Page - 25 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 370 which leads to ρ= exp(−1−a−bH) . Bywriting that ∫ M ρdλω=1,wesee that a isdeterminedbyb: exp(1+a)=P(b)= ∫ M exp(−bH)dλω . Definition18. LetH :M→RbeasmoothHamiltonianonasymplecticmanifold (M,ω). For eachb∈R such that the integral on the right sideof the equality P(b)= ∫ M exp(−bH)dλω converges, the smoothprobabilitymeasureonMwithdensity (with respect to theLiouvillemeasure) ρ(b)= 1 P(b) exp (−bH) is called theGibbs statistical state associated tob. The functionP : b →P(b) is called thepartition function. Thefollowingpropositionshowsthat theentropyfunction,notonly isstationaryatanyGibbs statistical state,but inacertainsenseattainsat that stateastrictmaximum. Proposition9. LetH : M→R be a smoothHamiltonian ona symplecticmanifold (M,ω) and b∈R be such that the integraldefining thevalueP(b)of thepartition functionPatb converges. Let ρb= 1 P(b) exp(−bH) be the probability density of theGibbs statistical state associated to b. We assume that theHamiltonian H is bounded by below, i.e., that there exists a constant m such that m ≤ H(z) for any z ∈ M. Then the integraldefining Eρb(H)= ∫ M ρbHdλω converges. Foranyother smoothprobabilitydensityρ1 such that Eρ1(H)=Eρb(H) , wehave s(ρ1)≤ s(ρb) , and the equality s(ρ1)= s(ρb)holds if andonly ifρ1= ρb. Proof. Sincem≤H, the functionρbexp(−bH) satisfies0≤ ρbexp(−bH)≤ exp(−mb)ρb, therefore is integrable on M. Let ρ1 be any smooth probability density on M satisfying Eρ1(H) = Eρb(H). The functiondefinedonR+ x → h(x)= ⎧⎪⎨⎪⎩x log ( 1 x ) ifx>0 0 ifx=0 beingconvex, itsgraphisbelowthe tangentatanyof itspoints ( x0,h(x0) ) .Wethereforehave, forall x>0andx0>0, h(x)≤ h(x0)−(1+ logx0)(x−x0)= x0−x(1+ logx0) . 25