Page - 23 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 370 Definition15. Astatistical stateμon the symplecticmanifold (M,ω) is said tobecontinuous (respectively, is said tobe smooth) if it hasa continuous (respectively, a smooth)densitywith respect to theLiouvillemeasure λω, i.e., if there exists a continuous function (respectively, a smooth function)ρ :M→R such that, for each Borel subsetAofM μ(A)= ∫ A ρdλω . Remark 12. The density ρ of a continuous statistical state on (M,ω) takes its values in R+ and of course satisfies ∫ M ρdλω=1. For simplicity we only consider in what follows continuous, very often even smooth statistical states. 6.1.2.Variation inTimeofaStatisticalState Let H be a smooth time independentHamiltonian on a symplecticmanifold (M,ω), XH the associatedHamiltonianvectorfieldandΦXH its reducedflow.Weconsider themechanical system whose timeevolution isdescribedbytheflowofXH. If thestateof thesystemat time t0, assumedtobeperfectlyknown, isapointz0∈M, its stateat time t1 is thepointz1=Φ XH t1−t0(z0). Let us nowassume that the state of the systemat time t0 is not perfectly known, but that a continuousprobabilitymeasureonthephasespaceM,whosedensitywithrespect to theLiouville measureλω isρ0,describes theprobabilitydistributionofpresenceof thestateof thesystemat time t0. In otherwords, ρ0 is thedensity of the statistical state of the systemat time t0. For anyother time t1, themapΦ XH t1−t0 isasymplectomorphism, therefore leaves invariant theLiouvillemeasureλω. Theprobabilitydensity ρ1 of the statistical stateof the systemat time t1 therefore satisfies, for any x0∈M forwhichx1=ΦXHt1−t0(x0) isdefined, ρ1(x1)= ρ1 ( ΦXHt1−t0(x0) ) = ρ0(x0) . Since ( ΦXHt1−t0 )−1 =ΦXHt0−t1,wecanwrite ρ1= ρ0◦ΦXHt0−t1 . Definition 16. Let ρ be the density of a continuous statistical state μ on the symplecticmanifold (M,ω). Thenumber s(ρ)= ∫ M ρ log ( 1 ρ ) dλω is called the entropyof the statistical stateμor,witha slightabuseof language, the entropyof thedensityρ. Remark13. 1. Byconventionwestate that0 log0=0.With that convention the functionx → x logx is continuouson R+. If the integral on the righthandside of the equalitywhichdefines s(ρ)doesnot converge,we state that s(ρ)=−∞.With these conventions, s(ρ) exists for anycontinuousprobabilitydensityρ. 2. TheaboveDefinition16of the entropyof a statistical state, foundedon ideasdevelopedbyBoltzmannin hisKineticTheory ofGases [46], specially in the derivation of his famous (and controversed)Theorem Êta, is too relatedwith the ideas ofClaudeShannon[47] on information theory. Theuseof information theory in thermodynamicswasmore recentlyproposedby Jaynes [48,49] andMackey [18]. Foraverynice discussionof theuseofprobability concepts inphysics andapplicationof information theory inquantum mechanics, the reader is referred to thepaperbyBalian [50]. 23