Page - 100 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 Wecanthenwrite theSouriau-Fishermetricas: Θ˜β(Z1,Z2)= J[Z1,Z2]− { JZ1, JZ2 } + 〈 ξˆ, [Z1,Z2] 〉 (269) Where theassociateddifferentiableapplication J, calledmomentmapis: J : M→ g∗ suchthat JX(x)= 〈J(x),X〉 , X∈ g x → J(x) (270) Thismomentmapcouldbe identifiedwiththeoperator that transformstherightalgebra toan elementof itsdualalgebragivenby: βM : g→ g∗ Z= [ N η 0 0 ] → J= [ N ( 1+mTR−1m ) +ηmTR−1 NR−1m+R−1η 0 0 ] (271) 10.Conclusions In thispaper,wehavedevelopedaSouriaumodelofLiegroupthermodynamics that recovers the symmetrybrokenby lackof covarianceofGibbsdensity in classical statisticalmechanicswith respect todynamicgroupsaction inphysics (GalileoandPoincarégroups, sub-groupofaffinegroup). TheontologicalmodelofSouriaugivesgeometricstatusto(Planck)temperature(elementofLiealebra), heat (elementofdualLie algebra) andentropy. Souriau said inoneofhispapers [30] on thisnew “Liegroupthermodynamics” that“these formulasareuniversal, in that theydonot involve the symplectic manifold, butonlygroupG, the symplectic cocycle. Perhaps thisLiegroup thermodynamics couldbeof interest formathematics”. For thisnewcovariant thermodynamics, the fundamental notion is the coadjoint orbit that is linkedtopositivedefiniteKKS(Kostant–Kirillov–Souriau)2-form[196]: ωw(X,Y)= 〈w, [U,V]〉withX= adwU∈TwMandY= adwV∈TwM (272) that is the Kähler-form of aG-invariant kähler structure compatiblewith the canonical complex structureofM,anddeterminesacanonicalsymplecticstructureonM.Whenthecocycleisequaltozero, theKKSandSouriau-Fishermetricareequal. This2-formintroducedbyJean-MarieSouriau is linked to thecoadjointactionandthecoadjointorbitsof thegrouponitsmomentspace. Souriauprovided a classification of the homogeneous symplecticmanifoldswith thismomentmap. The coadjoint representationofaLiegroupG is thedualof theadjoint representation. Ifgdenotes theLiealgebra ofG, thecorrespondingactionofGong∗, thedualspace tog, is calledthecoadjointaction. Souriau provedbasedon themomentmap that a symplecticmanifold is alwaysacoadjointorbit, affineof itsgroupofHamiltonian transformations,deducing that coadjointorbits are theuniversalmodels of symplecticmanifolds: a symplecticmanifoldhomogeneousunder the actionof aLie group, is isomorphic,uptoacovering, toacoadjointorbit. So the linkbetweenSouriau-FishermetricandKKS 2-formwill providea symplectic structure and foundation to informationmanifolds. For Souriau thermodynamics, theSouriau-Fishermetric is thecanonical structure linkedtoKKS2-form,modified by the cocycle (its symplectic leaves are the orbits of the affine action thatmakes equivariant the momentmap). This last property allowsus todetermine all homogeneous spaces of aLie group admittinganinvariantsymplecticstructurebytheactionofthisgroup: forexample, therearetheorbits ofthecoadjointrepresentationofthisgrouporofacentralextensionofthisgroup(thecentralextension allowingsuppressingthecocycle). Foraffinecoadjointorbits,wemakereference toAliceTumpach Ph.D. [197–199]whohasdevelopedpreviousworksofNeeb[200],BiquardandGauduchon[201–204]. 100