Page - 67 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 Figure7.Fourierheatequation inseminalmanuscriptof JosephFourier [88]. For β = 1 kT , K =−∂Q ∂β =−∂Q ∂T ( ∂(1/kT) ∂T )−1 = kT2 ∂Q ∂T linking the geometric capacity to calorificcapacity, thenFishermetriccanbe introducedinFourierheatequation(seeFigure7): ∂T ∂t = κ C ·DΔTwith ∂Q ∂T =C ·D⇒ ∂β −1 ∂t = κ [( β2/k ) · IFisher(β) ]−1 Δβ−1 (48) Wecanalsoobserve thatQ is relatedto themean,andK to thevarianceofU: K= I(β)=−∂Q ∂β =var(U)= M U(ξ)2 ·pβ(ξ)dω− ( M U(ξ) ·pβ(ξ)dω )2 (49) Weobserve that theentropy s isunchanged,andΦ is changedbutwith lineardependence toβ, with theconsequence thatFisherSouriaumetric is invariant: s [ Q ( Adg(β) )] = s(Q(β))and I ( Adg(β) ) =−∂ 2(Φ−〈θ(g−1) ,β〉) ∂β2 =−∂ 2Φ ∂β2 = I(β) (50) Wehaveobservedthat theconceptof“heatcapacity” is important in theSouriaumodelbecause itgivesageometricmeaningto itsdefinition. Thenotionof“heatcapacity”hasbeengeneralizedby PierreDuheminhisgeneralequationsof thermodynamics. Souriau[34]proposedtodefinea thermometer (θε μóσ)deviceprinciple thatcouldmeasure this geometric temperatureusing“relative idealgas thermometer”basedonatheoryofdynamicalgroup thermometryandhasalsorecoveredthe (geometric)Laplacebarometric law 5. Euler-PoincaréEquationsandVariationalPrincipleofSouriauLieGroupThermodynamics WhenaLiealgebraacts locally transitivelyontheconfigurationspaceofaLagrangianmechanical system,HenriPoincaréprovedthat theEuler-Lagrangeequationsareequivalent toanewsystemof differentialequationsdefinedontheproductof theconfigurationspacewiththeLiealgebra.Marlehas writtenabout theEuler-Poincaréequations [104],underan intrinsic form,withoutanyreference toa particular systemof local coordinates,proving that theycanbeconvenientlyexpressed in termsof theLegendreandmomentmapsof the lift to thecotangentbundleof theLiealgebraactionon the configurationspace. TheLagrangian isasmoothrealvaluedfunctionLdefinedonthe tangentbundle 67