Page - 69 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 Back to Koszul model of information geometry, we can then deduce an equivalent of the Euler-Poincaréequationforstatisticalmodels dx∗ dt = ad∗xx∗ and ⎧⎨⎩ Φ ∗(x∗)= 〈x,x∗〉−Φ(x) x= ∂Φ ∗(x∗) ∂x ∈Ω , x∗= ∂Φ(x)∂x ∈Ω∗ (60) We can use this Euler-Poincaré equation to deduce an associated equation on entropy: ds dt = 〈 dβ dt ,Q 〉 + 〈 β,ad∗βQ 〉 − dΦ dt that reduces to ds dt = 〈 dβ dt ,Q 〉 − dΦ dt (61) dueto 〈ξ,adVX〉=− 〈 ad∗Vξ,X 〉⇒〈β,ad∗βQ〉= 〈Q,adββ〉=0. Withthesenewequationofthermodynamics dQ dt = ad∗βQand d dt (Ad∗gQ)=0,wecanobservethat thenewimportantnotion is relatedtoco-adjointorbits, thatareassociatedtoasymplecticmanifoldby SouriauwithKKS2-form. We will then define the Poincaré-Cartan integral invariant for Lie group thermodynamics. Classically inmechanics, thePfaffianformω= p ·dq−H ·dt is relatedtoPoincaré-Cartan integral invariant [107].Dedeckerhasobserved,basedontherelation[108]: ω= ∂ .qL ·dq− ( ∂ .qL · . q−L ) ·dt=L ·dt+∂ .qL with = dq− . q ·dt (62) that theproperty thatamongall formsχ≡ L ·dtmod the formω= p ·dq−H ·dt is theonlyone satisfyingdχ≡0mod , isaparticularcaseofmoregeneralLepagecongruence. AnalogiesbetweengeometricmechanicsandgeometricLiegroupthermodynamics,provides the followingsimilaritiesof structures: { . q↔ β p↔Q , ⎧⎪⎪⎨⎪⎪⎩ L( . q)↔Φ(β) H(p)↔ s(Q) H= p · .q−L↔ s= 〈Q,β〉−Φ and ⎧⎪⎪⎪⎨⎪⎪⎪⎩ . q= dq dt = ∂H ∂p ↔ β= ∂s ∂Q p= ∂L ∂ . q ↔Q= ∂Φ ∂β (63) WecanthenconsiderasimilarPoincaré-Cartan-SouriauPfaffian form: ω= p ·dq−H ·dt↔ω= 〈Q,(β ·dt)〉−s ·dt=(〈Q,β〉−s) ·dt=Φ(β) ·dt (64) ThisanalogyprovidesanassociatedPoincaré-Cartan-Souriau integral invariant. Poincaré-Cartan integral invariant Ca p ·dq−H.dt= Cb p ·dq−H ·dt isgivenforSouriauthermodynamicsby: Ca Φ(β) ·dt= Cb Φ(β) ·dt (65) We can then deduce an Euler-Poincaré-Souriau variational principle for thermodynamics: The variationalprincipleholdson g, forvariations δβ= . η+[β,η],whereη(t) is anarbitrarypath that vanishesat theendpoints,η(a)=η(b)=0: 69