Page - 56 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 relativity in relativisticmechanics. Themaximumentropyprinciple [42–51] is preserved, and the Gibbsdensity isgivenbythedensityofmaximumentropy(amongtheequilibriumstates forwhich theaveragevalueof theenergytakesaprescribedvalue, theGibbsmeasuresare thosewhichhave the largestentropy),butwithanewprinciple“If adynamical systemis invariantunderaLie subgroupG’of theGalileogroup, then thenatural equilibria of the systemforms theGibbs ensemble of thedynamicalgroup G’”[10]. TheclassicalnotionofGibbscanonicalensemble isextendedforahomogneoussymplectic manifold onwhich aLie group (dynamic group) has a symplectic action. When the group is not abelian(non-commutativegroup), thesymmetry isbroken,andnew“cohomological”relationsshould beverifiedinLiealgebraof thegroup[52–55].Anaturalequilibriumstatewill thusbecharacterizedby anelementof theLiealgebraof theLiegroup,determiningtheequilibriumtemperatureβ. Theentropy s(Q), parametrizedbyQ thegeometricheat (meanofenergyU, elementof thedualLiealgebra) is definedbytheLegendre transform[56–59]of theMassieupotentialΦ(β)parametrizedbyβ (Φ(β) is theminus logarithmof thepartitionfunctionψΩ(β)): s(Q)= 〈β,Q〉−Φ(β)with ⎧⎪⎪⎨⎪⎪⎩ Q= ∂Φ ∂β ∈ g∗ β= ∂s ∂Q ∈ g (19) pGibbs(ξ)= eΦ(β)−〈β,U(ξ)〉= e−〈β,U(ξ)〉 M e−〈β,U(ξ)〉dω , Q= ∂Φ(β) ∂β = M U(ξ)e−〈β,U(ξ)〉dω M e−〈β,U(ξ)〉dω = M U(ξ)p(ξ)dω withΦ(β)=−log M e−〈β,U(ξ)〉dω (20) Souriaucompletedhis“geometricheat theory”by introducinga2-formintheLiealgebra, that is aRiemannianmetric tensor in thevalues of adjoint orbit of β, [β,Z]withZ an element of theLie algebra. Thismetric isgivenfor (β,Q): gβ([β,Z1] , [β,Z2])= 〈Θ(Z1) , [β,Z2]〉+〈Q, [Z1, [β,Z2]]〉 (21) whereΘ isacocycleof theLiealgebra,definedbyΘ=Teθwithθacocycleof theLiegroupdefined byθ(M)=Q(AdM(β))−Ad∗MQ.Wehaveobservedthat thismetricgβ isalsogivenbythehessian of theMassieupotential gβ =−∂ 2Φ ∂β2 = ∂logψΩ ∂β2 asFishermetric in classical informationgeometry theory [60], andso this isageneralizationof theFishermetric forhomogeneousmanifold.Wecall this newmetric theSouriau-Fishermetric.Asgβ=−∂Q∂β , Souriaucompareditbyanalogywithclassical thermodynamics toa“geometric specificheat” (geometriccalorificcapacity). The potential theory of thermodynamics and the introduction of “characteristic function” (previousfunctionΦ(β)=−logψΩ(β) inSouriautheory)wasinitiatedbyFrançoisJacquesDominique Massieu [61–64]. Massieuwas the sonofPierreFrançoisMarieMassieuandThérèseClaireCastel. Hemarried in1862withMlleMorandandhad2children.HegraduatedfromEcolePolytechnique in1851andEcoledesMinesdeParis in 1956, hehas integrated“CorpsdesMines”. Hedefended hisPh.D. in1861on“Sur les intégrales algébriquesdesproblèmesdemécanique”andon“Sur lemodede propagationdesondesplanes et la surfacede l’onde élémentairedans les cristauxbiréfringents àdeuxaxes” [65] with the jury composed of Lamé, Delaunay et Puiseux. In 1870, FrançoisMassieu presented his paper to theFrenchAcademyofScienceson“characteristic functionsof thevariousfluidsandthe theory of vapors” [61]. Thedesignof the characteristic function is thefinest scientific title ofMr. Massieu. Aprominent judge, JosephBertrand, donot hesitate todeclare, in a statement read to theFrench AcademyofSciences25 July1870, that“the introductionof this function in formulas that summarizeall the possible consequencesof the two fundamental theoremsseems, for the theory, a similar service almost equivalent 56