysics withthemathematics. Physics refers totherealworldof experimentsandresultsof measurement, thelatter quantiedintheformof numbers.Mathematicsrefers toa logicalstructure and torules of calculation;usually theseare built aroundnumbers, butnotalways. Thus, mathematicshastwofunctions: oneistoprovideatransparentlogical structurewithwhichtoviewphysics andinspireexperiment. Theother is tobelikea3mill intowhichthe miller pours the grainof experiment andout of whichcomes the our ofveriablepredictions. Itisastonishingthatthis paradigmworkstoperfectioninthermodynamics.(AnothergoodexampleisNewtonianmechanics, inwhichtherelevantmathematical structureisthe calculus.)Our theory of the second law concerns the mathematical structure, primarily. As suchitstartswithsomeaxiomsandproceedswithrulesoflogictouncoversomenon-trivial theoremsabouttheexistenceof entropyandsomeof itsproperties. Wedo, however, explainhowphysicsleadsustotheseparticularaxiomsandweexplainthephysicalapplicabilityofthetheorems.As noted in I.C below, we have a total of 15 axioms, which might seem like a lot. We can assurethereaderthatanyothermathematical structurethatderivesentropywithminimalassumptionswillhaveatleastthatmany, andusuallymore. (Wecouldroll several axiomsintoone, asothersoftendo, byusingsub-headings, e.g., ourA1-A6mightperfectlywell bedenotedbyA1(i)-(vi).)Thepointisthatweleavenothingtotheimaginationortosilentagreement;itisalllaidout.It must alsobeemphasizedthat our desiretoclarifythestructureof classical equilibriumthermodynamics is not merely pedagogical and not merely nit-picking. If the law of entropy increaseisevergoingtobederivedfromstatistical mechanicsagoal thathassofareludedthedeepestthinkersthenitisimportanttobeabsolutelyclearaboutwhatitisthatonewantstoderive.Manyattemptshavebeenmadeinthelastcenturyandahalf toformulatethesecondlawprecisely and to quantify it by means of an entropy function. Three of these formulations are classic(Kestin, 1976), (see also Clausius (1850),Thomson (1849)) and they can be paraphrased as follows:Clausius: No process is possible, the sole result of which is that heat is transferred from a bodytoahotterone.Kelvin(andPlanck): Noprocessispossible,thesoleresultofwhichisthatabodyiscooledandworkisdone.Caratheodory: In anyneighborhood ofanystatetherearestatesthatcannot be reachedfromitbyanadiabaticprocess.Thecrowninggloryofthermodynamicsisthequanticationofthesestatementsbymeansofaprecise, measurablequantitycalledentropy. Therearetwokindsofproblems, however. Oneistogiveaprecisemeaningtothewordsabove. Whatisheat ? Whatishotandcold ? Whatisadiabatic ? Whatisaneighborhood ? Justabouttheonlywordthatisrelativelyunambiguousisworkbecauseitcomesfrommechanics.Thesecondsortofprobleminvolvestherulesof logicthatleadfromthesestatementstoanentropy. Is it really necessary to draw pictures, some of which are false, or at least not self evident?Whatareall thehiddenassumptionsthatenterthederivationof entropy? For instance, weallknowthat discontinuities cananddooccur at phasetransitions, butalmost everypresentationof classical thermodynamicsis basedonthedierential calculus (whichpresupposescontinuousderivatives),especially(Caratheodory,1925)and(Truesdell-Bharata,1977,p.xvii).Wenote, inpassing,thattheClausius,Kelvin-PlanckandCaratheodoryformulationsareallassertions about impossible processes. Our formulation will rely, instead, mainly on assertions aboutpossibleprocessesandthusisnoticeablydierent. AttheendofSectionVII,whereeverythingissuccintlysummarized, therelationshipoftheseapproachesisdiscussed. Thisdiscussionislefttotheendbecauseit it cannot bedonewithout rstpresentingour resultsinsomedetail. SomereadersmightwishtostartbyglancingatSectionVII.Of coursewe areneither therst nor, presumably, thelast topresent aderivationof thesecondlaw(inthesenseofanentropyprinciple)thatpretendstoremoveallconfusionand,atthesametime,toachieveanunparalleledprecisionoflogicandstructure. Indeed,suchattemptshave4multipliedinthepastthreeorfour decades. These othertheories,reviewedinSect. I.B,appealtotheir creators as much as ours does to us and we must therefore conclude that ultimately a questionoftasteisinvolved.It is not easy to classify other approaches to the problem that concerns us. We shall attempt todosobriey,but rst letus statetheproblem clearly. Physicalsystemshavecertainstates(whichalwaysmeanequilibriumstatesinthispaper)and, bymeansof certainactions, calledadiabaticprocesses, itispossibletochangethestateofasystemtosomeotherstate. (Warning: Thewordadiabatic isusedinseveral waysinphysics. Sometimesitmeansslowandgentle, whichmightconjureuptheideaof aquasi-staticprocess, butthisiscertainlynotourintention. Theusagewehaveinthebackof ourmindsiswithoutexchangeof heat, butweshall avoiddeningtheword heat. The operational meaning of adiabatic will be dened later on, but for now the readershould simply accept it as singling out a particular class of processes about which certain physicallyinterestingstatementsaregoingtobemade.) Adiabaticprocessesdonothavetobeverygentle,and they certainlydo not have to be describable by a curve in the space of equilibrium states. Oneisallowed,likethegorillainawell-knownadvertisementforluggage,tojumpupanddownonthesystemandevendismantleittemporarily, providedthesystemreturnstosomeequilibriumstateattheendof theday. Inthermodynamics, unlikemechanics, notall conceivabletransitionsareadiabatic and it is a nontrivial problem to characterize the allowed transitions. We shall characterizethem as transitions that have no net eect on other systems except that energy has been exchangedwithamechanical source. Thetrulyremarkablefact, whichhasmanyconsequences, isthatforeverysystem thereis afunction,S, onthe spaceof its(equilibrium) states,withtheproperty thatonecangoadiabaticallyfromastateXtoastateYifandonlyifS(X) S(Y ). This, inessence,istheentropyprinciple(EP)(seesubsectionII.B).TheSfunctioncanclearlybemultipliedbyanarbitraryconstant andstill continuetodoits job, andthus it is not at all obvious that thefunctionS1for system1has anythingtodowiththefunctionS2forsystem2. ThesecondremarkablefactisthattheSfunctionsforall thethermodynamicsystemsintheuniversecanbesimultaneouslycalibrated(i.e., themultiplicativeconstantscanbedetermined)insuchawaythattheentropiesareadditive, i.e., theSfunctionforacompoundsystemisobtainedmerelybyaddingtheSfunctionsof theindividual systems,S1,2=S1+ S2. (Compound doesnotmeanchemical compound; acompoundsystemisjustacollectionofseveralsystems.) Toappreciatethisfactitisnecessarytorecognizethatthesystemscomprisingacompoundsystemcaninteract witheachother inseveral ways, andthereforethepossible adiabatic transitions in a compound are far more numerous than those allowed for separate,isolatedsystems. Nevertheless, the increase of the functionS1+S2continues todescribe theadiabatic processes exactlyneither allowingmore nor allowingless thanactuallyoccur. ThestatementS1(X1) +S2(X2) S1(X1) +S2(X2)doesnotrequireS1(X1) S1(X1).Themainproblem, fromourpointof view, isthis: What propertiesof adiabaticprocessespermitustoconstructsuchafunction? Towhatextentisitunique? Andwhatpropertiesoftheinteractionsofdierentsystemsinacompoundsystemresultinadditiveentropyfunctions?Theexistenceof anentropyfunctioncanbediscussedinprinciple, asinSectionII, withoutparametrizing the equilibriumstates byquantities suchas energy, volume, etc.. But it is anadditional factthatwhenstates areparametrizedintheconventional waysthenthederivativesofSexistandcontainall theinformationabouttheequationofstate, e.g., thetemperatureTisdenedbyS(U, V )/U[V= 1/T.5Inourapproachtothesecondlawtemperatureisneverformallyinvokeduntil theveryendwhenthedierentiabilityofSisprovednoteventhemoreprimitiverelativenotionsofhotnessand coldness are used. The priority of entropy is common in statistical mechanics and in some otherapproachestothermodynamicssuchasin(Tisza,1966)and(Callen,1985),buttheeliminationofhotnessandcoldnessisnotusualinthermodynamics,astheformulationsofClausiusandKelvinshow. The laws of thermal equilibrium (Section V), in particular the zeroth law of thermodynamics,do play acrucial rolefor us by relatingone system toanother (andthey are ultimatelyresponsiblefor thefact that entropies canbeadjustedtobeadditive), but thermal equilibriumis onlyanequivalencerelationand, inourform, itisnotastatement abouthotness. Itseemstousthattemperatureisfarfrombeinganobviousphysicalquantity. Itemerges,nally,asaderivativeofentropy, andunlikequantitiesinmechanicsorelectromagnetism, suchasforcesandmasses, itisnotvectorial, i.e.,itcannotbeaddedormultipliedbyascalar. Evenpressure,whileitcannotbeaddedinanunambiguous way,canatleastbemultipliedbya scalar. (Here,wearenotspeakingaboutchangingatemperaturescale; wemeanthatonceascalehasbeenxed, itdoesnotmeanverymuchtomultiplyagiventemperature, e.g., theboilingpointof water, bythenumber17.Whatevermeaningonemightattachtothisissurelynotindependentofthechosenscale. Indeed,isTtherightvariableorisit1/T?Inrelativitytheorythisquestionhasledtoanongoingdebateabout the natural quantity to choose as the fourth component of a four-vector. On the other hand,it does meansomethingunambiguous, tomultiplythepressureintheboiler by17. Mechanicsdictatesthemeaning.)Anothermysteriousquantityisheat. Noonehasever seenheat, norwill itever beseen,smelled or touched. Clausius wrote about the kind of motion we call heat, but thermodynamicseitherpractical ortheoreticaldoes notrelyforitsvalidityonthenotionof moleculesjumpingaround. Thereisnowaytomeasureheatuxdirectly(otherthanbyitseectonthesourceandsink) and, while we do not wish to be considered antediluvian, it remains true that caloric accountsforphysicsatamacroscopiclevel justaswell asheat does. Thereaderwill ndnomentionofheatinourderivationofentropy,exceptasamnemonicguide.Toconcludethisverybrief outlineof themainconceptual points, theconcept of convexityhastobementioned. Itiswell known, asGibbs(Gibbs1928), Maxwell andothersemphasized,thatthermodynamicswithoutconvexfunctions(e.g.,freeenergyperunitvolumeasafunctionofdensity)mayleadtounstablesystems. (Agooddiscussionofconvexityisin(Wightman, 1979).)Despitethisfact,convexityisalmostinvisibleinmostfundamentalapproachestothesecondlaw.Inourtreatmentitisessential forthedescriptionofsimplesystemsinSectionIII,whicharethebuildingblocksofthermodynamics.The concepts and goals we have just enunciated will be discussed in more detail in the followingsections. ThereaderwhoimpatientlywantsaquicksurveyofourresultscanjumptoSectionVIIwhereitcanbefoundincapsuleform. Wealsodrawthereadersattentiontothearticle(Lieb-Yngvason1998),whereasummary ofthisworkappeared. Letus nowturntoabrief discussionofothermodesofthoughtaboutthequestionswehaveraised.B.OtherapproachesThesimplestsolutiontotheproblemofthefoundationofthermodynamicsisperhapsthatofTisza(1966), andexpandedbyCallen(1985)(seealso(Guggenheim, 1933)), who, followingthetraditionof Gibbs(1928), postulatetheexistenceof anadditiveentropyfunctionfromwhichall6equilibriumpropertiesofasubstancearethentobederived. Thisapproachhastheadvantageofbringingonequicklytotheapplicationsofthermodynamics,butitleavesunstatedsuchquestionsas: Whatphysicalassumptionsareneededinordertoinsuretheexistenceofsuchafunction?Bynomeansdowewishtominimizetheimportanceofthisapproach, forthemanifoldimplicationsofentropyarewell knowntobenon-trivial andhighlyimportanttheoreticallyandpractically, asGibbswasoneofthersttoshowindetailinhisgreatwork(Gibbs,1928).Amongthemanyfoundational worksontheexistenceof entropy, themostrelevantforourconsiderations and aims here are those that we might, for want of a better word, call order theoret-ical becausetheemphasisisonthederivationofentropyfrompostulatedpropertiesofadiabaticprocesses. Thislineof thoughtgoesbacktoCaratheodory(1909and1925), althoughtherearesome precursors (see Planck, 1926) and was particularly advocated by (Born, 1921 and 1964). Thisbasicidea, if notCaratheodorys implementationof it withdierential forms, was developedinvariousmutationsintheworksofLandsberg(956), Buchdahl(1958, 1960, 1962, 1966), BuchdahlandGreve(1962), FalkandJung(1959), Bernstein(1960), Giles(964), Cooper(1967), Boyling,(1968, 1972), Roberts and Luce (1968), Duistermaat (1968), Hornix (1968), Rastall (1970), Zeleznik(1975)andBorchers(1981). TheworkofBoyling(1968, 1972), whichtakesofromtheworkofBernstein(1960)isperhapsthemostdirectandrigorousexpressionof theoriginal Cartheodoryideaofusingdierentialforms. SeealsothediscussioninLandsberg(1970).Planck (1926) criticized some of Caratheodorys work for not identifying processes that are notadiabatic. Hesuggestedbasingthermodynamicsonthefactthatrubbingisanadiabaticprocessthat is not reversible, an idea he already had in his 1879 dissertation. From this it follows that whileonecanundoarubbingoperationbysomemeans,onecannotdosoadiabatically. WederivethisprincipleofPlanckfromouraxioms. Itisveryconvenientbecauseitmeansthatinanadiabaticprocessonecaneectivelyaddasmuchheat (colloquiallyspeaking)asonewishes, buttheonethingonecannotdoissubtractheat,i.e.,usearefrigerator.Mostauthorsintroducetheideaofanempirical temperature, andlaterderivetheabsolutetemperaturescale. Inthesameveintheyoftenalsointroduceanempirical entropy andlaterderiveametric, oradditive, entropy, e.g., (FalkandJung, 1959)and(Buchdahl, 1958, etseq.,1966),(Buchdahl and Greve,1962),(Cooper, 1967). Weavoidall this;one of our results, as statedabove, isthederivationofabsolutetemperaturedirectly, withoutevermentioningevenhot andcold.One of the key concepts that is eventuallyneeded, although it is not obvious at rst, is that ofthecomparisonprinciple(orhypothesis), (CH). Itconcernsclassesof thermodynamicstatesandassertsthatforanytwostatesXandY withinaclassonecaneithergoadiabaticallyfromXtoY ,whichwewriteasX Y,(pronouncedXprecedes Y or Y follows X) or else one cango fromY to X, i.e., Y X. Obviously, thisisnotalwayspossible(wecannottransmuteleadintogold, althoughwecantransmutehydrogenplusoxygenintowater),sowewouldliketobeabletobreakuptheuniverseofstatesintoequivalenceclasses, insideeachofwhichthehypothesisholds. Itturnsoutthatthekeyrequirementforanequivalencerelationisthatif X Y andZ Y theneitherX ZorZ X. Likewise, if YXandYZbytheneither X ZorZ X. WendthisrstclearlystatedinLandsberg(1956)anditisalsofoundinoneformoranotherinmanyplaces,seee.g., (FalkandJung, 1959), (Buchdahl, 1958, 1962), (Giles, 1964). However, all authors, exceptfor Duistermaat(1968),seem to takethis postulatefor grantedand do not feel obligedto obtainit7from somethingelse. One ofthecentralpoints inour workistoderivethecomparisonhypothesis.Thisisdiscussedfurtherbelow.TheformulationofthesecondlawofthermodynamicsthatisclosesttooursisthatofGiles(Giles, 1964). His bookis full of deepinsights andwerecommendit highlytothereader. Itis aclassicthat does not appear tobeas knownandappreciatedas it should. His derivationof entropyfromafewpostulates about adiabatic processes is impressive andwas the startingpointforanumberoffurtherinvestigations. TheoverlapofourworkwithGilessisonlypartial(thefoundational parts, mainlythoseinoursectionII)andwherethereisoverlaptherearealsodierences.Todenetheentropyof astate, thestartingpoint inbothapproaches is tolet aprocessthatbyitselfwouldbeadiabaticallyimpossibleworkagainstanotheronethatispossible,sothatthetotal processisadiabaticallypossible. TheprocessesusedbyusandbyGilesare, however,dierent;for instanceGilesuses axedexternalcalibratingsystem,whereaswedene the entropyofastatebylettingasysteminteractwithacopyofitself. (AccordingtoR.E.Barieau(quotedin(Hornix,1967-1968))Gileswasunawareofthefactthatpredecessorsoftheideaofanexternalentropymetercanbediscernedin(LewisandRandall, 1923).) Tobeabitmoreprecise, Gilesusesastandardprocessasareferenceandcountshowmanytimesareferenceprocesshastoberepeatedtocounteractsomemultipleof theprocesswhoseentropy(orratherirreversibility)istobedetermined. Incontrast, weconstructtheentropyfunctionforasinglesystemintermsoftheamountofsubstanceinareferencestateofhighentropythatcanbeconvertedintothestateunderinvestigationwiththehelpofareferencestateoflowentropy. (Thisisreminiscentofanold denition of heat by Laplace and Lavoisier (quoted in (Borchers, 1981))in terms of the amountof icethatabodycanmelt.) Wegiveasimpleformulafortheentropy; Gilessdenitionislessdirect, inourview. However, whenwecalibratetheentropyfunctionsof dierentsystemswitheachother,wedonditconvenienttouseathirdsystemasastandardofcomparison.Giles workandoursuseverylittleof thecalculus. Contrarytoalmostall treatments, andcontrarytothe assertion(Truesdell-Bharata,1977)thatthedierentialcalculusis theappropriatetool forthermodynamics, weandheagreethatentropyanditsessential propertiescanbestbedescribed by maximum principles insteadof equations among derivatives. To be sure, real analysisdoes eventuallycomeintothediscussion, but onlyatanadvancedstage(SectionsIII andV inourtreatment).In Giles, too, temperature appears as a totally derived quantity,but Giless derivation requiressomeassumptions, suchasdierentiabilityoftheentropy. Weprovetherequireddierentiabilityfromnaturalassumptionsaboutthepressure.Amongthedierences, it canbementionedthat thecancellationlaw, whichplays akeyroleinourproofs,istakenbyGilestobeanaxiom, whereaswederiveitfromtheassumptionofstability,whichiscommontobothapproaches(seeSectionIIfordenitions).Themost important point of contact, however, andat thesametimethemost signicantdierence, concernsthecomparisonhypothesiswhich, asweemphasizedabove, isaconceptthatplays an essentialrole, althoughthis may not be apparent at rst. This hypothesis serves to dividethe universe nicely into equivalence classes of mutually accessible states. Giles takes the comparisonpropertyasanaxiomanddoesnotattempttojustifyitfromphysicalpremises. Themainpartofourworkisdevotedtojust thatjustication,andtoinquire whathappens ifitisviolated. (Thereis also a discussion of this point in (Giles,1964, Sect 13.3) in connection with hysteresis.) To get anideaofwhatisinvolved, notethatwecaneasilygoadiabaticallyfromcoldhydrogenplusoxygen8to hot water and we can go from ice to hot water, but can we go either from the cold gases to ice orthe reverseas the comparison hypothesis demands?It would appear that the only real possibility,if thereisoneatall, istoinvokehydrolysistodissociatetheice, butwhatif hydrolysisdidnotexist? Inotherexamplestherequisitemachinerymightnotbeavailabletosavethecomparisonhypothesis. For this reason we prefer to derive it, when needed, from properties of simple systemsand not to invokeit when considering situations involving variable composition or particle number,asinSectionVI.Another point of dierence is the factthat convexityis centralto our work. Gilesmentions it,butitisnotcentral inhisworkperhapsbecauseheisconsideringmoregeneral systemsthanwedo. Toalargeextentconvexityeliminatestheneedforexplicittopological considerationsaboutstatespaces,whichotherwisehastobeputinbyhand.FurtherdevelopmentsoftheGilesapproacharein(Cooper, 1967),(RobertsandLuce,1968)and(Duistermaat, 1968). Cooperassumestheexistenceof anempirical temperatureandintro-ducestopologicalnotionswhichpermitscertainsimplications. RobertsandLucehaveanelegantformulationof theentropyprinciple, whichismathematicallyappealingandisbasedonaxiomsabouttheorderrelation, , (inparticularthecomparisonprinciple, whichtheycall conditionalconnectedness), but these axioms are not physically obvious, especially axiom 6 and the comparisonhypothesis. Duistermaatis concernedwith general statementsabout morphisms of order relations,thermodynamicsbeingbutoneapplication.Aline of thought that is entirelydierent fromthe above starts withCarnot (1824) andwas ampliedinthe classics of Clausius andKelvin(cf. (Kestin, 1976)) andmanyothers. Ithas dominatedmost textbookpresentations of thermodynamics to this day. The central ideaconcernscyclicprocessesandtheeciencyofheatengines; heatandempiricaltemperatureenteras primitiveconcepts. Someof themoderndevelopments alongtheselines gowell beyondthestudy ofequilibrium statesandcyclicprocesses anduse some sophisticatedmathematicalideas. ArepresentativelistofreferencesisArens(1963),ColemanandOwen(1974,1977),Coleman,OwenandSerrin(1981), Dafermos (1979), Day(1987, 1988), FeinbergandLavine(1983), GreenandNaghdi (1978),Gurtin (1975),Man(1989),Owen (1984),Pitteri(1982),Serrin (1979,1983,1986),Silhavy(1997), TruesdellandBharata(1977), Truesdell(1980,1984). Undoubtedlythisapproachisimportantforthepractical analysisofmanyphysical systems, butweneitheranalyzenortakeapositiononthevalidityoftheclaimsmadebyitsproponents. Someoftheseare, quitefrankly,highlypolemicalandareoftwokinds: claimsofmathematicalrigorandphysicalexactnessontheonehandandassertions thatthesequalities arelackinginother approaches. See, for example,Truesdellscontributionin(Serrin,1986, Chapter5). Thechief reasonweomitdiscussionofthisapproachis that it does not directlyaddress thequestions wehave set for ourselves. Namely,usingonlytheexistenceofequilibriumstatesandtheexistenceofcertainprocessesthattakeoneintoanother,whencanitbesaidthatthelistofallowedprocessesischaracterizedexactlybytheincreaseofanentropyfunction?Finally, wementionaninterestingrecentpaperbyMacdonald(1995)thatfallsinneitherofthetwocategoriesdescribedabove. Inthispaperheat andreversibleprocesses areamongtheprimitive concepts andtheexistence of reversible processes linkinganytwostates of asystemis takenas apostulate. Macdonaldgives asimpledenitionof entropyof astate interms ofthe maximal amount of heat, extractedfromaninnite reservoir, that the systemabsorbs inprocessesterminatinginthegivenstate. Thereservoirthusplaystheroleof anentropymeter.9Thefurtherdevelopmentofthetheoryalongtheselines, however, reliesonunstatedassumptionsaboutdierentiabilityofthesodenedentropythatarenotentirelyobvious.C.OutlineofthepaperInSectionIIweformallyintroducetherelation andexplainitmorefully, butitistobeemphasized,inconnectionwithwhatwassaidaboveaboutanidealphysicaltheory,that hasawell denedmathematical meaningindependentofthephysical contextinwhichitmaybeused.Theconceptof anentropyfunction, whichcharacterizes thisaccessibilityrelation, isintroducednext; attheendofthesectionitwill beshowntobeuniqueuptoatrivial anetransformationof scale. Weshowthattheexistenceofsuchafunctionisequivalenttocertainsimplepropertiesof therelation , whichwecall axiomsA1toA6andthehypothesis CH. Anyformulationofthermodynamicsmust implicitlycontaintheseaxioms, sincetheyareequivalent totheentropyprinciple,anditisnotsurprising thattheycanbe foundinGiles,for example. Wedo believethatour presentationhas the virtue of directness and clarity,however. We givea simple formula for theentropy,entirelyintermsoftherelation withoutinvokingCarnotcyclesoranyothergedankenexperiment. AxiomsA1toA6arehighlyplausible; itisCH(thecomparisonhypothesis)thatisnot obvious but is crucial for the existence of entropy. We call it a hypothesis rather than an axiombecauseourultimategoal istoderiveitfromsomeadditional axioms. Inacertainsenseitcanbesaidthattherestofthepaperisdevotedtoderivingthecomparisonhypothesisfromplausibleassumptions. Thecontentof SectionII, i.e., thederivationof anentropyfunction, standsonitsown feet; the implementation of it via CH is an independent question and we feel it is pedagogicallysignicanttoisolatethemaininputinthederivationfromthederivationitself.SectionIIIintroducesoneofourmostnovel contributions. Weprove thatcomparisonholdsforthestatesinsidecertainsystemswhichwecallsimplesystems. Toobtainitweneedafewnewaxioms, S1 to S3. These axioms are mainly about mechanical processes, and not about the entropy.In short, our most important assumptions concern the continuity of the generalized pressure and theexistence of irreversible processes. Given the other axioms, the latter is equivalent to Caratheodorysprinciple.Thecomparisonhypothesis,CH,doesnotconcernsimplesystemsalone, butalsotheirprod-ucts, i.e.,compound systems composed of possibly interacting simple systems. In order to comparestates in dierent simple systems (and, in particular, to calibrate the various entropies so that theycanbeaddedtogether) thenotionof athermal joinisintroducedinSectionIV. Thisconcernsstatesthatareusuallysaidtobeinthermal equilibrium, butweemphasizethattemperatureisnotmentioned. Thethermal joinis, byassumption, asimplesystemand, usingthezerothlawand three other axioms about the thermal join,we reduce the comparisonhypothesis among statesof compoundsystemstothepreviouslyderivedresultforsimplesystems. Thisderivationisan-othernovel contribution. Withtheaidof thethermal joinwecanprovethatthemultiplicativeconstantsof theentropiesof all systemscanbechosensothatentropyisadditive, i.e., thesumof theentropiesof simplesystemsgivesacorrectentropyfunctionforcompoundsystems. Thisentropycorrectlydescribesall adiabaticprocessesinwhichthereisnochangeoftheconstituentsof compoundsystems. Whatremainselusivearetheadditiveconstants, discussedinSectionVI.Theseareimportantwhenchanges(duetomixingandchemicalreactions)occur.SectionVestablishesthecontinuousdierentiabilityof theentropyanddenesinversetem-peratureasthederivativeof theentropywithrespecttotheenergyintheusual way. Nonew10assumptions areneededhere. The factthattheentropyofasimplesystemis determineduniquelybyitsadiabatsandisothermsisalsoprovedhere.InSectionVI we discuss the vexedquestionof comparingstates of systems that dier inconstitutionorinquantityofmatter. Howcantheentropyofabottleofwaterbecomparedwiththesumoftheentropiesofacontainerofhydrogenandacontainerofoxygen? Todosorequiresbeingabletotransformoneintotheother, butthismaynotbeeasytodoreversibly. Theusualtheoretical underpinninghereistheuseof semi-permeablemembranesinavantHobox butsuchmembranesareusuallyfarfromperfectphysical objects, iftheyexistatall. Weexamineindetail justhowfaronecangoindeterminingtheadditiveconstantsfortheentropiesofdierentsystemsinthetherealworldinwhichperfectsemi-permeablemembranesdonotexist.InSectionVIIwecollectallouraxiomstogetherandsummarizeourresultsbriey.D.AcknowledgementsWearedeeplyindebtedtoJanPhilipSolovej formanyuseful discussionsandimportantin-sights, especiallyinregardtoSections III andVI. Our thanksalsogotoFredrickAlmgrenforhelpingusunderstandconvexfunctions,toRoyJackson,PierluigiContucci, ThorBakandBern-hardBaumgartner forcriticallyreadingourmanuscriptandtoMartinKruskal for emphasizingtheimportanceof Giles booktous. WethankRobinGilesforathoughtful anddetailedreviewwithmanyhelpful comments. WethankJohnC. Wheelerforaclarifyingcorrespondenceaboutthe relationship between adiabatic processes, as usually understood, and our denition of adiabaticaccessibility. Someof theroughspots inour story werepointedouttous by variouspeople duringvariouspubliclectureswegave,andthatisalsoverymuchappreciated.Asignicantpartof thisworkwascarriedoutatNorditainCopenhagenandattheErwinSchr odingerInstituteinVienna;wearegratefulfortheirhospitalityandsupport.11II.ADIABATICACCESSIBILITYANDCONSTRUCTIONOFENTROPYThermodynamics concerns systems, their states and an order relation among these states. Theorderrelationisthatofadiabaticaccessibility,which,physically,isdenedbyprocesseswhoseonlyneteectonthesurroundingsisexchangeofenergywithamechanical source. Thegloryofclassicalthermodynamics isthattherealwaysis anadditivefunction,calledentropy,onthestatespaceofanysystem,thatexactlydescribestheorderrelationintermsoftheincreaseofentropy.Additivity is very important physically and is certainly not obvious; it tells us that the entropyofacompoundsystemcomposedoftwosystemsthatcaninteractandexchangeenergywitheachotheristhesumof theindividual entropies. Thismeansthatthepairsof statesaccessiblefromagivenpair of states, whichisafar larger set thanmerelythepairsindividuallyaccessiblebythesystemsinisolation, isgivenbystudyingthesumof theindividual entropyfunctions. Thisisevenmoresurprisingwhenweconsiderthattheindividual entropieseachhaveundeterminedmultiplicativeconstants;thereis awaytoadjust, or calibratethe constantsin suchawaythatthesum givesthecorrectresult for theaccessiblestatesandthis canbe done onceand for allso thatthesamecalibrationworksforallpossiblepairsofsystems. Wereadditivitytofailwewouldhavetorewritethesteamtableseverytimeanewsteamengineisinvented.Theotherimportantpointaboutentropy,whichisoftenoverlooked, isthatentropynotonlyincreases,butentropyalsotellsusexactlywhichprocessesareadiabaticallypossibleinanygivensystem;statesofhighentropyinasystemarealways accessiblefromstatesoflowerentropy. Asweshall seethis is generallytrue but it couldconceivablyfailwhen there arechemicalreactionsormixing,asdiscussedinSectionVI.Inthissectionwebeginbydeningthesebasicconceptsmoreprecisely,andthenwepresenttheentropyprinciple. Next,weintroducecertainaxioms,A1-A6, relatingtheconcepts. Alltheseaxiomsarecompletelyintuitive. However, oneotherassumptionwhichwecall thecomparisonhypothesisisneeded for the construction of entropy. It is not at all obvious physically, but it is anessentialpartofconventionalthermodynamics. Eventually,inSectionsIIIandIV,thishypothesiswill bederivedfromsomemoredetailedphysical considerations. For thepresent, however, thishypothesiswillbeassumedand,usingit,theexistenceofanentropyfunctionwillbeproved. Wealsodiscuss theextenttowhichtheentropyfunctionisuniquely determinedbytheorder relation;thecomparisonhypothesisplaysakeyrolehere.TheexistenceofanentropyfunctionisequivalenttoaxiomsA1-A6inconjunctionwithCH,neither morenor lessisrequired. Thestatespaceneednothaveanystructurebesidestheoneimpliedbytheorderrelation. However,statespacesparametrizedbytheenergyandworkcoordi-nateshaveanadditional, convexstructure, whichimpliesconcavityoftheentropy, providedthattheformationofconvexcombinationofstatesisanadiabaticprocess. WeaddthisrequirementasaxiomA7toourlistofgeneralaxiomsabouttheorderrelation.Theaxioms inthis sectionare sogeneral that theyencompass situations whereall statesinawholeneighborhoodof agivenstateareadiabaticallyaccessiblefromit. Caratheodorysprincipleis thestatement that this does not happenfor physical thermodynamicsystems. Incontrast,ideal mechanical systems have the property that every state is accessible from every otherone(bymechanicalmeansalone),andthus theworldofmechanicalsystemswilltriviallyobeytheentropyprincipleinthesensethateverystatehasthesameentropy. Inthelast subsectionwediscuss the connectionbetweenCaratheodorys principle and the existenceof irreversible processes12starting from a givenstate. This principle will againbe invokedwhen, in SectionIII, we derive thecomparisonhypothesisforsimplethermodynamicsystems.Temperaturewill notbeusedinthissection, noteventhenotionof hot andcold. Therewillbenocycles, Carnotorotherwise. Theentropyonlydependson,andisdenedbytheorderrelation. Thus, whiletheapproachgivenhereisnottheonlypathtothesecondlaw, ithastheadvantageofacertainsimplicityandclaritythatatleasthaspedagogicandconceptualvalue. Weask the readers patience with our syllogisms, the point being that everything is here clearly spreadoutinfull view. There areno hidden assumptions,as oftenoccurinmanytextbookpresentations.Finally, wehopethatthereaderwillnotbeconfusedbyoursometimeslengthyasidesaboutthemotivationandheuristicmeaningofourvariousdenitionsandtheorems. Wealsohopetheseremarks will not beconstruedas part of thestructureof thesecondlaw. Thedenitions andtheoremsareself-contained, aswestatethem, andtheremarksthatsurroundthemareintendedonlyasahelpfulguide.A.Basicconcepts1. Systems and their state spacesPhysicallyspeaking athermodynamic systemconsistsof certainspeciedamountsof dierentkindsof matter; itmightbedivisibleintopartsthatcaninteractwitheachotherinaspeciedway. Aspecialclassofsystemscalledsimplesystemswillbediscussedinthenextchapter. Inanycasethepossibleinteractionof thesystemwithitssurroundingsisspecied. Itisablackboxinthesensethatwedonotneedtoknowwhatisinthebox,butonlyitsresponsetoexchangingenergy, volume, etc. withothersystems. Thestatesofasystemtobeconsideredherearealwaysequilibriumstates, buttheequilibriummaydependupontheexistenceofinternalbarriersinthesystem. Intermediate, non-equilibriumstatesthatasystempassesthroughwhenchangingfromone equilibrium state to another will not be considered. The entropy of a system not in equilibriummay,likethe temperature of such a system, havea meaning as anapproximate and useful concept,butthisisnotourconcerninthistreatment.Our systemscanbe quitecomplicatedandtheoutsideworldcanactonthem inseveralways,e.g., bychangingthevolumeandmagnetization, orremovingbarriers. Indeed, weareallowedtochopasystemintopiecesviolentlyandreassembletheminseveralways,eachtimewaitingfortheeventualestablishmentofequilibrium.Oursystemsmustbemacroscopic,i.e,nottoosmall. Tinysystems(atoms,molecules,DNA)exist,tobe sure,but wecannotdescribetheirequilibriathermodynamically,i.e.,theirequilibriumstates cannot be described in terms of the simple coordinates we use later on. There is a gradual shiftfrom tiny systems to macroscopic ones, and the empirical fact is that large enough systems conformtotheaxiomsgivenbelow. Atsomestageasystembecomesmacroscopic; wedonotattempttoexplainthisphenomenonortogiveanexactruleaboutwhichsystemsaremacroscopic.Ontheotherhand,systemsthataretoolargearealsoruledoutbecausegravitational forcesbecomeimportant. Twosunscannotunitetoformonebiggersunwiththesameproperties(theway two glasses of water can unite to become one large glass of water). A star with two solar massesis intrinsically dierent from a sun of one solar mass. In principle, the two suns could be kept apartand regarded as one system, but then this would only be a constrained equilibrium because of thegravitational attraction. Inotherwordstheconventional notionsofextensivity andintensivityfail for cosmic bodies. Nevertheless, it is possible to dene an entropy for such systems by measuring13itseectonsomestandardbody. Giles methodisapplicable, andourformula(2.20)inSectionII.E(which, inthecontextof ourdevelopment, isusedonlyforcalibratingtheentropiesdenedby(2.14)inSectionII.D,butwhichcouldbetakenasanindependentdenition)wouldallowit,too. (Thenice systemsthatdosatisfysize-scalingarecalledperfect byGiles.) Theentropy,sodened, wouldsatisfyadditivitybutnotextensivity, intheentropyprinciple ofSectionII.B.However,to prove this would requires a signicant enhancement of the basic axioms. In particular,we would have to take the comparison hypothesis, CH, for all systems as an axiom as Giles does.Itislefttotheinterestedreadertocarryoutsuchanextensionofourscheme.Abasicoperationiscompositionoftwoormoresystemstoformanewsystem. Physically,thissimplymeansputtingtheindividualsystemssidebysideandregardingthemasonesystem.Wethenspeakof eachsystemintheunionasasubsystem. Thesubsystemsmayormaynotinteractforawhile, byexchangingheatorvolumeforinstance, buttheimportantpointisthatastate of the total system(wheninequilibrium) is describedcompletelybythe states of thesubsystems.Fromthemathematical pointof viewasystemisjustacollectionof pointscalledastatespace, usuallydenotedby. Theindividual points of astatespacearecalledstatesandaredenoted here by capital Roman letters, X, Y, Z, etc. From the next section on we shall build up ourcollectionof statessatisfyingouraxiomsfromthestatesof certainspecial systems, calledsimplesystems. (Tojumpaheadforthemoment, thesearesystemswithoneormoreworkcoordinatesbut withonlyoneenergycoordinate.) Inthepresentsection,however,themanner inwhichstatesare described (i.e., the coordinates one uses, such as energy and volume, etc.) are of no importance.Noteventopological propertiesareassumedhereaboutoursystems, asisoftendone. Inasenseitisamazingthatmuchofthesecondlawfollowsfromcertainabstractpropertiesoftherelationamongstates, independentof physical details(andhenceof conceptssuchasCarnotcycles). InapproacheslikeGiles, whereitistakenasanaxiomthatcomparablestatesfall intoequivalenceclasses, itisevenpossibletodowithoutthesystemconceptaltogether, ordeneitsimplyasanequivalence class of states. In our approach, however, one of the main goals is to derive the propertywhichGilestakesasanaxiom,andsystemsarebasicobjectsinouraxiomaticscheme.Mathematically, thecompositionoftwospaces,1and2issimplytheCartesianproductofthestatespaces1 2. Inotherwords, thestatesin1 2arepairs(X1, X2)withX1 1andX2 2. Fromthephysical interpretationofthecompositionitisclearthatthetwospaces1 2and2 1aretobeidentied. Likewise, whenformingmultiplecompositionsof statespaces, the order and the grouping of the spaces is immaterial. Thus (12) 3, 1(23)and1 2 3aretobeidentiedasfarascompositionofstatespacesisconcerned. Strictlyspeaking, asymbol like(X1, . . . , XN)withstatesXiinstatespacesi, i=1, . . . , Nthusstandsforanequivalenceclassofn-tuples,correspondingtothedierentgroupingsandpermutationsofthestatespaces. Identicationsofthistypearenotuncommoninmathematics(theformationofdirectsumsofvectorspacesisanexample).A further operation we shall assume is the formation of scaledcopies of a given system whosestatespaceis . If t > 0 issome xednumber (thescalingparameter)thestatespace(t)consistsofpointsdenotedtXwithX . Ontheabstractlevel tXismerelyasymbol,ormnemonic,todene points in (t), but the symbol acquires meaning through the axioms givenlater in Sect. II.C.Inthephysical world, andfromSect. IIIonward, thestatespaceswillalwaysbesubsetsofsomeRn(parametrizedbyenergy,volume,etc.). InthiscasetXhastheconcreterepresentationastheproduct oftherealnumber tandthevectorX Rn. Thus inthiscase(t)issimplytheimageof14theset Rnunderscalingbythereal parametert. Hence, weshall sometimesdenote(t)byt.Physically, (t)isinterpretedasthestatespaceofasystemthathasthesamepropertiesasthesystemwithstatespace, exceptthattheamountofeachchemical substanceinthesystemhasbeenscaledbythefactortandtherangeof extensivevariableslikeenergy, volumeetc. hasbeenscaledaccordingly. Likewise,tXisobtainedfromXbyscalingenergy,volumeetc.,butalsothemattercontentof astateXisscaledbytheparametert. Fromthisphysical interpretationitisclearthats(tX)=(st)Xand((t))(s)=(st)andwetaketheserelationsalsoforgrantedontheabstract level. Thesameapples totheidentications (1)=and1X=X, andalso(12)(t)= (t)1(t)2andt(X, Y ) = (tX, tY ).Theoperationof formingcompoundstates is thus anassociative andcommutative binaryoperation on the set of all states, and the group of positive real numbers acts by the scaling operationonthissetinawaycompatiblewiththebinaryoperationandthemultiplicativestructureoftherealnumbers. Thesameistrueforthesetofallstatespaces. Fromanalgebraicpointofviewthesimplesystems,tobediscussedinSectionIII,areabasisforthisalgebraicstructure.Whiletherelationbetweenand(t)isphysicallyandintuitivelyfairlyobvious, therecanbesurprises. Electromagneticradiationinacavity(photongas), whichismentionedafter(2.6),isaninterestingcase; thetwostatespacesand(t)andthethermodynamicfunctionsonthesespacesareidentical inthiscase! Moreover, thetwospacesarephysicallyindistinguishable. ThiswillbeexplainedinmoredetailinSectionII.B.Theformationofscaledcopiesinvolvesacertainphysical idealizationbecauseitignoresthemolecular structureof matter. Scalingtoarbitrarilysmall sizes brings quantumeects totheforeandmacroscopicthermodynamicsisnolonger applicable. Attheotherextreme, scalingtoarbitrarilylargesizesbringsinunwantedgravitationaleectsasdiscussedabove. Inspiteofthesewell known limitations the idealization of continuous scaling is common practice in thermodynamicsand simplies things considerably. (In the statistical mechanics literature this goes under the rubricof the thermodynamic limit.)It should be noted that scaling is quite compatible with the inclusionofsurfaceeectsinthermodynamics. ThiswillbediscussedinSectionIII.A.Bycomposingscaledcopies of Nsystems withstate spaces 1, . . . , N, onecanform, fort1, . . . , tN> 0,theirscaledproduct(t1)1 (tN)Nwhosepointsare(t1X1, t2X2, . . . , tNXN).Intheparticularcasethatthejsareidentical, i.e., 1=2= =, weshall call anyspaceof thetheform(t1) (tN)amultiplescaledcopyof . Aswill beexplainedlaterinconnectionwithEq. (2.11), itis sometimes convenient incalculations toallowt =0as scalingparameter (andevennegativevalues). For themoment let usjust notethat if (0)occurs thereaderisaskedtoregarditastheemptysetornosystem. Inotherwords,ignoreit.Someexamplesmayhelpclarifytheconceptsofsystemsandstatespaces.(a) a: 1mole of hydrogen, H2. Thestate space canbeidentiedwithasubset of R2withcoordinatesU(=energy),V (=volume).(b) b:12moleofH2. IfaandbareregardedassubsetsofR2thenb= (1/2)a= (12U,12V ) :(U, V ) a.(c) c: 1moleofH2and12moleofO2(unmixed). c=a (12moleO2). Thisisacompoundsystem.(d) d: 1moleofH2O.15(e) e: 1 moleof H2 +12moleof O2(mixed). Notethate ,= dand e ,= c. This system showstheperilsinherentintheconceptofequilibrium. Thesystememakessenseaslongasonedoes not drop in a piece of platinum or walk across the laboratory oor too briskly. Real worldthermodynamics requires that we admit such quasi-equilibrium systems, although perhaps notquiteasdramaticasthisone.(f) f: All theequilibriumstatesof onemoleof H2andhalf amoleof O2(plusatinybitofplatinumtospeedupthereactions)inacontainer. AtypicalstatewillhavesomefractionofH2O,somefractionofH2andsomeO2. Moreover,thesefractionscanexistinseveralphases.2. The order relationThebasicingredientofthermodynamicsistherelationofadiabaticaccessibilityamongstatesofasystemorevendierentsystems. ThestatementX Y ,whenXandYarepointsinsome(possiblydierent)statespaces,meansthatthereisanadiabatictransition,inthesenseexplainedbelow,thattakesthepointXintothepointY .Mathematically, wedonothavetoaskthemeaningof adiabatic. All thatmattersisthatalistof all possiblepairsof statesXsandY ssuchthatX Y isregardedasgiven. Thislisthastosatisfycertainaxiomsthatweprescribe belowinsubsectionC.Amongotherthingsitmustbereexive, i.e., X X, andtransitive, i.e., X Y andYZimpliesX Z. (Technically,instandardmathematicalterminologythisiscalledapreorderrelationbecausewecanhavebothX Y andYXwithoutX=Y .) Ofcourse, inordertohaveaninterestingthermodynamicsresult from our relationitis essentialthatthere arepairs ofpoints X, Yforwhich X Yis nottrue.Althoughthephysical interpretationof therelation is not neededfor themathematicaldevelopment, for applications it isessential tohaveaclear understandingof its meaning. It isdicult to avoid some circularity when dening the concept of adiabatic accessibility. The followingversion(whichisinthespiritof Plancksformulationof thesecondlaw(Planck, 1926))appearstobesucientlygeneral andpreciseandappealstous. Ithasthegreatvirtue(asdiscoveredbyPlanck)thatitavoidshavingtodistinguishbetweenworkandheatorevenhavingtodenetheconceptofheat; heat, intheintuitivesense, canalwaysbegeneratedbyrubbinginaccordancewith Count Rumfords famous discovery while boring cannons! Weemphasize, however,that otherdenitionsarecertainlypossible. Ourphysicaldenitionisthefollowing:Adiabaticaccessibility: AstateY is adiabaticallyaccessiblefromastateX, insymbolsX Y , if it ispossibletochangethestatefromXtoY bymeansof aninteractionwithsomedevice(whichmayconsist of mechanical andelectrical parts aswell asauxiliarythermodynamicsystems)andaweight, insuchawaythat thedevicereturnstoitsinitial stateat theendof theprocesswhereastheweightmayhavechangeditspositioninagravitational eld.LetuswriteX Y if X Y but Y , X. (2.1)Inthereal worldY isadiabaticallyaccessiblefromXonlyif X Y . WhenX Y andalsoYXthenthestatechangecanonlyberealizedinanidealizedsense, foritwill takeinnitelylongtimetoachieveitinthemannerdecribed. Analternativewayistosaythatthedevicethat16appearsinthedenitionofaccessibilityhastoreturntowithin ofitsoriginal state(whateverthatmaymean)andwetakethelimit 0. Toavoidthiskindofdiscussionwehavetakenthedenitionasgivenabove, butweemphasizethatitiscertainlypossibletoredothewholetheoryusing only the notion of . An emphasis on appears in Lewis and Randalls discussion of thesecondlaw(LewisandRandall,1923,page116).Remark: It should be notedthat the operationaldenition above is a denition of the conceptof adiabaticaccessibility andnottheconceptof anadiabaticprocess. Astatechangeleadingfrom XtoYcanbe achievedinmany dierentways(usuallyinnitely many),andnot allof themwill be adiabatic processes in the usual terminology. Our concern is not the temporal developmentof the state change which, in real processes, always leads out of the space of equilibrium states. Onlythe endresult for thesystem andfor the restof theworldinterestsus. However,it isimportant toclarify the relation between our denition of adiabatic accessiblity and the usual textbook denitionof anadiabaticprocess. Thiswill bediscussedinSectionCafterTheorem2.1andagaininSec.III; cf. Theorem3.8. Thereitwill beshownthatourdenitionindeedcoincideswiththeusualnotionbasedonprocessestakingplacewithinanadiabaticenclosure. Afurtherpointtonoticeisthatthewordadiabatic issometimesusedtomeansloworquasi-static, butnothingofthesortismeanthere. Indeed,anadiabaticprocesscanbequiteviolent. Theexplosionofabombinaclosedcontainerisanadiabaticprocess.Herearesomefurtherexamplesofadiabaticprocesses:1. Expansion or compression of a gas, with or without the help of a weight being raised or lowered.2. Rubbingorstirring.3. Electricalheating. (Notethattheconceptofheatisnotneededhere.)4. Natural processesthat occur withinanisolatedcompoundsystemafter somebarriershavebeenremoved. Thisincludesmixingandchemicalornuclearprocesses.5. Breakingasystemintopieceswithahammerandreassembling(Fig. 1).6. Combinationsofsuchchanges.Intheusual parlance, rubbingwouldbeanadiabatic process, but not electrical heating,becausethelatter requirestheintroductionof apairof wiresthroughtheadiabaticenclosure.Forus,bothprocessesareadiabaticbecausewhatisrequiredisthatapartfromthechangeofthesystem itself, nothing more than the displacement of a weight occurs. To achieveelectricalheating,one drills a hole in the container,passes a heater wire through it,connects the wires to a generatorwhich, in turn, is connectedto a weight. After the heating the generator is removedalong with thewires,theholeisplugged,andthesystemisobservedtobeinanewstate. Thegenerator,etc. isinitsoldstateandtheweightislower.-(InsertFigure1here)-WeshallusethefollowingterminologyconcerninganytwostatesXandY . Thesestatesaresaidtobecomparable(withrespecttotherelation ,ofcourse)ifeitherX Y orY X. IfbothrelationsholdwesaythatXandY areadiabaticallyequivalentandwriteX AY. (2.2)Thecomparisonhypothesis referredtoaboveis thestatement that anytwostates inthesamestatespacearecomparable. Inthe examplesof systems (a)to(f)above,allsatisfy thecomparisonhypothesis. Moreover, everypointincisintherelation tomany(butnotall)pointsind.17States in dierent systems may or may not be comparable. An example of non-comparable systemsisonemoleofH2andonemoleofO2. AnotherisonemoleofH2andtwomolesofH2.Onemightthinkthatifthecomparisonhypothesis, whichwillbediscussedfurtherinSects.II.CandII.E, weretofail for somestatespacethenthesituationcouldeasilyberemediedbybreakingupthestatespaceintosmallerpiecesinsideeachof whichthehypothesisholds. This,generally, isfalse. Whatisneededtoaccomplishthisistheextrarequirementthatcomparabilityisanequivalencerelation; this,inturn,amountstosayingthattheconditionX ZandY ZimpliesthatXandY arecomparableand,likewise,theconditionZ XandZ Y impliesthatXandY arecomparable. (Thisaxiomcanbefoundin(Giles,1964),seeaxiom2.1.2,andsimilarrequirementsweremadeearlierbyLandsberg(1956), FalkandJung(1959)andBuchdahl (1962,1966).) While these two conditions are logicallyindependent, they can be shown to be equivalent iftheaxiomA3inSectionII. Cis adopted. Inanycase,wedo notadoptthecomparisonhypothesisasanaxiombecausewendithardtoregarditasaphysicalnecessity. Inthesamevein, wedonotassumethatcomparabilityisanequivalencerelation(whichwouldthenleadtothevalidityofthecomparisonhypothesisforsuitablydenedsubsystems). Ourgoal istoprovethecomparisonhypothesisstartingfromaxiomsthatwendmoreappealingphysically.B.TheentropyprincipleGiventherelation forall possiblestatesof all possiblesystems, wecanaskwhetherthisrelation can be encoded in an entropy function according to the following principle, which expressesthesecondlawofthermodynamicsinapreciseandquantitativeway:Entropyprinciple: Thereisareal-valuedfunctiononall states of all systems (includingcompoundsystems),calledentropyanddenotedbySsuchthata) Monotonicity:WhenXandY arecomparablestatesthenX Y ifandonlyif S(X) S(Y ). (2.3)(See(2.6)below.)b) Additivity and extensivity:If XandY arestates of some(possiblydierent) systemsandif(X, Y )denotesthecorrespondingstateinthecompositionofthetwosystems,thentheentropyisadditiveforthesestates,i.e.,S((X, Y )) = S(X) +S(Y ). (2.4)Sisalsoextensive,i.e.,foreacht > 0andeachstateXanditsscaledcopytX,S(tX) = tS(X). (2.5)[Note: Fromnowonweshall omitthedoubleparenthesisandwritesimplyS(X, Y )inplaceofS((X, Y )).]Alogicallyequivalent formulationof (2.3), that doesnot usethewordcomparable is thefollowingpairofstatements:X AY=S(X) = S(Y ) andX Y=S(X) < S(Y ). (2.6)18The lastline is especiallynoteworthy. It says thatentropy must increase inanirreversible process.Our goalistoconstructanentropyfunctionthatsatisesthecriteria(2.3)-(2,5),andtoshowthatitisessentiallyunique. Weshall proceedinstages, therstbeingtoconstructanentropyfunctionforasinglesystem, ,anditsmultiplescaledcopies(inwhichcomparabilityisassumedtohold). Havingdone this, the problemof relatingdierent systems will thenarise, i.e., thecomparisonquestionfor compound systems. Inthe present SectionII (and onlyinthis section)weshallsimplycompletetheprojectbyassumingwhatweneedbywayofcomparability. InSectionIV,thethermalaxioms(thezerothlawofthermodynamics, inparticular)willbeinvokedtoverifyourassumptionsaboutcomparabilityincompoundsystems. Intheremainderof thissubsectionwediscusshesignicanceofconditions(2.3)-(2.5).The physical content of (2.3) was already commented on; adiabatic processes not only increaseentropybutanincreaseof entropyalsodictateswhichadiabaticprocessesarepossible(betweencomparablestates,ofcourse).Thecontentofadditivity, (2.4), isconsiderablymorefarreachingthanonemightthinkfromthe simplicityof the notationas we mentionedearlier. Consider four states X, X, Y, YandsupposethatX Y andXY. Then(andthiswillbeoneofouraxioms)(X, X) (Y, Y),and(2.4)containsnothingnewinthiscase. Ontheotherhand, thecompoundsystemcanwellhaveanadiabaticprocessinwhich(X, X) (Y, Y)but X , Y . Inthiscase,(2.4)conveysmuchinformation. Indeed, bymonotonicity,therewillbemanycasesofthiskindbecausetheinequalityS(X) + S(X) S(Y ) + S(Y)certainlydoesnotimplythatS(X) S(Y ). ThefactthattheinequalityS(X) + S(X) S(Y ) + S(Y)tellsusexactly whichadiabaticprocessesareallowedinthecompoundsystem(assumingcomparability), independentofanydetailedknowledgeofthemannerinwhichthetwosystemsinteract,isastonishingandisattheheartofthermodynamics.Extensivity, (2.5), is almost aconsequence of (2.4) alonebut logically it is independent.Indeed, (2.4) impliesthat(2.5) holdsforrational numberst providedoneaccepts thenotionofrecombinationasgiveninAxiomA5below, i.e., onecancombinetwosamplesofasysteminthesamestateintoabiggersysteminastatewiththesameintensiveproperties. (Forsystems,suchascosmicbodies, thatdonotobeythisaxiom, extensivityandadditivityaretrulyindependentconcepts.) Ontheotherhand,usingtheaxiomofchoice, onemayalwayschangeagivenentropyfunctionsatisfying(2.3)and(2.4)insuchawaythat(2.5)isviolatedforsomeirrational t, butthenthefunction t S(tX)wouldendup being unbounded ineveryt interval. Such pathologicalcasescouldbeexcludedbysupplementing(2.3)and(2.4)withtherequirementthatS(tX)shouldlocallybe a bounded function of t, either from below or above. This requirement, plus (2.4),wouldthenimply(2.5). Foradiscussionrelatedtothispointsee(Giles,1964),whoeectivelyconsidersonlyrationalt. Seealso(Hardy,Littlewood,Polya1934)foradiscussionoftheconceptofHamelbaseswhichisrelevantinthiscontext.Theextensivityconditioncansometimeshavesurprising results,asinthecaseofelectromag-neticradiation(thephotongas). Asiswell known(LandauandLifschitz, 1969, Sect. 60), thephasespaceofsuchagas(whichweimaginetoresideinaboxwithapistonthatcanbeusedtochangethevolume)isthequadrant = (U, V ) : 0 < U< , 0 < V< . Thus,(t)= assets, whichisnotsurprisingorevenexceptional. Whatisexceptional isthatS, whichgivestheentropyofthestatesin,satisesS(U, V ) = (const.) V1/4U3/4.19It ishomogeneousof rst degreeinthecoordinatesand, therefore,the extensivitylawtellsus thattheentropyfunctiononthescaledcopy(t)isS(t) (U, V ) = tS(U/t, V/t) = S(U, V ).Thus, all the thermodynamic functions on the two state spaces are the same! This unusual situationcould, in principle, happen for an ordinary material system, but we know of no example besides thephoton gas. Here, the result can be traced to the fact that particle number is not conserved, as it isfor materialsystems,but itdoes show that one should not jump toconclusions. There is, however,afurtherconceptualpointaboutthephotongaswhichisphysicalratherthanmathematical. Ifamaterialsystemhadahomogeneousentropy(e.g.,S(U, V ) = (const.)V1/2U1/2)weshouldstillbeabletodistinguish(t)from, eventhoughthecoordinatesandentropywereindistinguishable.This could be done by weighing the two systems and nding out that one weighs t times as much astheother. Butthephotongasisdierent: noexperimentcantellthetwoapart. However,weightperse plays no role in thermodynamics, so the dierence betweenthe material and photon systemsisnotthermodynamicallysignicant.There are two points of view one could take about this anomalous situation. One is to continuetousethestatespaces(t), eventhoughtheyhappentorepresentidentical systems. Thisisnotreallyaproblembecausenoonesaidthat(t)hadtobedierentfrom. Theonlyconcernistocheck the axioms, and in this regard there is no problem. We could even allow the additive entropyconstanttodependont,provideditsatisestheextensivitycondition(2.5). Thesecondpointofviewistosaythatthereisonlyoneandno(t)satall. Thiswouldcauseustoconsiderthephotongasasoutsideourformalismandtorequirespecialhandlingfromtimetotime. Therstalternativeismoreattractivetousforobviousreasons. Thephotongaswill bementionedagaininconnectionwithTheorem2.5.C.AssumptionsabouttheorderrelationWe now list our assumptions for the order relation . As always,X, Y , etc. will denote states(thatmaybelongtodierentsystems), andif Xisastateinsomestatespace, thentXwitht > 0isthecorrespondingstateinthescaledstatespace(t).A1) Reexivity. X AX.A2) Transitivity. X Y andY ZimpliesX Z.A3) Consistency. X XandY Yimplies(X, Y ) (X, Y).A4) Scalinginvariance. IfX Y ,thentX tY forall t > 0.A5) Splittingandrecombination. For0 < t < 1X A(tX, (1 t)X). (2.7)(IfX ,thentherightsideisinthescaledproduct(t)(1t),ofcourse.)A6) Stability. If,forsomepairofstates,XandY ,(X, Z0) (Y, Z1)holdsforasequenceofstendingtozeroandsomestatesZ0,Z1,thenX Y.20Remark: Stabilitymeanssimply thatonecannotincreasethesetofaccessiblestateswithaninnitesimalgrainofdust.Besides these axioms the following property of state spaces, the comparison hypothesis, playsacrucial roleinouranalysisinthissection. Itwill eventuallybeestablishedforall statespacesafterwehaveintroducedsomemorespecicaxiomsinlatersections.CH) Denition: Wesaythecomparisonhypothesis(CH)holds forastatespaceif anytwostatesXandY inthespacearecomparable, i.e.,X Y orY X.Inthenextsubsectionweshallshowthat, foreverystatespace, , assumptionsA1-A6, andCHforall two-foldscaledproducts,(1 ), notjustitself, areinfactequivalenttotheexistenceof anadditive andextensiveentropy functionthat characterizesthe order relationonthestatesinall scaledproductsof . Moreover, foreach, thisfunctionisunique, uptoananetransformationofscale,S(X) aS(X) +B. BeforeweproceedtotheconstructionofentropywederiveasimplepropertyoftheorderrelationfromassumptionsA1-A6,whichisclearlynecessaryiftherelationistobecharacterizedbyanadditiveentropyfunction.THEOREM2.1(Stabilityimpliescancellationlaw). AssumepropertiesA1-A6, espe-ciallyA6thestabilitylaw. Thenthecancellationlawholdsasfollows. IfX, Y andZarestatesofthree(possiblydistinct)systemsthen(X, Z) (Y, Z) implies X Y (CancellationLaw).Proof: Let = 1/nwithn = 1, 2, 3, . . .. Thenwehave(X, Z) A((1 )X, X, Z) (byA5) ((1 )X, Y, Z) (byA1,A3andA4)A((1 2)X, X, Y, Z) (byA5) ((1 2)X, 2Y, Z) (byA1,A3,A4andA5).Bydoingthisn=1/timeswendthat(X, Z) (Y, Z). BythestabilityaxiomA6wethenhaveX Y .Remark: Undertheadditional assumptionthatY andZarecomparablestates(e.g., iftheyareinthesamestatespaceforwhichCHholds),thecancellationlawislogicallyequivalenttothefollowingstatement(usingtheconsistencyaxiomA3):If X Y then(X, Z) (Y, Z)forall Z.Thecancellationlawlooks innocent enough, but it is reallyrather strong. It is apartialconverseoftheconsistencyconditionA3anditsaysthatalthoughtheorderingin1 2isnotdeterminedsimplybytheorderin1and2,therearelimitstohowmuchtheorderingcanvarybeyondtheminimal requirementsof A3. Itshouldalsobenotedthatthecancellationlawisinaccordwithourphysical interpretationoftheorderrelationinSubsectionII.A.2.; aspectator,namelyZ,cannotchangethestatesthatareadiabaticallyaccessiblefromX.21Remarkabout AdiabaticProcesses: Withtheaidof thecancellationlawwecannowdis-cusstheconnectionbetweenournotionofadiabaticaccessibilityandthetextbookconceptofanadiabaticprocess. Oneproblemwefaceisthatthislatterconceptishardtomakeprecise(thiswas our reasonfor avoidingit inour operational denition) andtherefore thediscussionmustnecssearilybesomewhatinformal. Thegeneral ideaofanadiabaticprocess, however, isthatthesystem of interestis lockedin a thermally isolatingenclosure that prevents heat from owing intooroutofoursystem. Hence,asfarasthesystemisconcerned,alltheinteractionithaswiththeexternal worldduringanadiabaticprocesscanbethoughtofasbeingaccomplishedbymeansofsomemechanicalorelectricaldevices. Ouroperationaldenitionoftherelation appearsatrstsighttobebasedonmoregeneral processes, sinceweallowanauxilarythermodynamical systemaspartofthedevice. Weshallnowshowthat, despiteappearances,ourdenitioncoincideswiththeconventionalone.Let us temporarily denote by the relationbetween states based on adiabatic processes, i.e.,X Yif and only if there is a mechanical/electricaldevice that starts in a state Mand ends up inastateMwhilethesystemchangesfromXtoY . Wenowassumethatthemechanical/electricaldevicecanberestoredtotheinitial stateMfromthenal stateMbyaddingorsubstractingmechanical energy, and this latter process can be reduced to the raising or lowering of a weight in agravitationaleld. (Thiscanbetakenasadenitionofwhat wemeanbyamechanical/electricaldevice. Notethatdeviceswithdissipation donothavethisproperty.) Thus, X Y meansthereisaprocessinwhichthemechanical/electrical devicestartsinsomestateMandendsupinthesamestate, aweightmovesfromheighthtoheighth, whilethestateofoursystemchangesfromXtoY . Insymbols,(X, M, h) (Y, M, h). (2.8)Inourdenitionofadiabaticaccessibility, ontheotherhand,wehavesomearbitrarydevice,whichinteractswithoursystemandwhichcangenerateorremoveheatif desired. Thereisnothermal enclosure. The important constraintis thatthe devicestarts insome stateDandends upinthesamestateD. Asbeforeaweightmovesfromheighthtoheighth,whileoursystemstartsinstateXandendsupinstateY . Insymbols,(X, D, h) (Y, D, h) (2.9).Itisclearthat(2.8)isaspecial caseof(2.9), soweconcludethatX Y impliesX Y . Thedevicein(2.9)mayconsistofathermalpartinsomestateZandelectrical andmechanical partsinsomestateM. ThusD = (Z, M),and(2.9)clearlyimpliesthat(X, Z) (Y, Z).Itisnatural toassumethat satisesaxiomsA1-A6, justas does. Inthatcasewecaninferthecancellationlawfor ,i.e.,(X, Z) (Y, Z, )impliesX Y . Hence,X Y (whichiswhat (2.9) says) implies X Y . Altogether we have thus shown that and are really the samerelation. Inwords: adiabaticaccessibilitycanalways beachievedbyanadiabaticprocess appliedtothesystemplusadeviceand,furthermore, theadiabaticprocesscanbesimplied(althoughthismaynotbeeasytodoexperimentally)byeliminatingall thermodynamicpartsof thedevice, thusmakingtheprocessanadiabaticoneforthesystemalone.22D.TheconstructionofentropyforasinglesystemGivenastatespacewemay, asdiscussedinSubsectionI.A.1., constructitsmultiplescaledcopies,i.e.,statesoftheformY= (t1Y1, . . . , tNYN)withti> 0,Yi . ItfollowsfromourassumptionA5thatifCH(comparisonhypothesis)holdsinthestatespace(t1) (tN)witht1, ..., tNxed, thenanyotherstateof thesameform,Y=(t1Y1, . . . , tMYM)withYi, iscomparabletoY provided

iti=

j tj(butnot, ingeneral, ifthesumsarenotequal). ThisisprovedasfollowsforN=M=2; theeasyextensiontothegeneral caseislefttothereader. Sincet1+ t2=t1+ t2wecanassume, withoutlossofgenerality, that t1t1= t2t2> 0, because the case t1t1= 0 is already covered by CH (which wasassumed) for (t1)(t2). By the splitting axiom, A5, we have (t1Y1, t2Y2) A(t1Y1, (t1t1)Y1, t2Y2)and(t1Y1, t2Y2) A(t1Y1, (t1 t1)Y2, t2Y2). ThecomparabilitynowfollowsfromCHonthespace(t1)(t1t1)(t2).Theentropyprincipleforthestatesinthemultiplescaledcopiesofasinglesystemwill nowbederived. Moreprecisely,weshallprovethefollowingtheorem:THEOREM2.2(EquivalenceofentropyandassumptionsA1A6, CH).Letbeastatespaceandlet bearelationonthemultiplescaledcopiesof. Thefollowingstatementsareequivalent.(1) Therelation satisesaxiomsA1A6,andCHholdsforall multiplescaledcopiesof.(2) Thereisafunction,Sonthatcharacterizestherelationinthesensethatift1 + +tN=t1 + +tM,(forall N 1andM 1)then(t1Y1, ..., tNYN) (t1Y1, ..., tMYM)holdsifandonlyifN

i=1tiS(Yi) M

j=1tjS(Yj). (2.10)ThefunctionSisuniquelydeterminedon, uptoananetransformation, i.e.,anyotherfunctionSonsatisfying(2.10)isof theformS(X)= aS(X) + Bwithconstantsa > 0andB.Denition. A function Son that characterizes the relation on the multiple scaled copiesofinthesensestatedinthetheoremiscalledanentropyfunctionon.WeshallsplittheproofofTheorem2.2intoLemmas2.1,2.2,2.3andTheorem2.3below.At this point it is convenient tointroducethefollowingnotionof generalizedordering.While(a1X1, a2X2, . . . , aNXN)hassofaronlybeendenedwhenall ai>0, wecandenethemeaningoftherelation(a1X1, . . . , aNXN) (a1X1, . . . , aMXM) (2.11)forarbitraryai R, ai R, NandMpositiveintegersandXi i, Xi iasfollows. Ifanyai(orai)iszerowejustignorethecorrespondingterm. Example: (0X1, X2) (2X3, 0X4)meansthesamethingasX2 2X3. Ifanyai(orai)isnegative, justmoveaiXi(oraiXi)totheothersideandchangethesignofai(orai). Example:(2X1, X2) (X3, 5X4, 2X5, X6)23meansthat(2X1, 5X4, X2) (X3, 2X5, X6)in(2)1(5)42and3(2)56. (Recallthatab= ba). Itiseasytocheck,usingthecancellationlaw,thatthesplittingandrecombinationaxiomA5extendstononpositivescalingparameters, i.e., axiomsA1-A6implythatX A(aX, bX)forall a, b Rwitha + b=1, if therelation fornonpositiveaandbisunderstoodinthesensejustdecribed.For the denition of the entropy function we need the following lemma, which depends cruciallyon the stabilityassumption A6 and on the comparison hypothesis CH for the statespaces (1)

().LEMMA2.1SupposeX0andX1aretwopointsinwithX0 X1. For Rdeneo= X : ((1 )X0, X1) X. (2.12)Then(i)ForeveryX thereisa RsuchthatX o.(ii)ForeveryX ,sup : X o < .Remark. SinceX A((1 )X, X)byassumptionA5,thedenitionof oreallyinvolvestheorderrelationondoublescaledcopiesof(oronitself,if = 0or1.)Proof of Lemma2.1. (i)IfX0 XthenobviouslyX o0byaxiomA2. Forgeneral Xweclaimthat(1 +)X0 (X1, X) (2.13)forsome 0andhence((1 )X0, X1) Xwith= . Theproofreliesonstability, A6,andthecomparisonhypothesisCH(whichcomesintoplayforthersttime): If (2.13)werenottrue,thenbyCHwewouldhave(X1, X) (1 +)X0forall > 0andso,byscaling,A4,andA5_X1,1X__X0,1X0_.BythestabilityaxiomA6thiswouldimplyX1 X0incontradictiontoX0 X1.(ii) If sup: X o= , thenfor somesequenceof s tendingtoinnitywewouldhave((1 )X0, X) Xandhence(X0, X1) (X, X0)byA3andA5. ByA4thisimplies_1X0, X1__1X, X0_andhenceX1 X0bystability,A6.We can now state our formulafortheentropyfunction. If all points in are adiabaticallyequivalentthereisnothingtoprove(theentropyisconstant), sowemayassumethattherearepointsX0,X1 withX0 X1. WethendeneforX S(X) := sup : ((1 )X0, X1) X. (2.14)(Thesymbol a:=bmeansthataisdenedbyb.) ThisSwill bereferredtoasthecanonicalentropyonwithreferencepointsX0andX1. ThisdenitionisillustratedinFigure2.-InsertFigure2here-24ByLemma2.1S(X)iswell denedandS(X)< forall X. (Notethatbystabilitywecould replace by in (2.14).) We shall now show that this Shas all the right properties. Therststepisthefollowingsimplelemma,whichdoesnotdependonthecomparisonhypothesis.LEMMA2.2(isequivalentto ). SupposeX0 X1arestatesanda0, a1, a0, a1arereal numberswitha0 +a1= a0 +a1. Thenthefollowingareequivalent.(i) (a0X0, a1X1) (a0X0, a1X1)(ii) a1 a1(andhencea0 a0).Inparticular, Aholdsin(i)ifandonlyifa1= a1anda0= a0.Proof: We give the proof assuming that the numbers a0, a1, a0, a1are all positive and a0+a1=a0 +a1= 1. Theothercasesaresimilar. Wewritea1= anda1= .(i) (ii). If>then, byA5andA3, ((1 )X0, X1, ( )X1) ((1 )X0, ( )X0, X1). Bythecancellationlaw, Theorem2.1, (( )X1) (( )X0). Byscalinginvariance,A5,X1 X0,whichcontradictsX0 X1.(ii) (i). Thisfollowsfromthefollowingcomputation.((1 )X0, X1) A((1 )X0, ()X0, X1) (byaxiomsA3andA5) ((1 )X0, ()X1, X1) (byaxiomsA3andA4)A((1 )X0, X1) (byaxiomsA3andA5).The next lemma will imply, among other things, that entropyis unique, upto ananetransformation.LEMMA2.3(Characterizationofentropy). LetSdenotethecanonical entropy(2.14)onwithrespecttothereferencepointsX0 X1. IfX thentheequality = S(X)isequivalenttoX A((1 )X0, X1).Proof: First,if = S(X)then,bythedenitionofsupremum,thereisasequence1 2 . . . 0convergingtozero,suchthat((1 ( n))X0, ( n)X1) Xforeachn. Hence,byA5,((1 )X0, X1, nX0) A((1 +n)X0, ( n)X1, nX1) (X, nX1),andthus((1 )X0, X1) XbythestabilitypropertyA6. Ontheotherhand, sinceisthesupremumwehaveX ((1 ( +)X0, ( +)X1)forall > 0bythecomparisonhypothesisCH.Thus,(X, X0) ((1 )X0, X1, X1),25so,byA6,X ((1 )X0, X1). ThisshowsthatX A((1 )X0, X1)when = S(X).Conversely, if [0, 1] is suchthat XA((1 )X0, X1), then((1 )X0, X1) A((1 )X0, X1)bytransitivity. Thus, = byLemma2.2.Remark1: Without the comparisonhypothesis we couldnd that S(X0) = 0 and S(X) = 1forallXsuchthatX0 X.Remark2: FromLemma2.3andthecancellationlawitfollowsthatthecanonical entropywithreferencepointsX0 X1satises0 S(X) 1if andonlyif Xbelongstothestrip(X0, X1)denedby(X0, X1) := X : X0 X X1 .LetusmakethedependenceofthecanonicalentropyonX0andX1explicitbywritingS(X) = S(X[X0, X1). (2.15)ForXoutsidethestripwecanthenwriteS(X[X0, X1) = S(X1[X0, X)1ifX1 XandS(X[X0, X1) = S(X0[X, X1)1 S(X0[X, X1)ifX X0.Proof of Theorem 2.2:(1) =(2): Put i=S(Yi), i=S(Yi ). ByLemma2.3 we knowthat YiA((1 i)X0, iX1)andYiA((1 i)X0, iX1). BytheconsistencyaxiomA3andtherecombinationaxiomA5itfollowsthat(t1Y1, . . . , tNYN) A(

iti(1 i)X0,

itiiX1)and(t1Y1, . . . , tNYN) A(

iti(1 i)X0,

itiiX1).Statement(2)nowfollowsfromLemma2.2. Theimplication(2)=(1)isobvious.TheproofofTheorem2.2isnowcompleteexceptfortheuniquenesspart. Weformulatethispart separately in Theorem 2.3 below, which is slightly stronger than the last assertion in Theorem2.2. It implies that anentropyfunctionfor themultiplescaledcopies of is alreadyuniquelydetermined, uptoananetransformation, bytherelationonstatesoftheform((1 )X, Y ),i.e.,itrequiresonlythecaseN= M= 2,inthenotationofTheorem2.2.THEOREM2.3(Uniquenessofentropy)IfSisafunctiononthatsatises((1 )X, Y ) ((1 )X, Y)ifandonlyif(1 )S(X) +S(Y ) (1 )S(X) +S(Y),forall RandX, Y, X, Y ,thenS(X) = aS(X) +B26witha = S(X1) S(X0) > 0, B= S(X0).HereSisthecanonical entropyonwithreferencepointsX0 X1.Proof: This follows immediately from Lemma 2.3, which says that for every Xthere is a unique,namely = S(X),suchthatX A((1 )X, X) A((1 )X0, X1).Hence,bythehypothesisonS,and = S(X),wehaveS(X) = (1 )S(X0) +S(X1) = [S(X1) S(X0)]S(X) +S(X0).ThehypothesisonSalsoimpliesthata := S(X1) S(X0) > 0,becauseX0 X1.Remark: NotethatSisdenedon andsatisesS(X) = aS(X) +Bthere. Onthespace(t)acorrespondingentropyis, bydenition, givenbyS(t)(tX)=tS(X)=atS(X) + tB=aS(t)(tX) + tB, whereS(t)(tX)isthecanonical entropyon(t)withreferencepointstX0, tX1.Thus,S(t)(tX) ,= aS(t)(tX) +B (unlessB= 0,ofcourse).Itisapparentfromformula(2.14)thatthedenitionofthecanonical entropyfunctiononinvolves only the relation on the double scaled products (1)()besides the reference pointsX0andX1. Moreover, thecanonical entropyuniquelycharacterizes therelationonall multiplescaled copies of , which implies in particular that CH holds for all multiple scaled copies. Theorem2.3maythereforeberephrasedasfollows:THEOREM2.4 (The relationondouble scaledcopies determines the relationeverywhere). Let and betworelationsonthemultiplescaledcopiesofsatisfyingaxiomsA1-A6, and also CH for (1)()for each xed [0, 1]. If and coincide on (1)()foreach [0, 1],then and coincideonall multiplescaledcopiesof,andCHholdsonallthemultiplescaledcopies.TheproofofTheorem2.2isnowcomplete.E.ConstructionofauniversalentropyintheabsenceofmixingIn theprevious subsectionweshowedhow toconstructanentropyfor asinglesystem,, thatexactlydescribestherelation withinthestatesobtainedbyformingmultiplescaledcopiesof.Itisuniqueuptoamultiplicativeconstanta>0andanadditiveconstantB, i.e., towithinananetransformation. Weremindthereaderthatthisentropywasconstructedbyconsidering justtheproductoftwoscaledcopiesof,butouraxiomsimpliedthatitautomaticallyworkedforallmultiplescaledcopiesof. WeshallrefertoaandBasentropyconstantsforthesystem.Ourgoal istoputtheseentropiestogether andshowthattheybehaveintherightwayonproductsofarbitrarilymanycopiesofdierentsystems. Moreover,thisuniversalentropywillbeuniqueuptoonemultiplicativeconstantbutstillmanyadditiveconstants. Thecentralquestionhereisoneof calibration , whichistosaythatthemultiplicativeconstantinfrontof eachele-mentaryentropyhastobe choseninsuchawaythattheadditivityrule(2.4)holds. Itisnotevenobviousyetthattheadditivitycanbemadetoholdatall,whateverthechoiceofconstants.27Letusnotethatthenumberofadditiveconstantsdependsheavilyonthekindsofadiabaticprocesses available. The system consistingofonemoleof hydrogenmixedwithonemoleof heliumand the system consisting of one mole of hydrogen mixed with two moles of helium are dierent. Theadditiveconstantsareindependent unlessaprocessexistsinwhichbothsystemscanbeunmixed,and thereby making the constants comparable. In nature we expect only 92 constants, one for eachelementof theperiodictable, unlessweallownuclearprocessesaswell, inwhichcasethereareonly two constants (for neutrons and for hydrogen). On the other hand, if un-mixing is not alloweduncountably many constants are undetermined. In SectionVI we address the question of adiabaticprocessesthatunmixmixturesandreversechemicalreactions. Thatsuchprocessesexistisnotsoobvious.Tobeprecise, theprincipalgoal ofthissubsectionistheproofofthefollowingTheorem2.5,whichisacaseoftheentropyprinciplethatisspecialinthatitisrestrictedtoprocessesthatdonotinvolvemixingorchemicalreactions. ItisageneralizationofTheorem2.2.THEOREM2.5(Consistententropyscales). Considerafamilyofsystemsfulllingthefollowingrequirements:(i) Thestatespacesofanytwosystemsinthefamilyaredisjointsets,i.e.,everystateofasysteminthefamilybelongstoexactlyonestatespace.(ii) All multiplescaledproductsofsystemsinthefamilybelongalsotothefamily.(iii) Everysysteminthefamilysatisesthecomparisonhypothesis.ForeachstatespaceofasysteminthefamilyletSbesomedeniteentropyfunctionon. ThenthereareconstantsaandBsuchthatthefunctionS,denedforallstatesinallsbyS(X) = aS(X) +BforX ,hasthefollowingproperties:a). IfXandY areinthesamestatespacethenX Y ifandonlyif S(X) S(Y ).b). Sisadditiveandextensive,i.e.,S(X, Y ) = S(X) +S(Y ). (2.4)and,fort > 0,S(tX) = tS(X). (2.5)Remark. Note that 1and 12are disjoint as sets for any (nonempty) statespaces 1and2.Proof: Fix some system 0and two points Z0 Z1in 0. In eachstate space choose somexedpointX insuchawaythattheidentitiesX12= (X1, X2) (2.16)Xt= tX(2.17)28hold. Withtheaidortheaxiomof choicethiscanbeachievedbyconsideringtheformal vectorspacespannedbyallsystemsandchoosingaHamelbasisofsystems inthisspacesuchthatevery system can be written uniquely as a scaled product of a nite number of the s. (See Hardy,LittlewoodandPolya, 1934). Thechoiceof anarbitrarystateXineachof theseelementarysystemsthendenesforeachauniqueXsuchthat(2.17)holds. (If thereaderdoesnotwishtoinvoketheaxiomofchoicethenanalternativeistohypothesizethateverysystemhasauniquedecompositionintoelementarysystems; thesimplesystemsconsideredinthenextsectionobviouslyqualifyastheelementarysystems.)ForX weconsiderthespace 0withitscanonicalentropyasdenedin(2.14),(2.15)relativetothepoints(X, Z0)and(X, Z1). UsingthisfunctionwedeneS(X) = S0((X, Z0) [(X, Z0), (X, Z1)). (2.18)Note: Equation(2.18)xestheentropyofXtobezero.LetusdenoteS(X)bywhich,byLemma2.3,ischaracterizedby(X, Z0) A((1 )(X, Z0), (X, Z1)).Bythecancellationlawthisisequivalentto(X, Z0) A(X, Z1)). (2.19)By(2.16)and(2.17)thisimmediatelyimpliestheadditivityandextensivityofS. Moreover,sinceX Y holdsifandonlyif(X, Z0) (Y, Z0)itisalsoclearthatSisanentropyfunctiononany. HenceSandSarerelatedbyananetransformation,accordingtoTheorem2.3.Denition(Consistent entropies). Acollectionof entropyfunctionsSonstatespacesiscalledconsistentiftheappropriatelinearcombinationofthefunctionsisanentropyfunctiononall multiplescaledproductsof thesestatespaces. Inotherwords, thesetisconsistentif themultiplicativeconstantsa,referredtoinTheorem2.5,canallbechosenequalto1.ImportantRemark:Fromthedenition,(2.14), ofthecanonical entropyand(2.19)itfollowsthattheentropy(2.18)isgivenbytheformulaS(X) = sup : (X, Z1) (X, Z0) (2.20)forX . Theauxiliarysystem0canthusberegardedasanentropymeter inthespiritof(LewisandRandall,1923)and(Giles, 1964). Sincewehavechosentodenetheentropyforeachsystemindependently,byequation(2.14), theroleof 0inourapproachissolelytocalibratetheentropyofdierentsystemsinordertomakethemconsistent.Remarkabout thephotongas: AswediscussedinSectionII.Bthephotongasisspecial andtherearetwowaystoviewit. Onewayistoregardthescaledcopies(t)asdistinctsystemsandthe other is to say that there is only one and the scaled copies are identical to it and, in particular,musthaveexactlythesameentropyfunction. Weshallnowseehowtherstpointofviewcanbereconciledwiththelatterrequirement. Note,rst,thatinourconstructionabovewecannottakethepoint(U, V ) = (0, 0)tobetheducialpointXbecause(0, 0)isnotinourstatespacewhich,accordingtothediscussioninSectionIIIbelow, hastobeanopensetandhencecannotcontainanyof itsboundarypointssuchas(0, 0). Therefore, wehavetomakeanotherchoice, soletus29takeX= (1, 1). Buttheconstructionintheproof abovesetsS(1, 1) = 0andthereforeS(U, V )will nothavethehomogeneousformShom(U, V )=V1/4U3/4. Nevertheless, theentropiesof thescaled copies will be extensive,as required by the theorem. If one feels that all scaled copies shouldhave the same entropy (because they represent the same physical system) then the situation can beremediedinthefollowingway: WithS(U, V )beingtheentropyconstructedasintheproof using(1, 1), wenotethatS(U, V )= Shom(U, V ) + BwiththeconstantBgivenbyB= S(2, 2).Thisfollowsfromsimplealgebraandthefactthatweknowthattheentropyof thephotongasconstructedinourproofmustequal Shomtowithinanadditiveconstant. (Thereadermightaskhowweknowthisandtheansweristhattheentropyof thegas isuniqueuptoadditiveandmultiplicativeconstants, thelatterbeingdeterminedbythesystemofunitsemployed. Thus,theentropydeterminedbyourconstructionmustbethecorrectentropy,uptoanadditiveconstant,and this correct entropy is what it is, as determined by physical measurement. Hopefully it agreeswiththefunctiondeducedin(LandauandLifschitz, 1969).) Letususeourfreedomtoaltertheadditive constants as we please, provided we maintain the extensivity condition (2.5). It will not beuntilSectionVIthatwehavetoworry abouttheadditiveconstantspersebecauseitisonlytherethat mixing and chemical reactions are treated. Therefore, we redene the entropy of the state spaceofthephotongastobeS(U, V ) := S(U, V ) + S(2, 2). whichisthesameasShom(U, V ). Wealsohavetoaltertheentropyofthescaledcopiesaccordingtotherulethatpreservesextensivity,namelyS(t) (U, V ) S(t) (U, V ) + tS(2, 2)=S(t) (U, V ) + S(t) (2t, 2t)=Shom(U, V ). Inthisway, all thescaledcopiesnowhavethesame(homogeneous)entropy, butweremindthereaderthatthesameconstructioncouldbecarriedoutforanymaterialsystemwithahomogeneous(or,moreexactlyanane)entropyfunctionifoneexisted. Fromthethermodynamicviewpoint,thephotongasisunusualbutnotspecial.F.ConcavityofentropyUptonowwehavenotused,orassumed,anygeometricpropertyofastatespace. Itisanimportantstabilitypropertyofthermodynamicalsystems,however,thattheentropyfunctionisaconcavefunctionofthestatevariablesarequirementthatwasemphasizedbyMaxwell, Gibbs,Callenandmanyothers. Concavityalsoplaysanimportantroleinthedenitionoftemperature,asinsectionV.Inordertohavethisconcavityitisrstnecessarytomakethestatespaceonwhichentropyisdenedintoaconvexset, andfor thispurposethechoiceof coordinates isimportant. Here,webeginthediscussionofconcavitybydiscussingthisgeometricpropertyoftheunderlyingstatespaceandsomeoftheconsequencesoftheconvexcombinationaxiomA7fortherelation ,tobegivenafterthefollowingdenition.Denition: By a statespacewithaconvexstructure, or simply a convexstatespace,wemeanastatespace,thatisaconvexsubsetofsomelinearspace,e.g.,Rn. Thatis,ifXandY areanytwopointsinandif0 t 1,thenthepointtX + (1 t)Y isawell-denedpointin. Aconcavefunction,S,onisonesatisfyingtheinequalityS(tX + (1 t)Y ) tS(X) + (1 t)S(Y ). (2.21)Ourbasicconvexcombinationaxiomfortherelation isthefollowing.30A7) Convexcombination. AssumeXandY arestatesinthesameconvexstatespace, . Fort [0, 1] lettXand(1 t)Y bethecorrespondingstatesoftheirtscaledand(1 t)scaledcopies,respectively. Thenthepoint(tX, (1 t)Y )intheproductspace(t)(1t)satises(tX, (1 t)Y ) tX + (1 t)Y . (2.22)Notethattherightsideof (2.22) isinandisdenedbyordinaryconvexcombinationofpointsintheconvexset.ThephysicalmeaningofA7ismoreorlessevident,but itisessentialtonotethattheconvexstructuredependsheavilyonthechoiceof coordinatesfor. A7meansthatif wetakeabottlecontaining1/4molesof nitrogenandonecontaining3/4moles(withpossiblydierentpressuresanddensities), andif wemixthemtogether, thenamongthestatesof onemoleofnitrogenthatcanbereachedadiabaticallythereisoneinwhichtheenergyisthesumofthetwoenergiesand,likewise,thevolumeisthe sum of thetwovolumes. Again,weemphasize thatthechoiceofenergyandvolumeasthe(mechanical)variableswithwhichwecanmakethisstatementisanimportantassumption. If, for example,temperature and pressure were used instead,the statementwouldnotonlynothold,itwouldnotevenmakemuchsense.The physical example aboveseems not exceptionablefor liquids and gases. On the other handitisnotentirelyclearhowtoascribeanoperationalmeaningtoaconvexcombinationinthestatespace of a solid, and the physical meaning of axiom A7 is not as obvious in this case. Note, however,thatalthoughconvexityisaglobal property, itcanoftenbeinferredfromalocal propertyoftheboundary. (Aconnectedsetwithasmoothboundary,forinstance,isconvexifeverypointontheboundary has a neighbourhood, whose intersectionwiththe setis convex.) In suchcasesit sucestoconsiderconvexcombinationsofpointsthatareclosetogetherandclosetotheboundary. Forsmall deformationof anisotropicsolidthesixstraincoordinates, multipliedbythevolume, canbetakenasworkcoordinates. Thus,A7amountstoassumingthataconvexcombinationofthesecoordinatescanalwaysbeachievedadiabatically. See,e.g.,(Callen,1985).IfX wedenotebyAXtheset Y : X Y . AXiscalledtheforwardsector ofXin. Moregenerally,ifisanothersystem,wecallthesetY : X Y ,theforwardsectorofXin.Usuallythisconceptisappliedtothecaseinwhichandareidentical, butitcanalsobeuseful incasesinwhichonesystemischangedintoanother; anexampleisthemixingof twoliquids intwocontainers(inwhich case is a compound system)intoa thirdvesselcontainingthemixture(inwhichcaseissimple).ThemaineectofA7isthatforwardsectorsareconvexsets.THEOREM2.6(Forwardsectors are convex). Let andbe state spaces of twosystems,withaconvexstatespace. AssumethatA1A5holdforandand,inaddition,A7holdsfor. ThentheforwardsectorofXin, denedabove, isaconvex subsetofforeachX .Proof: SupposeXY1andXY2andthat 0 0dependingonthesystem(butnotonthestate).A point in Rn+1is written as X= (U, V ) with Ua distinguished coordinate called the internalenergyandwithV= (V1, . . . , Vn) Rn. ThecoordinatesViarecalledtheworkcoordinates.We could, if we wished, consider the case n=0, inwhichcase we wouldhave asystemwhosestatesareparametrizedbytheenergyalone. Suchasystemiscalledathermometeroradegenerate simple system. These systems must be (and will be in Section IV) treated separatelybecausetheywill fail tosatisfythetransversalityaxiomT4, introducedinSectionIV. Fromthepointof viewof theconvexit