Page - 22 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 370 ThePoincaré-Cartan1-form ̂L intheLagrangianformalism,or ̂H intheHamiltonianformalism, depends on the choice of a particular reference frame, made for using the Lagrangian or the Hamiltonianformalism.But theirexteriordifferentials, thepresymplectic formsd̂Lord̂H,donot dependonthat choice,moduloasimplechangeofvariables in theevolutionspace. Souriaudefinedthispresymplectic forminaframeworkmoregeneral thanthoseofLagrangian orHamiltonianformalisms,andcalled it theLagrange form. In thismoregeneral setting, itmaynot beanexact2-form. SouriauproposedasanewPrinciple, theassumption that it alwaysprojectson thespaceofmotionsof thesystemsasa symplectic form, even inrelativisticmechanics inwhich the definitionofanevolutionspace isnotclear.Hecalledthisnewprinciple theMaxwellPrinciple. Bargmannprovedthat thesymplecticcohomologyof theGalileangroupisofdimension1,and Souriauproved that the cohomology class of its action on themanifold ofmotions of an isolated classical (non-relativistic)mechanical systemcanbe identifiedwith the totalmassof thesystem[14], Chapter III,p. 153. Readers interested in theGalileangroupandmomentummapsof itsactionsarereferredto the recentbookbydeSaxcéandVallée [45]. 6. StatisticalMechanicsandThermodynamics 6.1. BasicConcepts inStatisticalMechanics DuringtheXVIII–thandXIX–thcenturies, the idea thatmaterialbodies (fluidsaswellassolids) areassembliesof avery largenumberof small,movingparticles, began tobeconsideredbysome scientists,notablyDanielBernoulli (1700–1782),RudolfClausius (1822–1888), JamesClerkMaxwell (1831–1879)andLudwigEduardoBoltzmann(1844–1906),asareasonablepossibility.Attempswere madetoexplain thenatureofsomemeasurablemacroscopicquantities (forexample the temperature ofamaterialbody, thepressureexertedbyagasonthewallsof thevessel inwhich it is contained), andthe lawswhichgovernthevariationsof thesemacroscopicquantities,byapplicationof the lawsof classicalmechanics to themotionsof theseverysmallparticles. Described in the frameworkof the Hamiltonianformalism, thematerialbodyisconsideredasaHamiltoniansystemwhosephasespace isaveryhighdimensional symplecticmanifold (M,ω), sinceanelementof that spacegivesaperfect informationabout thepositionsandthevelocitiesofall theparticlesof thesystem.Theexperimental determinationof theexact stateof thesystembeing impossible,oneonlycanuse theprobabilityof presence, at each instant, of the state of the system invariousparts of thephase space. Scientists introducedtheconceptofa statistical state,definedbelow. Definition 14. Let (M,ω) be a symplecticmanifold. A statistical state is a probabilitymeasure μ on the manifoldM. 6.1.1. TheLiouvilleMeasureonaSymplecticManifold On each symplecticmanifold (M,ω), with dimM = 2n, there exists a positivemeasure λω, called the Liouville measure. Let us briefly recall its definition. Let (U,ϕ) be aDarboux chart of (M,ω)Section4.4.1. TheopensubsetUofM is,bymeansof thediffeomorphismϕ, identifiedwith an open subset ϕ(U) ofR2n onwhich the coordinates (Darboux coordinates)will be denoted by (p1, . . . ,pn,x1, . . . ,xn).With this identification, theLiouvillemeasure (restricted toU) is simply the Lebesguemeasureon theopen subset ϕ(U)ofR2n. In otherwords, for eachBorel subset Aof M containedinU,wehave λω(A)= ∫ ϕ(A) dp1 . . . dpndx1 . . . dxn . Onecaneasilycheckthat thisdefinitiondoesnotdependonthechoiceof theDarbouxcoordinates (p1, . . . ,pn,x1, . . . ,xn)onϕ(A). ByusinganatlasofDarbouxchartson (M,ω), onecaneasilydefine λω(A) foranyBorel subsetAofM. 22