Page - 101 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 Other promising domains of research are theory of generatingmaps [205–208] and the link with Poisson geometry through affinePoisson group. As observed byPierreDazord [209] in his paper“GroupedePoissonAffines”, theextensionof aPoissongroup toanaffinePoissongroupdue toDrinfel’d [210] includes theaffinestructuresofSouriauondualLiealgebra. ForanaffinePoisson group, itsuniversal coveringcouldbe identifiedtoavectorspacewithanassociatedaffinestructure. If this vector space is an abelian affinePoissongroup,we canfind the affine structure of Souriau. For theabeliangroup (R3,+), affinePoissongroupsare theaffinestructuresofSouriau. SouriaumodelofLiegroupthermodynamicscouldbeapromisingwaytoachieveRenéThom’s dreamtoreplace thermodynamicsbygeometry [211,212], andcouldbeextendedto thesecondorder extensionof theGibbsstate [213,214]. We could explore the links between “stochastic mechanics” (mécanique alétoire) developed by Jean-Michel Bismut based on Malliavin Calculus (stochastic calculus of variations) and Souriau “Liegroupthermodynamics”,especially toextendcovariantSouriauGibbsdensityonthestochastic symplecticmanifold (e.g., tomodelcentrifugewithrandomvibratingaxeandtheGibbsdensity). We have seen that Souriau has replaced classicalMaximumEntropy approach by replacing Lagrangeparametersbyonlyonegeometric“temperaturevector”aselementofLiealgebra. Inparallel, as refered in [15], Ingardenhas introduced [213,214] second andhigher order temperature of the Gibbs state that could be extended to Souriau theory of thermodynamics. Ingardenhigher order temperaturescouldbedefinedinthecasewhennovariational is considered,butwhenaprobability distributiondependingonmore thanoneparameter. Ithasbeenobserved that Ingardencan fail if the followingassumptionsarenot fulfilled: thenumberof componentsof the sumgoes to infinity and the components of the sumare stochastically independent. Gibbs hypothesis can also fail if stochastic interactionswith theenvironmentarenot sufficientlyweak. Inall these cases,wenever observeabsolute thermalequilibriumofGibbs typebutonlyflowsor turbulence.Nonequilibrium thermodynamicscouldbe indirectlyaddressedbymeansof theconceptofhighorder temperatures. MomentumQ = ∂Φ(β)∂β should be replaced byhigher ordermoments given by the relationQk = ∂Φ(β1,...,βn) ∂βk = M Uk(ξ) ·e − n∑ k=1 〈βk,Uk(ξ)〉 dω M e − n∑ k=1 〈βk,Uk(ξ)〉 dω defined by extendedMassieu characteristic function Φ(β1,...,βn) =−log M e − n∑ k=1 〈βk,Uk(ξ)〉 dω. Entropy isdefinedbyLegendre transformof thisMassieu characteristic functionS(Q1,...,Qn)= n ∑ k=1 〈βk,Qk〉−Φ(β1,...,βn)whereβk= ∂S(Q1,...,Qn)∂Qk .Weare able also to define high order thermal capacities given byKk = −∂Qk∂βk . TheGibbs density could be thenextendedwith respect tohighorder temperaturesby pGibbs(ξ) = e n ∑ k=1 〈βk,Uk(ξ)〉−Φ(β1,...,βn) = e − n∑ k=1 〈βk,Uk(ξ)〉 M e − n∑ k=1 〈βk,Uk(ξ)〉 dω . We also have tomake reference to theworks of Streater [16],Nencka [215] andBurdet [216]. NenckaandStreater [215], forcertainunitaryrepresentationsofaLiealgebrag,definethestatistical manifoldMofstatesas theconvexconeofX∈ g forwhichthepartitionfunctionZ=Tr [exp(−X)] isfinite. TheHessianof logZdefinesaRiemannianmetricgondualLiealgebrag∗. Theyobserve that g∗ foliates into theunionofcoadjointorbits, eachofwhichcanbegivenacomplexKostantstructure (thatofKostant). 101