Page - 57 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 to that Clausius hasmade by linking the Carnot’s theorem to entropy” [66]. The finalmanuscriptwas publishedbyMassieu in1873, “Exposédesprincipes fondamentauxde la théoriemécaniquede la chaleur (Notedestinée à servird’introductionauMémoirede l’auteur sur les fonctions caractéristiquesdesdiversfluides et la théoriedesvapeurs)” [63]. Massieu introducedthe followingpotentialΦ(β), called“characteristic function”,as illustrated inFigure2, that is thepotentialusedbySouriautogeneralize the theory: s(Q)= 〈β,Q〉−Φ(β) ⇒ β= 1T Φ= QT −S. However, inhis thirdpaper,Massieuwas influencedbyM.Bertrand, as illustrated in Figure3, to replace thevariableβ= 1T (thatheused inhis twofirstpapers)byT. Wehave then to wait50yearsmore for thepaperofPlanck,whointroducedagain thegoodvariableβ= 1T, andthen generalizedbySouriau,givingtoPlancktemperatureβanontologicalandgeometricstatusaselement of theLiealgebraof thedynamicgroup. Figure2.Extract fromthesecondpaperofFrançoisMassieutotheFrenchAcademyofSciences[61,62]. Figure3. RemarkofMassieu in1876paper [64],whereheexplainedwhyhe took intoaccount the “goodadvice”ofBertrandtoreplacevariable1/T,used inhis initialpaperof1869,bythevariableT. ThisLiegroupthermodynamicsofSouriau isable toexplainastronomicalphenomenon(rotation ofcelestialbodies: theEarthandthestars rotatingabout themselves). Thegeometric temperatureβ canbealso interpretedasa space-timevector (generalizationof the temperaturevectorofPlanck), where the temperaturevectorandentropyfluxare indualityunifyingheatconductionandviscosity (equationsofFourierandNavier). Incaseofcentrifugesystem(e.g.,usedforenrichmentofuranium), theGibbsEquilibriumstate [60,67]aregivenbySouriauequationsas thevariation inconcentrationof thecomponentsofaninhomogeneousgas.Classicalstatisticalmechanicscorrespondstothedynamical groupof timetranslations, forwhichwerecover fromSouriauequations theconceptsandprinciples of classical thermodynamics (temperature, energy,heat,work, entropy, thermodynamicpotentials) andof thekinetic theoryofgases (pressure, specificheats,Maxwell’svelocitydistribution,etc.). Souriaualsostudiedcontinuousmediumthermodynamics,where the“temperaturevector” isno longerconstrainedtobe inLiealgebra,butonlycontrainedbyphenomenologicequations(e.g.,Navier equations, etc.). For thermodynamicequilibrium, the“temperaturevector” is thenaKillingvector 57