Page - 54 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 thermodynamic concepts. In Section 3, we develop and revisit the Lie group thermodynamics model of Jean-Marie Souriau in modern notations. In Section 4, we make the link between SouriauRiemannianmetric and Fishermetric defined as a geometric heat capacity of Lie group thermodynamics. InSection5,weelaborateEuler-LagrangeequationsofLiegroupthermodynamics andavariationalmodelbasedonPoincaré-Cartan integral invariant. InSection6,weexploreSouriau affine representationofLiegroupandLiealgebra (including thenotionsof: affine representations andcocycles,Souriaumomentmapandcocycles, equivarianceofSouriaumomentmap,actionofLie grouponasymplecticmanifoldanddualspacesoffinite-dimensionalLiealgebras)andweanalyzethe linkandparallelismswithKoszulaffinerepresentation,developedinanothercontext (comparison is synthetized ina table). InSection7,we illustrateKoszulandSouriauLiegroupmodelsof information geometry formultivariateGaussiandensities. InSection8,after identifyingtheaffinegroupacting for these densities, we compute the Souriaumomentmap to obtain the Euler-Poincaré equation, solvedbygeodesic shootingmethod. InSection9, SouriauRiemannianmetricdefinedbycocycle formultivariateGaussiandensities is computed. Wegiveaconclusion inSection10with research prospects in the frameworkof affinePoissongeometry [13], Bismut stochasticmechanics [14] and secondorderextensionof theGibbsstate [15,16].Wehavethreeappendices:AppendixAdevelops the Clairaut(-Legendre)equationofMauriceFréchetassociatedto“distinguishedfunctions”asaseminal equationof informationgeometry;AppendixBisaboutaBalianGaugemodelof thermodynamicsand its compliancewith theSouriaumodel;AppendixCisdevotedto the linkofCasalis-Letac’sworkson affinegroupinvariance fornaturalexponential familieswithSouriau’sworks. 2. PositionofSouriauSymplecticModelofStatisticalPhysics inHistoricalDevelopmentsof ThermodynamicConcepts In thisSection,wewill explain theemergenceof thermodynamicconcepts thatgiverise to the generalizationof theSouriaumodelofstatisticalphysics. TounderstandSouriau’s theoreticalmodel ofheat,wehave toconsiderfirsthiswork ingeometricmechanicswherehe introducedtheconceptof “momentmap”and“symplecticcohomology”.Wewill then introduce theconceptof“characteristic function”developedbyFrançoisMassieu,andgeneralizedbySouriauonhomogeneoussymplectic manifolds. Inhisstatisticalphysicsmodel,Souriauhasalsogeneralizedthenotionof“heatcapacity” thatwas initially extendedbyPierreDuhemas akey structure to jointly considermechanics and thermodynamicsunder theumbrellaof the same theory. PierreDuhemhasalso integrated, in the corpus, theMassieu’scharacteristic functionasa thermodynamicpotential. Souriau’s idea todevelop acovariantmodelofGibbsdensityonhomogeneousmanifoldwasalso influencedby theseminal workofConstantinCarathéodory thataxiomatizedthermodynamics in1909basedonCarnot’sworks. Souriauhasadaptedhisgeometricmechanicalmodel for the theoryofheat,whereHenriPoincarédid notsucceed inhispaperonattemptsofmechanicalexplanationfor theprinciplesof thermodynamics. Lagrange’s works on “mécanique analytique (analyticmechanics)” has been interpreted by Jean-MarieSouriau in the frameworkofdifferentialgeometryandhas initiatedanewdisciplinecalled afterSouriau,“mécaniquegéométrique (geometricmechanics)” [17–19]. Souriauhasobservedthat the collectionofmotionsofadynamical systemisamanifoldwithanantisymmetricflat tensor that isa symplectic formwhere thestructurecontainsall thepertinent informationof thestateof thesystem (positions, velocities, forces, etc.). Souriausaid: “CequeLagrange avu, quen’a pas vuLaplace, c’était la structure symplectique (What Lagrange saw, that Laplace didn’t see, was the symplectic structure” [20]. Using the symmetries of a symplecticmanifold, Souriau introduced amappingwhich he called the“momentmap” [21–23],which takes its values ina spaceattached to thegroupof symmetries (in thedual spaceof itsLiealgebra). He [10] calleddynamicalgroupseverydimensionalgroupof symplectomorphisms (an isomorphismbetween symplecticmanifolds, a transformationof phase space that isvolume-preserving),andintroducedGalileogroupforclassicalmechanicsandPoincaré groupforrelativisticmechanics (botharesub-groupsofaffinegroup[24,25]). For instance,aGalileo 54