Seite - 121 - in Differential Geometrical Theory of Statistics

entropy Article LinkbetweenLieGroupStatisticalMechanicsand ThermodynamicsofContinua GérydeSaxcé LaboratoiredeMécaniquedeLille,CNRSFRE3723,UniversitédeLille1,Villeneuved’AscqF59655,France; gery.desaxce@univ-lille1.fr;Tel.: +33-3-2033-7172 AcademicEditors: FrédéricBarbarescoandFrankNielsen Received: 29April2016;Accepted: 7 July2016;Published: 12 July2016 Abstract: In thiswork,weconsider thevalueof themomentummapof thesymplecticmechanics asanaffine tensor calledmomentumtensor. Fromthispointofview,weanalyze theunderlying geometric structureof the theoriesofLiegroupstatisticalmechanicsandrelativistic thermodynamics ofcontinua, formulatedbySouriau independentlyofeachother.Webridge thegapbetweenthemin theclassicalGalileancontext. Thesegeometric structuresof the thermodynamicsare richandwe think theymightbe a sourceof inspiration for thegeometric theoryof informationbasedon the conceptofentropy. Keywords: Lie groups; symplectic geometry; affine tensors; continuum thermodynamics; statisticalmechanics 1. Introduction In [1],Souriauproposes torevisitmechanics,emphasizing itsaffinenature. It is thisviewpoint thatwewill adopthere, starting fromageneralizationof theconceptofmomentumunder the formof anaffineobject [2].Ourstartingpoint is closelyrelatedtoSouriau’sapproachonthebasisof twokey ideas: anewdefinitionofmomentaandthecrucialpartplayedbytheaffinegroupofRn. Thisgroup proposesan intentionallypoorgeometrical structure. Indeed, thischoice isguidedbythefact that it containsbothGalileoandPoincarégroups[3,4],whichallowsthesimultaneous involvementof the Galileanandrelativisticmechanics. In the follow-up,weshalldetailonly theapplications toclassical mechanicsandthermodynamics. Aclassof tensorscorrespondstoeachgroup. Thecomponentsof these tensorsare transformed accordingto theactionof theconsideredgroup. Thestandardtensorsdiscussedin the literatureare thoseof the lineargroupofRn. Wewill call them linear tensors. A fruitful standpoint consists of consideringtheclassof theaffinetensors, correspondingto theaffinegroup[2,5]. Thisviewpoint is closelyrelatedtosymplecticmechanics [3,4,6] in thesense that thevaluesof themomentummapare just thecomponentsof themomentumtensors. Thepresentpaper is structuredas follows. In Section2,wepresent briefly the affine tensors, startingwith themostsimpleones: thepointsofanaffinespacewhichare1-contravariantandthereal affinefunctionsonthis spaceoraffine formswhichare1-covariant.AsasubgroupGof theaffinegroup ofRnnaturallyactsonto theaffinetensorsbyrestrictiontoGof their transformation law,wedefine thecorrespondingG-tensor. InSection3,weuse this framework,definingthemomentumasamixed 1-covariantand1-contravariantaffinetensor. IfG isaLiegroup,wedemonstrate the important fact that its transformationlaw isnothingother thanthecoadjointrepresentationofG inthedualg∗of itsLie algebra. InSection4,werecallclassical toolsofsymplecticmechanicsaroundtheconceptofsymplectic action and amomentummap. An important result called theKirillov–Kostant–Souriau theorem reveals theorbit symplectic structure. InSection5,werecall shortly themainconceptsof theLiegroup statisticalmechanicsproposedbySouriau in [3,4], usinggeometric tools. InSection6,wepresent Entropy2016,18, 254 121 www.mdpi.com/journal/entropy