Page - 122 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 254 brieflythecornerstoneresultsof theGalileanversionofa thermodynamicsofcontinuacompatible withgeneral relativityproposedbySouriau in [7,8] independently of his statisticalmechanics. In Section7,wereveal the linkbetweenthepreviousrelativistic thermodynamicsofcontinuaandLie groupstatisticalmechanics in theclassicalGalileancontext,working insevensteps. 2.AffineTensors Pointsofanaffinespace.LetAT beanaffinespaceassociatedtoalinearspaceT offinitedimension n. By thechoiceofanaffineframe f composedofabasisofT andanorigina0,wecanassociate to eachpointaasetofn (affine)componentsVigathered in then-columnV∈Rn. Forachangeofaffine frames, the transformation lawfor thecomponentsofapoint reads: V=C+PV′ , (1) which isanaffinerepresentationof theaffinegroupofRndenotedAf f(n). It is clearlydifferent from theusual transformation lawofvectorsV=PV′. Affine forms.TheaffinemapsΨ fromAT intoRarecalledaffineformsandtheirset isdenoted A∗T . Inanaffineframe,Ψ is representedbyanaffinefunctionΨ fromRn intoR.Hence, itholds: Ψ(a)=Ψ(V)=χ+ΦV , whereχ=Ψ(0)=Ψ(a0)andΦ= lin(Ψ) is an-row. WecallΦ1,Φ2, · · · ,Φn,χ the componentsof Ψor, equivalently, the coupleofχ and the rowΦ collecting theΦα. The setA∗T is a linear space ofdimension (n+1) called thevectordualofAT . Ifwechange theaffine frame, thecomponents ofanaffine formaremodifiedaccording to the inducedactionofAf f(n), that leads to, taking into account (1): χ′=χ−ΦP−1C, Φ′=ΦP−1 , (2) which isa linearrepresentationofAf f(n). Affine tensors.Wecangeneralize thisconstructionanddefineanaffinetensorasanobject: • thatassignsasetofcomponents toeachaffineframe f ofanaffinespaceAT offinitedimensionn, • witha transformation law,whenchangingof frames,which isanaffineora linear representation ofAf f(n). With thisdefinition, theaffine tensorsareanaturalgeneralizationof theclassical tensors thatwe shall call linear tensors, these lastonesbeingtrivialaffinetensors forwhichtheaffinetransformation a=(C,P)acts throughits linearpartP= lin(a). Anaffinetensorcanbeconstructedasamapwhich isaffineor linearwithrespect toeachof itsarguments. Similar to the linear tensors, theaffineonescan beclassified in three families: covariants, contravariantandmixed. Themostsimpleaffine tensorsare thepointswhichare1-contravariantandtheaffineformswhichare1-covariantbutwecanconstruct morecomplexoneshavingastrongphysicalmeaning: the torsors (proposedin[5]), the co-torsorsand themomentaextensivelydetailledin[2]. Formoredetailsontheaffinedualspace,affinetensorproduct, affinewedgeproductandaffinetangentbundles, thereader interested in this topic is referredto the so-calledAV-differentialgeometry [9]. G-tensors.AsubgroupGofAf f(n)naturallyactsonto theaffine tensorsbyrestriction toGof their transformation law.LetFGbeasetofaffineframesofwhichG isa transformationgroup.The elementsofFG arecalledG-frames.AG-tensor isanobject: • thatassignsasetofcomponents toeachG-frame f, • witha transformation law,whenchangingof frames,which isanaffineora linear representation ofG. 122