Page - 50 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 1. Introduction ThisMDPIEntropySpecial Issueon“DifferentialGeometrical Theoryof Statistics” collects a limitednumberofselected invitedandcontributedtalkspresentedduringtheGSI’15conferenceon “GeometricScienceof Information” inOctober2015. Thispaper isanextendedversionof thepaper [3] “Symplectic Structure of InformationGeometry: FisherMetric andEuler-PoincaréEquation of SouriauLie GroupThermodynamics”publishedinGSI’15Proceedings.AtGSI’15conference,aspecial sessionwas organizedon“liegroupsandgeometricmechanics/thermodynamics”,dedicatedtoJean-MarieSouriau’s works instatisticalphysics,organizedbyGerydeSaxcéandFrédéricBarbaresco,andaninvitedtalk on“ActionsofLiegroupsandLiealgebrasonsymplectic andPoissonmanifolds.Application toLagrangian andHamiltoniansystems”byCharles-MichelMarle,addressing“Souriau’s thermodynamicsofLiegroups”. Inhonorof Jean-MarieSouriau,whodied in2012andClaudeVallée [4–6],whopassedawayin2015, thisSpecial IssuewillpublishthreepapersonSouriau’s thermodynamics:Marle’spaperon“FromTools inSymplectic andPoissonGeometry toSouriau’sTheories ofStatisticalMechanicsandThermodynamics” [7], deSaxcé’spaperon“LinkbetweenLieGroupStatisticalMechanicsandThermodynamicsofContinua”[8]and thispublicationbyBarbaresco. Thispaperalsoproposesnewdevelopments, comparedtopaper [9] whererelationsbetweenSouriauandKoszulmodelshavebeen initiated. Thispaper, similar tothegoalof thepapersofMarleanddeSaxcé inthisSpecial Issue, is intended tohonorthememoryoftheFrenchPhysicist Jean-MarieSouriauandtopopularizehisworks,currently littleknown,onstatisticalphysicsandthermodynamics. Souriau iswellknownforhisseminaland major contributions ingeometricmechanics, thedisciplinehe created in the 1960s, fromprevious Lagrange’sworksthatheconceptualizedintheframeworkofsymplecticgeometry,butveryfewpeople knoworhaveexploitedSouriau’sworkscontainedinChapter IVofhisbook“Structuredes systèmes dynamiques”publishedin1970[10]andonly translated intoEnglish in1995 in thebook“Structureof DynamicalSystems:ASymplecticViewofPhysics” [11], inwhichheappliedthe formalismofgeometric mechanics tostatisticalphysics. Thepersonalauthor’s contribution is toplace theworkofSouriau in thebroader context of the emerging“Geometric Science of Information” [12] (addressed inGSI’15 conference), forwhich theauthorwill showthat theSouriaumodelof statisticalphysics isparticularly welladaptedtogeneralize“informationgeometry”, that theauthor illustrates forexponentialdensities family andmultivariate gaussian densities. The authorwill observe that the Riemannianmetric introducedbySouriau isageneralizationofFishermetric,used in“informationgeometry”, asbeing identifiedto thehessianof the logarithmof thegeneralizedpartition function(Massieucharacteristic function), for the case of densities on homogeneous manifolds where a non-abelian group acts transively. For a groupof time translation,we recover the classical thermodynamics and for the Euclidean space,we recover the classical Fishermetric used in Statistics. The author elaborates a newEuler-Poincaré equation for Souriau’s thermodynamics, action on “geometric heat” variable Q (element of dual Lie algebra), and parameterized by “geometric temperature” (element of Lie algebra). Theauthorwill integrateSouriau thermodynamics inavariationalmodelbydefiningan extendedCartan-Poincaré integral invariantdefinedbySouriau“geometriccharacteristic function” (the logarithmof thegeneralizedSouriaupartition functionparameterizedbygeometric temperature). These results are illustrated for multivariate Gaussian densities, where the associated group is identifiedtocomputeaSouriaumomentmapandreducetheEuler-Poincaréequationofgeodesics. In addition, the symplectic cocycle and Souriau-Fisher metric are deduced from a Lie group thermodynamicsmodel. Themaincontributionsof theauthor in thispaperare the following: • TheSouriaumodelofLiegroup thermodynamics ispresentedwith standardnotationsofLie grouptheory, inplaceofSouriauequationsusing less classical conventions (thathave limited understandingofhisworkbyhiscontemporaries). • Weprove thatSouriauRiemannianmetric introducedwithsymplectic cocycle isageneralization of Fisher metric (called Souriau-Fisher metric in the following) that preserves the property 50