Page - 26 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 370 Withx= ρ1(z)andx0= ρb(z),zbeinganyelement inM, that inequalitybecomes h ( ρ1(z) ) = ρ1(z) log ( 1 ρ1(z) ) ≤ ρb(z)− ( 1+ logρb(z) ) ρ1(z) . ByintegrationoverM,usingthefact thatρb is theprobabilitydensityof theGibbsstateassociated tob,weobtain s(ρ1)≤1−1− ∫ M ρ1 logρbdλω= s(ρb) . Wehaveproventhe inequality s(ρ1)≤ s(ρb). Ifρ1= ρb,wehaveofcourse theequality s(ρ1)= s(ρb). Conversely if s(ρ1)= s(ρb), the functionsdefinedonM z →ϕ1(z)= ρ1(z) log ( 1 ρ1(z) ) and z →ϕ(z)= ρb(z)− ( 1+ logρb(z) ) ρ1(z) arecontinuousonMexcept,maybe, forϕ, atpointszatwhichρb(z)=0andρ1(z) =0,but thesetof suchpoints isofmeasure0sinceϕ is integrable. Theysatisfytheinequalityϕ1≤ϕ. Bothare integrable onMandhavethesameintegral. Thefunctionϕ−ϕ1 iseverywhere≥0, is integrableonMandits integral is0. That function is thereforeeverywhereequal to0onM.Wecanwrite, for anyz∈M, ρ1(z) log ( 1 ρ1(z) ) = ρb(z)− ( 1+ logρb(z) ) ρ1(z) . (6) Foreachz∈M suchthatρ1(z) =0,wecandivide thatequalitybyρ1(z).Weobtain ρb(z) ρ1(z) − log ( ρb(z) ρ1(z) ) =1. Since the function x → x− logx reaches itsminimum,equal to1, forauniquevalueof x> 0, thatvaluebeing1,wesee that foreach z∈Matwhichρ1(z)>0,wehaveρ1(z)= ρb(z). Atpoints z∈Matwhichρ1(z)=0,Equation(6) showsthatρb(z)=0. Thereforeρ1= ρb. Remark14. Themaximality property of the entropy functionρ → s(ρ) at aGibbs state densityρb proven inProposition9of course implies the stationarity of that functionatρb with respect to smooth infinitesimal variationsofρwithfixedmeanvalueofH,proven inProposition8. ThatProposition therefore couldbeomitted. Wechose tokeep it because itsproof ismucheasier than that ofProposition9, andexplainswhy it is interesting to lookatprobabilitydensitiesproportional to exp(−bH) for someb∈R. ThefollowingpropositionshowsthataGibbsstatistical state remains invariantunder theflowof theHamiltonianvectorfieldXH.OnecanthereforesaythataGibbsstate isastatisticalequilibrium state.Ofcourse thereexist statisticalequilibriumstatesother thanGibbsstates. Proposition10. LetHbeasmoothHamiltonianboundedbybelowonasymplecticmanifold (M,ω), b∈R be such that the integral defining the value P(b) of the partition functionPat b converges. TheGibbs state associated tob remains invariantunder theflowofof theHamiltonianvectorfieldXH. Proof. Thedensityρbof theGibbsstateassociatedtob,withrespect to theLiouvillemeasureλω, is ρb= 1 P(b) exp(−bH) . SinceH is constantalongeach integral curveofXH,ρb too isconstantalongeach integral curve ofXH. Moreover, theLiouvillemeasureλω remains invariantunder theflowofXH. Therefore the Gibbsprobabilitymeasureassociatedtob tooremains invariantunder thatflow. 26