Page - 24 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 370 Theentropy s(ρ)ofaprobabilitydensityρhasveryremarkablevariationalpropertiesdiscussed in the followingdefinitionsandproposition. Definition17. Letρbe thedensityof a smoothstatistical state onasymplecticmanifold (M,ω). 1. For each function f defined on M, taking its values inR or in some finite-dimensional vector space, such that the integral on the righthandsideof the equality Eρ(f)= ∫ M fρdλω converges, thevalueEρ(f)of that integral is called themeanvalueof f with respect toρ. 2. Let f be a smooth function on M, taking its values inR or in some finite-dimensional vector space, satisfying theproperties statedabove.Asmooth infinitesimalvariationofρwithfixedmeanvalueof f is a smoothmap,definedontheproduct ]−ε,ε[×M,withvalues inR+,where ε>0, (τ,z) → ρ(τ,z) , τ∈]−ε,ε[, z∈M , such that • forτ=0andanyz∈M,ρ(0,z)= ρ(z), • for eachτ∈]−ε,ε[ , z → ρτ(z)= ρ(τ,z) is a smoothprobabilitydensityonMsuch that Eρτ(f)= ∫ M ρτ fdλω=Eρ(f) . 3. The entropy function s is said to be stationary at the probability density ρ with respect to smooth infinitesimal variations of ρ with fixed mean value of f, if for any smooth infinitesimal variation (τ,z) → ρ(τ,z)ofρwithfixedmeanvalueof f ds(ρτ) dτ ∣∣∣ τ=0 =0. Proposition 8. Let H : M→R be a smoothHamiltonian on a symplecticmanifold (M,ω) and ρ be the densityofasmoothstatistical stateonMsuchthat the integraldefiningthemeanvalueEρ(H)ofHwithrespect toρ converges. The entropy functions is stationaryatρwithrespect to smooth infinitesimalvariationsofρwith fixedmeanvalueofH, if andonly if there exists a real b∈R such that, for all z∈M, ρ(z)= 1 P(b) exp (−bH(z)) , with P(b)= ∫ M exp(−bH)dλω . Proof. Letτ → ρτbeasmoothinfinitesimalvariationofρwithfixedmeanvalueofH. Since ∫ M ρτdλω and ∫ M ρτHdλω donotdependonτ, it satisfies, forallτ∈]−ε,ε[ ,∫ M ∂ρ(τ,z) ∂τ dλω(z)=0, ∫ M ∂ρ(τ,z) ∂τ H(z)dλω(z)=0. Moreoveraneasycalculation leads to ds(ρτ) dτ ∣∣∣ τ=0 =− ∫ M ∂ρ(τ,z) ∂τ ∣∣∣ τ=0 (1+ log ( ρ(z) ) dλω(z) . Awellknownresult incalculusofvariationsshowsthat theentropyfunction s is stationaryatρ withrespect tosmooth infinitesimalvariationsofρwithfixedmeanvalueofH, if andonly if there exist tworealconstants aandb, calledLagrangemultipliers, suchthat, forallz∈M, 1+ log(ρ)+a+bH=0, 24