Page - 37 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 370 Foreachb∈Ω, thedifferential at b of themapEJ (which is a linearmapdefinedonG,withvalues in its dualG∗) isgivenby 〈 DEJ(b)(Y),Z 〉 = 〈 EJ(b),Y 〉〈 EJ(b),Z 〉−Eρb(〈J,Y〉〈J,Z〉) , withYandZ∈G , wherewehavewritten, as inDefinition17, Eρb (〈J,Y〉〈J,Z〉)= 1 P(b) ∫ M 〈J,Y〉〈J,Z〉exp(−〈J,b〉)dλω . Ateachb∈Ω, thedifferential of the entropy functionSProposition12,which is a linearmapdefinedon G,withvalues inR, inotherwordsanelementofG∗, is givenby 〈 DS(b),Y 〉 = 〈 DEJ(b)(Y),b 〉 , Y∈G . Proof. ByassumptionsSection7.2, thedifferentialsofPandEJ canbecalculatedbydifferentiation under thesign ∫ M. Easy (but tedious)calculations leadto the indicatedresults. Corollary3. With the sameassumptionsandnotationsas those inProposition14, for anyb∈ΩandY∈G, 〈 DEJ(b)(Y),Y 〉 =− 1 P(b) ∫ M 〈 J−EJ(b),Y 〉2dλω≤0. Proof. Thisresult followsfromthewellknownresult inProbabilitytheoryalreadyusedintheproofof Proposition11. Themomentummap Jof theHamiltonianactionΦ isnotuniquelydetermined: foranyconstant μ ∈ G∗, J1 = J+μ too is amomentummap forΦ. The followingproposition indicates how the generalizedthermodynamic functionsP,EJ andSchangewhen J is replacedby J1. Proposition15. Withthesameassumptionsandnotationsas those inProposition14, letμ∈G∗ beaconstant. Whenthemomentummap J is replacedby J1= J+μ, theopensubsetΩofG remainsunchanged,while the generalized thermodynamic functionsP,EJ andS, are replaced, respectively, byP1, EJ1 andS1, givenby P1(b)= exp (−〈μ,b〉)P(b), EJ1(b)=EJ(b)+μ , S1(b)=S(b) . TheGibbs satistical state and itsdensityρbwith respect to theLiouvillemeasureλω remainunchanged. Proof. Wehave exp (−〈J+μ,b〉)= exp(−〈μ,b〉)exp(−〈J,b〉) . The indicatedresults followbyeasycalculations. Thefollowingproposition indicateshowthegeneralizedthermodynamic functionsP,EJ andS varyalongorbitsof theadjointactionof theLiegroupGonitsLiealgebraG. Proposition16. Theassumptionsandnotationsare the sameas those inProposition14. TheopensubsetΩ ofG is anunion of orbits of the adjoint action ofG onG. In otherwords, for each b∈Ω and each g∈G, Adgb∈Ω.Moreover, let θ :G→G∗ be the symplectic cocycle ofG for the coadjoinactionofGonG∗ such that, for anyg∈G, J◦Φg=Ad∗g−1◦ J+θ(g) . 37