Page - 35 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 370 7.GeneralizationforHamiltonianActions 7.1.GeneralizedGibbsStates Inhisbook[15]andinseveralpapers [13,16,17],Souriauextends theconceptofaGibbsstate for aHamiltonianactionofaLiegroupGonasymplecticmanifold (M,ω).UsualGibbsstatesdefined inSection6 for a smoothHamiltonianHonasymplecticmanifold (M,ω)appearas special cases, inwhich the Lie group is a one-parameter group. If the symplecticmanifold (M,ω) is the phase spaceof theHamiltonian system, that one-parameter group,whoseparameter is the time t, is the groupof evolution, as a functionof time, of the stateof the system, starting fromits state at some arbitrarilychosen initial time t0. If (M,ω) is thesymplecticmanifoldofall themotionsof thesystem, thatone-parametergroup,whoseparameter isarealτ∈R, is the transformationgroupwhichmaps onemotionof thesystemwithsomeinitial stateat time t0 onto themotionof thesystemwiththesame initial stateatanother time (t0+τ).Wediscussbelowthisgeneralization. NotationsandConventions In this section,Φ :G×M→M isaHamiltonianaction (forexampleonthe left)ofaLiegroupG onasymplecticmanifold (M,ω).WedenotebyG theLiealgebraofG, byG∗ itsdual spaceandby J :M→G∗ amomentummapof theactionΦ. Definition19. Let b∈G be such that the integrals on the righthandsidesof the equalities P(b)= ∫ M exp (−〈J,b〉)dλω and EJ(b)=Eρb(J)= 1 P(b) ∫ M Jexp (−〈J,b〉)dλω converge. The smoothprobabilitymeasureonMwithdensity (with respect to theLiouvillemeasureλω onM) ρb= 1 P(b) exp (−〈J,b〉) is called the generalizedGibbs statistical state associated to b. The functions b → P(b) and b → EJ(b) so definedonthesubsetofGmadebyelementsb forwhichthe integralsdefiningP(b)andEJ(b)convergearecalled thepartition functionassociated to themomentummap J and themeanvalueof J atgeneralizedGibbs states. ThefollowingPropositiongeneralizes9. Proposition12. Let b∈G be such that the integralsdefiningP(b)andEJ(b) inDefinition19converge, and ρb be thedensityof thegeneralizedGibbs stateassociated tob. Theentropys(ρb),whichwill bedenotedbyS(b), exists and isgivenby S(b)= log ( P(b) ) + 〈 EJ(b),b 〉 = log ( P(b) )−〈D(logP(b)),b〉 . (7) Moreover, for anyother smoothprobabilitydensityρ1 such that Eρ1(J)=Eρb(J)=EJ(b) , wehave s(ρ1)≤ s(ρb) , and the equality s(ρ1)= s(ρb)holds if andonly ifρ1= ρb. 35