Page - 66 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 It is obvious that one can only define average values on objects belonging to a vector (or affine) space;Therefore—so this assertionmayseemBourbakist—thatwewill observeandmeasureaverage values only as quantity belonging to a set havingphysically an affine structure. It is clear that this structure is necessarilyunique—ifnot the averagevalueswouldnot bewell defined. (Il est évident que l’onnepeutdéfinirdevaleursmoyennesque surdes objets appartenant àunespace vectoriel (ouaffine);donc—sibourbakistequepuisse semblercetteaffirmation—que l’onn’observera etnemesureradevaleursmoyennesquesurdesgrandeursappartenantàunensemblepossédant physiquementune structureaffine. Il est clair que cette structure estnécessairementunique—sinon lesvaleursmoyennesne seraientpasbiendéfinies.). 4.TheSouriau-FisherMetricasGeometricHeatCapacityofLieGroupThermodynamics We observe that Souriau Riemannian metric, introduced with symplectic cocycle, is a generalizationof theFishermetric, thatwecall theSouriau-Fishermetric, thatpreserves theproperty tobedefinedasahessianof thepartitionfunction logarithmgβ=−∂ 2Φ ∂β2 = ∂2logψΩ ∂β2 as inclassical information geometry. We will establish the equality of two terms, between Souriau definition based on Lie group cocycleΘ and parameterized by “geometric heat” Q (element of dual Lie algebra)and“geometric temperature”β (elementofLiealgebra)andhessianofcharacteristic function Φ(β)=−logψΩ(β)withrespect to thevariableβ: gβ([β,Z1] , [β,Z2])= 〈Θ(Z1) , [β,Z2]〉+〈Q, [Z1, [β,Z2]]〉= ∂ 2logψΩ ∂β2 (43) If we differentiate this relation of Souriau theorem Q ( Adg(β) ) = Ad∗g(Q)+ θ(g), this relationoccurs: ∂Q ∂β (− [Z1,β] , .)= Θ˜(Z1, [β, .])+ 〈 Q,Ad.Z1([β, .]) 〉 = Θ˜β(Z1, [β, .]) (44) − ∂Q ∂β ([Z1,β] ,Z2.)= Θ˜(Z1, [β,Z2])+ 〈 Q,Ad.Z1([β,Z2]) 〉 = Θ˜β(Z1, [β,Z2]) (45) ⇒−∂Q ∂β = gβ([β,Z1] , [β,Z2]) (46) As the entropy is defined by the Legendre transform of the characteristic function, this Souriau-Fishermetric is also equal to the inverse of the hessian of “geometric entropy” s(Q) withrespect to thevariableQ: ∂2s(Q) ∂Q2 For themaximumentropydensity (Gibbsdensity), the followingthree termscoincide: ∂2logψΩ ∂β2 thatdescribestheconvexityofthelog-likelihoodfunction, I(β)=−E [ ∂2logpβ(ξ) ∂β2 ] theFishermetricthat describesthecovarianceofthelog-likelihoodgradient,whereas I(β)=E [ (ξ−Q)(ξ−Q)T ] =Var(ξ) thatdescribes thecovarianceof theobservables. Wecanalsoobserve that theFishermetric I(β) =−∂Q ∂β is exactly theSouriaumetricdefined throughsymplecticcocycle: I(β)= Θ˜β(Z1, [β,Z2])= gβ([β,Z1] , [β,Z2]) (47) TheFishermetric I(β)=−∂ 2Φ(β) ∂β2 =−∂Q ∂β hasbeenconsideredbySouriauasageneralizationof “heat capacity”. Souriaucalled itK the“geometric capacity”. 66