Page - 59 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 der Thermodynamik) by Constantin Carathéodory based on Carnot’s works [77]. Carathéodory introducedentropythroughamathematicalapproachbasedonthegeometricbehaviorofacertain classofpartialdifferential equationscalledPfaffians.Carathéodory’s investigationsstartbyrevisiting the first law and reformulating the second law of thermodynamics in the form of two axioms. Thefirstaxiomapplies toamultiphasesystemchangeunderadiabaticconditions (axiomofclassical thermodynamics due to Clausius [78,79]). The second axiom assumes that in the neighborhood of any equilibrium state of a system (of any number of thermodynamic coordinates), there exist states that are inaccessible by reversible adiabatic processes. In the book of Misha Gromov “Metric Structures for Riemannian andNon-Riemannian Spaces”,written and edited byPierre Pansu and JacquesLafontaine, a newmetric is introduced called “Carnot-Carathéodorymetric”. In one of his papers,MishaGromov [80,81] gives historical remarks “This result (which seems obvious by the modern standards) appears (in amore general form) in the 1909-paper by Carathéorody on formalization of the classical thermodynamicswhere horizontal curves roughly correspond to adiabatic processes. In fact, theaboveproofmaybeperformed in the languageofCarnot (cycles) and for this reason themetrisdistHwere christened ‘Carnot-Carathéodory’ inGromov-Lafontaine-Pansubook”[82].WhenIaskthisquestiontoPierre Pansu,hegavemetheanswer that“Thesection4of [76], entitledHilfsatzausderTheoriedesPfaffschen Gleichungen(Lemmafromthe theoryofPfaffianequations)openswithastatement relating to thedifferential 1-forms. Carathéodorysays, If aPfaffianequationdx0+X1dx1+X2dx2+ ...+Xndxn=0 isgiven, inwhich theXiarefinite, continuous,differentiable functionsof thexi, andoneknows that inanyneighborhoodof an arbitrarypointPof the spaceofxi there is apoint that one cannot reachalongacurve that satisfies this equation then the expressionmustnecessarily possess amultiplier thatmakes it into a complete differential”. This is confirmedinthe introductionofhispaper [76],whereCarathéodorysaid“Finally, inorder tobeable to treat systemswitharbitrarilymanydegrees of freedomfromtheoutset, insteadof theCarnot cycle that is almost alwaysused, but is intuitiveandeasy tocontrol only for systemswith twodegreesof freedom,onemust employa theoremfromthe theoryofPfaffiandifferential equations, forwhicha simpleproof isgiven in the fourth section”. Wehavealso tomakereference toHenriPoincaré [83] thatpublishedthepaper“Onattemptsof mechanical explanation for theprinciples of thermodynamics (Sur les tentativesd’explicationmécaniquedes principes de la thermodynamique)” at the Comptes rendus de l’Académie des sciences in 1889 [84], in which he tried to consolidate links between mechanics and thermomechanics principles. TheseelementswerealsodevelopedinPoincaré’s lectureof1892[85]on“thermodynamique” inChapter XVII“Reductionof thermodynamicsprinciples to thegeneralprinciples ofmechanics (Réductiondesprincipes de laThermodynamiqueauxprincipesgénérauxde lamécanique)”. Poincaréwrites inhisbook[85]“It is otherwisewith the second lawof thermodynamics. Clausiuswas thefirst toattempt tobring it to theprinciples ofmechanics, butnot succeedsatisfactorily.Helmholtz inhismemoirontheprinciple of least actionsdevelopeda theorymuchmoreperfect thanthatofClausius.However, it cannotaccount for irreversiblephenomena. (Il enest autrementdusecondprincipede la thermodynamique. Clausius, a lepremier, tentéde le ramenerauxprincipes de laMécanique,mais sansyréussird’unemanière satisfaisante.Helmoltzdans sonmémoire sur leprincipede lamoindreaction, adéveloppéune théoriebeaucoupplusparfaite quecelledeClausius; cependant ellenepeut rendrecomptedesphénomènes irréversibles.)”.AboutHelmoltzwork,Poincaréobserves [85]“It follows from these examples that theHelmholtz hypothesis is true in the case of body turning around an axis; So it seemsapplicable tovortexmotionsofmolecules (Il résultede ces exemplesque l’hypothèsed’Helmoltz est exacte dans le casde corps tournantautourd’unaxe; elleparaitdoncapplicable auxmouvements tourbillonnairesdes molecules.)”,butheadds in the followingthat theHelmoltzmodel isalso true in thecaseofvibrating motionsasmolecularmotions.However,hefinallyobserves that theHelmoltzmodelcannotexplain the increasingofentropyandconcludes [85]“All attemptsof thisnaturemustbeabandoned; theonlyones thathaveanychanceof successare thosebasedonthe interventionof statistical laws, for example, thekinetic theoryofgases. Thisview,which I cannotdevelophere, canbe summedup ina somewhatvulgarwayas follows: Supposewewant toplaceagrainof oats in themiddleof aheapofwheat; itwill be easy; thensupposewewanted tofind it andremove it;wecannotachieve it. All irreversiblephenomena, according to somephysicists,would bebuilt on thismodel (Toutes les tentativesdecettenaturedoiventdoncêtreabandonnées; les seulesquiaient 59