Page - 55 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 groupcouldberepresented inamatrix formby(withA rotation,b theboost, cspace translationand e timetranslation):⎡⎢⎣ x′t 1 ⎤⎥⎦= ⎡⎢⎣ A b c0 1 e 0 0 1 ⎤⎥⎦ GALILEOGROUP ⎡⎢⎣ xt 1 ⎤⎥⎦with ⎧⎪⎨⎪⎩ A∈SO(3) b,c∈R3 e∈R , LieAlgebra ⎡⎢⎣ ω η γ0 0 ε 0 0 0 ⎤⎥⎦with ⎧⎪⎨⎪⎩ ω∈ so(3) η,γ∈R3 ε∈R+ (18) Souriau associated to thismomentmap, the notion of symplectic cohomology, linked to the fact that such amoment is defined up to an additive constant that brings into play an algebraic mechanism(calledcohomology). Souriauprovedthat themomentmapisaconstantof themotion, andprovidedgeometricgeneralizationofEmmyNoether invariant theorem(invariantsofE.Noether theoremare thecomponentsof themomentmap). For instance,Souriaugaveanontologicaldefinition ofmass in classicalmechanics as themeasure of the symplectic cohomology of the action of the Galileogroup(themass isno longeranarbitraryvariablebutacharacteristicof the space). This is no longer true forPoincarégroupinrelativisticmechanics,where thesymplecticcohomologyisnull, explaining the lackof conservationofmass in relativity. All thedetailsof classicalmechanics thus appearasgeometricnecessities, asontologicalelements. Souriauhasalsoobservedthat thesymplectic structurehas theproperty tobeable tobereconstructedfromitssymmetriesalone, througha2-form (calledKirillov–Kostant–Souriau form)definedoncoadjointorbits. Souriau said that thedifferent versionsofmechanical sciencecanbeclassifiedbythegeometry thateach implies forspaceandtime; geometry isdeterminedbythecovarianceofgrouptheory. Thus,Newtonianmechanics iscovariant by thegroupofGalileo, the relativityby thegroupofPoincaré;General relativityby the“smooth” group(thegroupofdiffeomorphismsofspace-time).However,Souriauadded“However, there are some statementsofmechanicswhose covariancebelongs toa fourthgrouprarely considered: theaffinegroup, agroup showninthe followingdiagramfor inclusion.Howis itpossible thataunitarypointofview(whichwouldbe necessarilya true thermodynamics), hasnotyet come tocrownthepicture?Mystery...” [26].SeeFigure1. Figure 1. Souriau Scheme about mysterious “affine group” of a true thermodynamics between Galileo group of classicalmechanics, Poincaré group of relativisticmechanics and Smooth group ofgeneral relativity. As early as 1966, Souriau applied his theory to statistical mechanics, developed it in the Chapter IV of his book “Structure of Dynamical Systems” [11], and elaborated what he called a “Lie group thermodynamics” [10,11,27–37]. Using Lagrange’s viewpoint, in Souriau statistical mechanics, a statistical state is a probabilitymeasure on themanifold ofmotions (andno longer inphase space [38]). Souriauobserved thatGibbsequilibrium[39] isnot covariantwith respect to dynamicgroupsofPhysics. Tosolve thisbrakingofsymmetry,Souriau introducedanew“geometric theoryofheat”wheretheequilibriumstatesareindexedbyaparameterβwithvaluesintheLiealgebra of thegroup,generalizing theGibbsequilibriumstates,whereβplays theroleofageometric (Planck) temperature. The invariancewith respect to thegroup, and the fact that the entropy s is a convex functionof thisgeometric temperatureβ, imposesverystrict,universalconditions (e.g., thereexists necessarily a critical temperature beyondwhichno equilibriumcanexist). Souriauobserved that thegroupof time translationsof theclassical thermodynamics [40,41] isnotanormal subgroupof theGalileigroup,proving that if adynamical systemisconservative inan inertial reference frame, itneednotbeconservative inanother. Basedonthis fact,Souriaugeneralizedthe formulationof the Gibbsprinciple tobecomecompatiblewithGalileorelativity inclassicalmechanicsandwithPoincaré 55