Page - 28 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 370 bemathematicallydescribedbymeansof aHamiltonian systemona symplecticmanifold (M,ω) whoseHamiltonianH satisfies theassumptionsSection6.2.1. Forphysicists, aGibbsstatistical state, i.e., aprobabilitymeasureofdensity ρb = 1 P(b) exp(−bH)onM, is a thermodynamic equilibriumof thephysical system.Thesetofpossible thermodynamicequilibriaof thesystemis therefore indexed by a real parameter b > 0. The following argumentwill showwhat physicalmeaning can have thatparameter. Let us consider two similar physical systems,mathematicallydescribedby twoHamiltonian systems,ofHamiltoniansH1 onthesymplecticmanifold (M1,ω1)andH2 onthesymplecticmanifold (M2,ω2). We first assume that they are independent and both in thermodynamic equilibrium, withdifferentvaluesb1 andb2 of theparameterb.WedenotebyE1(b1)andE2(b2) themeanvalues ofH1 on themanifoldM1withrespect to theGibbsstateofdensityρ1,b1 andofH2 on themanifold M2withrespect to theGibbsstateofdensityρ2,b2.Weassumenowthat the twosystemsarecoupled in awayallowing an exchangeof energy. For example, the twovessels containing the twogases canbeseparatedbyawall allowingaheat transferbetween them. Coupled together, theymakea newphysical system,mathematicallydescribedbyaHamiltoniansystemonthesymplecticmanifold (M1×M2,p∗1ω1+ p∗2ω2), where p1 : M1×M2 → M1 and p2 : M1×M2 → M2 are the canonical projections. TheHamiltonianofthisnewsystemcanbemadeasclosetoH1◦p1+H2◦p2asonewishes, bymakingverysmall thecouplingbetweenthe twosystems. Themeanvalueof theHamiltonianof thenewsystemis thereforeveryclose toE1(b1)+E2(b2).Whenthe total systemwill reachastateof thermodynamicequilibrium, theprobabilitydensitiesof theGibbsstatesof its twoparts,ρ1,b′ onM1 andρ2,b′ onM2willbe indexedbythesamerealnumberb′>0,whichmustbesuchthat E1(b′)+E2(b′)=E1(b1)+E2(b2) . ByProposition11,wehave, forallb>0, dE1(b) db ≤0, dE2(b) db ≤0. Thereforeb′must liebetweenb1 andb2. If, forexample,b1< b2,wesee thatE1(b′)≤E1(b1)and E2(b′)≥E2(b2). Inorder toreachastateof thermodynamicequilibrium,energymustbe transferred from thepart of the systemwhere bhas the smallest value, towards thepart of the systemwhere b has the highest value, until, at thermodynamic equilibrium, b has the same value everywhere. Everydayexperienceshowsthat thermalenergyflowsfrompartsofasystemwhere the temperature ishigher, towardspartswhere it is lower. For this reasonphysicists consider the realvariable bas awaytoappreciate the temperatureofaphysical systeminastateof thermodynamicequilibrium. Moreprecisely, theystate that b= 1 kT whereT is theabsolute temperatureandkaconstantdependingonthechoiceofunitsofenergyand temperature, calledBoltzmann’s constant in honour of the greatAustrian scientist LudwigEduard Boltzmann(1844–1906). For a physical system mathematically described by a Hamiltonian system on a symplectic manifold (M,ω),withHasHamiltonian, inastateof thermodynamicequilibrium,E(b)andS(b)are the internal energyandthe entropyof thesystem. 6.2.3. TowardsThermodynamicEquilibrium Everydayexperienceshowsthataphysical system,whensubmitted toexternal conditionswhich remainunchangedforasufficientlylongtime,veryoftenreachesastateofthermodynamicequilibrium. Atfirst look, it seemsthatLagrangianorHamiltoniansystemswith time-independentLagrangiansor Hamiltonianscannotexhibitasimilarbehaviour. Letus indeedconsideramechanical systemwhose 28