Page - 61 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 3.RevisitedSouriauSymplecticModelofStatisticalPhysics In thisSection,wewill revisit theSouriaumodelof thermodynamicsbutwithmodernnotations, replacingpersonalSouriauconventionsusedinhisbookof1970bymoreclassicalones. In1970,Souriau introducedtheconceptofco-adjointactionofagrouponitsmomentumspace (or“momentmap”:mapping inducedbysymplecticmanifoldsymmetries),basedontheorbitmethod works, that allows todefinephysical observables like energy, heat andmomentumormoment as puregeometrical objects (themomentmap takes its values ina spacedeterminedby thegroupof symmetries: thedualspaceof itsLiealgebra). Themoment(um)mapisaconstantof themotionandis associatedtosymplecticcohomology(assignmentofalgebraic invariants toa topological space that arises fromthealgebraicdualizationof thehomologyconstruction). Souriau introducedthemoment mapin1965ina lecturenotesatMarseilleUniversityandpublishedit in1966. Souriaugavetheformal definitionanditsnamebasedonitsphysical interpretation in1967. Souriauthenstudied itsproperties ofequivariance,andformulatedthecoadjointorbit theoreminhisbook in1970.However, inhisbook, Souriaualsoobserved inChapter IV thatGibbsequilibriumstates arenot covariantbydynamical groups (GalileoorPoincarégroups)andthenhedevelopedacovariantmodel thathecalled“Liegroup thermodynamics”,where equilibriumsare indexedbya“geometric (Planck) temperature”, givenbya vectorβ that lies in theLiealgebraof thedynamicalgroup. ForSouriau, all thedetails of classical mechanicsappearasgeometricnecessities (e.g.,mass is themeasureof thesymplectic cohomologyof theactionofaGalileogroup). BasedonthisnewcovariantmodelofthermodynamicGibbsequilibrium, Souriauhas formulatedstatisticalmechanicsandthermodynamics in the frameworkofsymplectic geometrybyuseofsymplecticmomentsanddistribution-tensorconcepts,givingageometricstatus for temperature,heatandentropy. There is a controversy about the name “momentum map” or “moment map”. Smale [92] referred to thismapas the“angularmomentum”,whileSouriauused theFrenchword“moment”. CushmanandDuistermaat [93]havesuggestedthat theproperEnglish translationofSouriau’sFrench wordwas “momentum”which fit betterwith standard usage inmechanics. On the other hand, GuilleminandSternberg [94]havevalidatedthenamegivenbySouriauandhaveused“moment” in English. In thispaper,wewill see thatname“moment”givenbySouriauwasthemostappropriate word. InhisChapter IVofhisbook[10], studyingstatisticalmechanics,Souriau[10]has ingeniously observedthatmomentsof inertia inmechanicsareequivalent tomoments inprobability inhisnew geometricmodelofstatisticalphysics.Wewill see that inSouriauLiegroupthermodynamicmodel, thesestatisticalmomentswillbegivenbytheenergyandtheheatdefinedgeometricallybySouriau, andwillbeassociatedwith“momentmap”indualLiealgebra. ThisworkhasbeenextendedbyClaudeVallée [5,6]andGerydeSaxcé [4,8,95,96].Morerecently, Kapranovhasalsogivenathermodynamical interpretationof themomentmapfor toricvarieties [97] andPavlov, thermodynamics fromthedifferentialgeometrystandpoint [98]. Theconservationof themomentofaHamiltonianactionwascalledbySouriauthe“symplectic or geometricNoether theorem”. Consideringphases spaceas symplecticmanifold, cotangentfiberof configurationspacewithcanonical symplectic form, ifHamiltonianhasLiealgebra, thenthemoment map is constant along the system integral curves. Noether theorem is obtained by considering independentlyeachcomponentof themomentmap. In a first step to establish new foundations of thermodynamics, Souriau [10] has defined a Gibbs canonical ensemble on a symplecticmanifoldM for a Lie group action onM. In classical statistical mechanics, a state is given by the solution of Liouville equation on the phase space, thepartitionfunction.Assymplecticmanifoldshaveacompletelycontinuousmeasure, invariantby diffeomorphisms, theLiouvillemeasureλ, all statistical stateswill be theproduct of theLiouville measurebythescalar functiongivenbythegeneralizedpartitionfunction eΦ(β)−〈β,U(ξ)〉definedby the energyU (defined in the dual of the Lie algebra of this dynamical group) and the geometric temperature β, whereΦ is a normalizing constant such the mass of probability is equal to 1, 61