Seite - 49 - in Differential Geometrical Theory of Statistics

entropy Article GeometricTheoryofHeat fromSouriauLieGroups ThermodynamicsandKoszulHessianGeometry: Applications inInformationGeometryfor ExponentialFamilies FrédéricBarbaresco AdvancedRadarConceptsBusinessUnit,ThalesAirSystems,Limours91470,France; frederic.barbaresco@thalesgroup.com AcademicEditor:AdomGiffin Received: 4August2016;Accepted: 27September2016;Published: 4November2016 Abstract: Weintroduce the symplectic structureof informationgeometrybasedonSouriau’sLie group thermodynamicsmodel,with a covariant definition ofGibbs equilibriumvia invariances throughco-adjointactionofagrouponitsmomentspace,definingphysicalobservables likeenergy, heat, andmoment as pure geometrical objects. Using geometric Planck temperature of Souriau model and symplectic cocycle notion, theFishermetric is identifiedas a Souriaugeometric heat capacity. TheSouriaumodel isbasedonaffinerepresentationofLiegroupandLiealgebra thatwe comparewithKoszulworksonG/Khomogeneousspaceandbijectivecorrespondencebetweenthe setofG-invariantflatconnectionsonG/Kandthesetofaffinerepresentationsof theLiealgebraof G. In theframeworkofLiegroupthermodynamics,anEuler-Poincaréequation iselaboratedwith respect to thermodynamicvariables, andanewvariationalprincipal for thermodynamics isbuilt throughaninvariantPoincaré-Cartan-Souriau integral. TheSouriau-Fishermetric is linkedtoKKS (Kostant–Kirillov–Souriau) 2-formthat associates a canonicalhomogeneous symplecticmanifold to the co-adjoint orbits. Weapply thismodel in the frameworkof informationgeometry for the actionof anaffinegroup for exponential families, andprovide some illustrationsofuse cases for multivariategaussiandensities. Informationgeometryispresentedinthecontextof theseminalwork ofFréchetandhisClairaut-Legendreequation. TheSouriaumodelof statisticalphysics isvalidated as compatiblewith theBaliangaugemodelof thermodynamics. Werecall theprecursorworkof Casalisonaffinegroupinvariance fornaturalexponential families. Keywords:Liegroupthermodynamics;momentmap;Gibbsdensity;Gibbsequilibrium;maximum entropy; informationgeometry; symplecticgeometry;Cartan-Poincaré integral invariant;geometric mechanics;Euler-Poincaréequation;Fishermetric;gaugetheory;affinegroup Lorsque le faitqu’onrencontreestenoppositionavecune théorie régnante, il fautaccepter le fait et abandonner la théorie, alorsmêmequecelle-ci, soutenuepardegrandsnoms, estgénéralementadoptée —ClaudeBernard in“Introductionà l’Étudede laMédecineExpérimentale” [1] Au départ, la théorie de la stabilité structurelle m’avait paru d’une telle ampleur et d’une telle généralité, qu’avec elle je pouvais espérer en quelque sorte remplacer la thermodynamiquepar lagéométrie,géométriserenuncertainsens lathermodynamique, éliminer des considérations thermodynamiques tous les aspects à caractèremesurable et stochastiques pour ne conserver que la caractérisation géométrique correspondante desattracteurs. —RenéThomin“Logoset théoriedesCatastrophes” [2] Entropy2016,18, 386 49 www.mdpi.com/journal/entropy