Page - 58 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 of Space-Time. For eachpointX, there is a “temperature vector” β(X), such it is an infinitesimal conformal transformof themetricof theuniversegij. Conservationequationscanthenbededuced for components of impulsion-energy tensor Tij and entropy flux Sj with ∂ˆiTij = 0 and ∂iSj = 0. Temperatureandmetricarerelatedbythe followingequations:⎧⎨⎩ ∂ˆiβj+ ∂ˆjβi=λgij∂iβj+∂jβi−2Γkijβk=λgij with { ∂ˆi. : covariantderivative βj : componentofTemperaturevector λ=0⇒ KillingEquation (22) Leon Brillouin made the link between Boltzmann entropy and Negentropie of information theory [68–71], but before Jean-Marie Souriau, only Constantin Carathéodory and Pierre Duhem[72–75] initiatedfirst theoreticalworks togeneralize thermodynamics. After threeyearsas lectureratLilleuniversity,Duhempublishedapaper in theofficial revue of theEcoleNormaleSupérieure, in1891, “Ongeneral equationsof thermodynamics” [72] (Sur les équationsgénéralesde laThermodynamique) inAnnalesScientifiquesde l’EcoleNormaleSupérieure. Duhemgeneralized theconceptof“virtualwork”under theactionof“externalactions”bytaking into accountbothmechanicalandthermalactions. In1894, thedesignofageneralizedmechanicsbased onthermodynamicswas furtherdeveloped: ordinarymechanicshadalreadybecome“aparticular caseofamoregeneral science”.Duhemwrites“Wemadedynamicsa special case of thermodynamics, a science that embraces commonprinciples inall changesof statebodies, changesofplacesaswell as changes in physical qualities” (Nousavons fait de ladynamiqueuncasparticulierde la thermodynamique,uneSciencequi embrassedansdesprincipes communs tous les changementsd’étatdes corps, aussi bien les changementsde lieu que les changementsdequalitésphysiques). Intheequationsofhisgeneralizedmechanics-thermodynamics, somenewtermshadtobe introduced, inorder toaccount for the intrinsicviscosityandfrictionof the system.AsobservedbyStefanoBordoni,Duhemaimedatwideningthescopeofphysics: thenewphysics couldnotconfineitself to“localmotion”buthadtodescribewhatDuhemqualified“motionsofmodification”. If Boltzmann had tried to proceed from “localmotion” to attain the explanation ofmore complex transformations,Duhemwas trying toproceed fromgeneral lawsconcerninggeneral transformation inorder to reach“localmotion”asasimplifiedspecific case. Four scientistswere creditedbyDuhem with having carried out “themost important researches on that subject”: Massieu hadmanaged to derive thermodynamics froma “characteristic function and its partial derivatives”; Gibbs had shownthatMassieu’s functions“couldplay theroleofpotentials in thedeterminationof thestatesof equilibrium”inagivensystem;vonHelmholtzhadput forward“similar ideas”;vonOettingenhad given“anexpositionof thermodynamicsof remarkablegenerality”basedongeneraldualityconcept in“Die thermodynamischenBeziehungenantithetischentwickelt”publishedatSt.Petersburg in1885. Duhemtookintoaccountasystemwhoseelementshadthesametemperatureandwhere thestateof thesystemcouldbecompletelyspecifiedbygivingits temperatureandnother independentquantities. He then introduced some “external forces”, andheld the system in equilibrium. Avirtualwork correspondedtosuchforces, andasetofn+1equationscorrespondedto theconditionofequilibrium of thephysical system. Fromthe thermodynamicpointofview,every infinitesimal transformation involving thegeneralizeddisplacementshad toobey to thefirst law,which couldbeexpressed in termsof the (n+1)generalizedLagrangianparameters. Theamountofheatcouldbewrittenasasum of (n+1) terms. Thenewalliancebetweenmechanicsandthermodynamics ledtoasortof symmetry betweenthermalandmechanicalquantities. Then+1functionsplayedtheroleofgeneralized thermal capacities, andthe last termwasnothingother thantheordinary thermal capacity. Theknowledgeof the “equilibriumequationsof a system”allowedDuhemtocompute thepartialderivativesof the thermal capacitywith regard to all theparameterswhichdescribed the state of the system, apart from its derivativewith regard to temperature. The thermal capacitieswere thereforeknown“except for an unspecified functionof temperature”. Theaxiomaticapproachofthermodynamicswaspublishedin1909inMathematischeAnnalen[76] under the title“Examinationof theFoundationsofThermodynamics” (UntersuchungenüberdieGrundlagen 58