Page - 63 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 Foreachtemperatureβ, elementof theLiealgebrag, Souriauhas introducedatensor Θ˜β, equal to thesumof thecocycle Θ˜andtheheatcoboundary(with [.,.] Liebracket): Θ˜β(Z1,Z2)= Θ˜(Z1,Z2)+ 〈 Q,adZ1(Z2) 〉 with adZ1(Z2)= [Z1,Z2] (29) This tensor Θ˜β has the followingproperties: • Θ˜(X,Y)= 〈Θ(X),Y〉where themapΘ is theone-cocycleof theLiealgebragwithvalues ing∗, withΘ(X)=Teθ(X(e))where θ theone-cocycleof theLiegroupG. Θ˜(X,Y) is constantonM andthemap Θ˜(X,Y) : g×g→ isaskew-symmetricbilinear form,andiscalled the symplectic cocycle ofLie algebragassociatedto themomentmap J,with the followingproperties: Θ˜(X,Y)= J[X,Y]−{JX, JY}with {., .} PoissonBracketand J theMomentMap (30) Θ˜([X,Y] ,Z)+Θ˜([Y,Z] ,X)+Θ˜([Z,X] ,Y)=0 (31) where JX linear application from g to differential function on M: g→C∞(M,R) X→ JX and the associateddifferentiableapplication J, calledmoment(um)map: J :M→ g∗ suchthat JX(x)= 〈J(x),X〉 , X∈ g x → J(x) (32) If insteadof Jwetake the followingmomentmap: J′(x)= J(x)+Q , x∈M whereQ∈ g∗ is constant, thesymplecticcocycleθ is replacedbyθ′(g)= θ(g)+Q−Ad∗gQ where θ′−θ =Q−Ad∗gQ is one-coboundaryofGwithvalues in g∗. Wealsohaveproperties θ(g1g2)=Ad∗g1θ(g2)+θ(g1)andθ(e)=0. • Thegeometric temperature,elementof thealgebrag, is in the thekernelof the tensor Θ˜β: β∈Ker Θ˜β, suchthat Θ˜β(β,β)=0 , ∀β∈ g (33) • Thefollowingsymmetric tensorgβ,definedonallvaluesof adβ(.)= [β, .] ispositivedefinite: gβ([β,Z1] , [β,Z2])= Θ˜β(Z1, [β,Z2]) (34) gβ([β,Z1] ,Z2)= Θ˜β(Z1,Z2) , ∀Z1∈ g,∀Z2∈ Im ( adβ(.) ) (35) gβ(Z1,Z2)≥0 , ∀Z1,Z2∈ Im ( adβ(.) ) (36) where the linearmap adX ∈ gl(g) is the adjoint representation of the Lie algebra gdefined by X,Y∈ g(=TeG) → adX(Y)= [X,Y], and the co-adjoint representation of the Lie algebra g the linearmap ad∗X ∈ gl(g∗)which satisfies, for each ξ ∈ g∗ and X,Y ∈ g:〈ad∗X(ξ),Y〉 = 〈ξ,−adX(Y)〉These equations are universal, because they are not dependent on the symplectic manifoldbutonlyonthedynamicalgroupG, thesymplecticcocycleΘ, the temperatureβand theheatQ. Souriaucalledthismodel“Liegroups thermodynamics”. Wewillgive themaintheoremofSouriaufor this“Liegroupthermodynamics”: Theorem1(SouriauTheoremofLieGroupThermodynamics).LetΩbe the largest openproper subset of g,Lie algebraofG, such that M e−〈β,U(ξ)〉dλand M ξ ·e−〈β,U(ξ)〉dλare convergent integrals, this setΩ is convexand is invariantunder every transformationAdg(.),where g →Adg(.) is theadjoint representationof G, such thatAdg=Teigwith ig : h → ghg−1 . Let a :G×g∗→ g∗ auniqueaffineaction a such that linear 63